Propeller Whirl Flutter Analysis of the Nasa All-Electric X-57 Through Multibody Dynamics Simulations

PROPELLER WHIRL FLUTTER ANALYSIS OF THE NASA ALL-ELECTRIC X-57 THROUGH MULTIBODY DYNAMICS SIMULATIONS

by CHRISTIAN BLAKE HOOVER JINWEI SHEN, COMMITTEE CHAIR PAUL HUBNER CHARLES O’NEILL WEIHUA SU STEVE SHEPARD

A DISSERTATION

Submitted in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy in the Department of Aerospace Engineering and Mechanics in the Graduate School of The University of Alabama

TUSCALOOSA, ALABAMA

With ridesharing companies researching into intracity air vehicles, aircraft efficiency has a significant impact on cost and resources. To increase efficiency, thinner and higher aspect ratio wings are used to reduce drag along with propellers due to their high propulsion efficiency. This research presents a whirl

ﬂutter analysis of the NASA all-electric X-57 and how it inﬂuences the design of the vehicle.

Multibody dynamics analysis (MBD) is used for this study, Dymore, with correlating data from CAMRAD II given by NASA. Along with numerous studies of the X-57 using these MBDs, two cases are presented for validating the use of multibody dynamics: a Goland wing undergoing torsion bending flutter and an isolated propeller in whirl flutter. The whirl flutter stability of the X-57 is conducted in stages, starting with an isolated propeller and closing with a full free-flying aircraft. The wing is input into Dymore as an equivalent beam derived from the full NASTRAN FEM of the X-57.

This dissertation is divided into four parts for the whirl flutter analysis: isolated propeller, semi-span, full-span, and a free-flying model. The isolated propeller has a large margin of safety with the pylon pitch and yaw stiffness values having to be reduced by two orders of magnitude for whirl flutter to occur. The semi-span undergoes three design revisions to ensure whirl flutter is

ii not encountered for any of the symmetric wing modes. The full-span does not experience whirl flutter for the wing symmetric or anti-symmetric modes, and a parametric study is performed on various design variables. The free-flying model is stable for all the wing modes and shows the longitudinal flight dynamics modes interact with the flexible wing modes. When the stiffness in one of the pylon mounts is reduced to simulate damage, the system becomes unstable when the short period mode crosses the wing symmetric out-of-plane bending mode.

This shows the need to include the ﬂight dynamic modes when considering the whirl ﬂutter stability of future aircraft.

iii DEDICATION

To my parents, for their willingness to become fans of whichever school I attend. To my siblings: Ryan, Scott, and Cayce for their understanding and support while facing challenges and new experiences of their own. And to my close friends that have helped me along the road to completing this dissertation.

iv LIST OF ABBREVIATIONS AND SYMBOLS

Symbols C Damping Term

CDF Fuselage Drag Coeﬃcient

CH H-Force Coeﬃcient D Drag, lb

DF use Fuselage Drag, lb H Propeller Drag Force, lb I Moment of Inertia

slug Iy Moment of Inertia About the T-Axis, ft2 L Lift, lb K Stiﬀness Term M Pitching Moment, Positive Pitch Up lb − ft

M{•} The Moment Derivative Along the Y-Axis, Nondimensional- ized by Moment of Inertia; ∂M/∂• Iy Ma Mass Term R Radius S Wing Area, ft2 X Axial Force, Positive Forward lb

X{•} The Force Derivative Along the X-Axis, Nondimensionalized ∂X/∂• by Mass; m Y Side Force, Positive Right lb Z Normal Force, Positive Up lb

v Z{•} The Force Derivative Along the Z-Axis, Nondimensionalized ∂Z/∂• by Mass; m g ft Accelerations due to Gravity, sec2 h Pivot Distance of the Propeller on the Pylon m Mass, slugs q radians Pitch Rate, sec q lb ∞ Dynamic Pressure, ft2 r Blade Disk Radial Coordinate u ft Freestream Velocity in the X-Direction, sec w ft Freestream Velocity in the Z-Direction, sec α Angle of Attack, radians α˙ radians Time Derivative of Angle of Attack, sec

αx Yaw Degree of Freedom

αy Pitch Degree of Freedom

β1c Longitudinal Tip-Path-Plane Tilt Angle

β1s Lateral Tip-Path-Plane Tilt Angle γ Blade Lock Number λ Rotor Inﬂow Ratio µ Rotor Advance Ratio ν Rotating Natural Frequency of Blade

νβ Rotating Natural Frequency of Blade Fundamental Flap Mode θ Pitch Angle, radians ζ Damping Ratio ω Frequency {•}˙ Time Derivative

vi Subscripts ρ Phugoid Mode sp Short Period Mode

Acronyms CAD Computer Aided Design DEP Distributed Electric Propulsion FE Finite Element FEM Finite Element Method MBD Multibody Dynamics Analysis TDT Transonic Dynamics Tunnel SCEPTOR Scalable Convergence Electric Propulsion Technology Opera- tions Research

vii ACKNOWLEDGMENTS I would like to thank the many people for their support and encouragement throughout this long and difficult process. I am also eternally grateful to my advisor, Dr. Jinwei Shen, for giving me the opportunity to work on this incredibly fascinating research topic. Since my coming to Alabama, Dr. Shen has been understanding, encouraging, and supportive through both academic and non-academic avenues. In addition, he has been a great resource going into the aerospace and rotorcraft community. I have been extremely fortunate to have him as my advisor. I would also like to thank my Ph.D. committee members, Dr. Paul Hubner, Dr. Charles O’Neill, Dr. Weihua Su, and Dr. Steve Shepard for their guidance and input both in this dissertation and throughout my time in graduate school. I am also thankful for my friends at The University of Alabama: Tony Wente, Cole Frederick, Chris New, Aaron Suttle, Andrew Greff, Zach Hagan, Marcos Canabal, and the rest of the Formula SAE members for their unconditional support and for having someone to share the good times and bad times with. This research was funded by NASA Langley Research Center through National Institute of Aerospace with Jennifer Heeg as the technical monitor. I would also like to thank Jennifer Heeg, Bret Stanford, Jeffrey Viken, and Nicholas Borer, (NASA Langley Research Center), Andrew Kreshock (Army Research Laboratory, Vehicle Technology Directorate) for their support of this study. The Dymore analysis is kindly provided by Olivier Bauchau (University of Maryland, College Park).

viii CONTENTS

ABSTRACT ...... ii

DEDICATION ...... iv

LIST OF ABBREVIATIONS AND SYMBOLS ...... v

ACKNOWLEDGMENTS ...... viii

LIST OF TABLES ...... xii

LIST OF FIGURES ...... xiii

1 INTRODUCTION ...... 1

1.1 Literature Review ...... 6

1.2 X-57 Maxwell ...... 10

1.3 Approach and Objectives ...... 15

2 ANALYTICAL MODELS ...... 20

2.1 Analyses ...... 20

2.1.1 Dymore ...... 20

2.1.2 CAMRAD II ...... 21

2.1.3 NASTRAN ...... 21

2.2 Multibody Dynamics Model of the X-57 ...... 21

2.2.1 Equivalent Beam Model of the Wing/Pylon System ...... 21

2.2.2 Semi-Span Dymore Model ...... 24

ix 2.2.3 Full-Span Dymore Model ...... 25

2.2.4 Free-Flying Aircraft Dymore Model ...... 27

2.2.5 Propeller Dymore Model ...... 29

3 FUNDAMENTAL THEORY OF PROPELLER WHIRL FLUTTER . . . . . 35

3.1 Whirl Flutter Equations ...... 35

4 VALIDATION OF MULTIBODY DYNAMICS ANALYSIS ...... 38

4.1 The Goland Wing Undergoing Torsion-Bending Flutter ...... 39

4.2 Isolated Propeller Undergoing Whirl Flutter ...... 43

4.2.1 Transient Results ...... 44

4.2.2 Comparison to Mathematical Model and Experimental Data . . . 46

5 ANALYTICAL RESULTS ...... 47

5.1 X-57 Standalone Propeller ...... 47

5.1.1 Equal Pitch and Yaw Stiﬀness ...... 47

5.1.2 Diﬀerent Pitch and Yaw Stiﬀness ...... 49

5.1.3 Eﬀect of Diﬀerent Airfoil ...... 49

5.1.4 Summary and Conclusions for Isolated Propelller ...... 53

5.2 Propeller Whirl Flutter Study with a Semi-Span X-57 Propeller/Wing/Py- lon Model ...... 53

5.2.1 Three Wing/Pylon Design Versions ...... 53

5.2.2 X-57 Maxwell Analytical Model Veriﬁcation ...... 55

5.2.3 Equivalent Beam Model of Three Fixed System Versions . . . . . 56

5.2.4 Whirl Flutter Stability ...... 59

5.2.5 Semi-Span Summary and Conclusions ...... 63

5.3 Propeller Whirl Flutter Study with a Full-Span X-57 Propeller/Wing/Py- lon Model ...... 71

5.3.1 Whirl Flutter Stability of the Full-Span X-57 ...... 71

x 5.3.2 Comparison of the Full-Span to the Semi-Span X-57 Model . . . . 72

5.3.3 Full-Span X-57 With Wing Aerodynamics ...... 73

5.3.4 Eﬀect of Change in Blade Stiﬀness on the Full-Span X-57 . . . . . 74

5.3.5 Eﬀect of Wing Stiﬀness on Whirl Flutter Stability of the Full-Span X-57 Model ...... 74

5.3.6 Full-Span Summary and Conclusions ...... 75

5.4 Whirl Flutter Analysis of a Free-Flying Electric Driven Propeller Aircraft 89

5.4.1 Whirl Flutter of the Free-Flying X-57 ...... 89

5.4.2 Eﬀect of Unsteady Aerodynamics on Whirl Flutter Stability . . . 90

5.4.3 Longitudinal Flight Dynamics ...... 96

5.4.4 Transient Results of the Longitudinal Flight Dynamics ...... 103

5.4.5 Severely Damaged Pylon ...... 110

5.4.6 Severely Damaged Pylon With Constrained Rigid Body Motion . 122

5.4.7 Free-Flying Summary and Conclusions ...... 124

6 CONCLUSIONS ...... 125

7 REFERENCES ...... 128

xi LIST OF TABLES

1.1 Key Aircraft Parameters ...... 12

2.1 Dymore X-57 Fuselage Model ...... 26

2.2 X-57 Propeller Properties ...... 29

2.3 Blade Frequency at Cruise RPM (2250 RPM) ...... 30

4.1 Goland Wing Properties ...... 39

4.2 Goland Wing Results Comparison ...... 41

5.1 Frequencies of Wing/Pylon Model; Version 3 ...... 58

5.2 Longitudinal Stability Derivatives with Appropriate Units ...... 97

xii LIST OF FIGURES

1.1 Comparison of the Wing Area of the X-57 to the Tecnam P2006T . . . . 2

1.2 Typical Propulsion Noise [1] ...... 2

1.3 Typical Propulsion Eﬃciencies [2] ...... 3

1.4 Diagram of Whirl Flutter [3] ...... 3

1.5 Electra Transonic Dynamics Tunnel Wind Tunnel Tests [4] ...... 5

1.6 Beechcraft 1900C Whirl Flutter Accident [5] ...... 6

1.7 Concept of X-57 Maxwell ...... 12

1.8 Multiview of the X-57 ...... 13

1.9 Tecnam P2006T to X-57 (Courtesy of NASA) ...... 14

1.10 X-57 Undeformed Wing ...... 17

1.11 X-57 Symmetric Wing Mode Shapes ...... 18

1.12 X-57 Anti-Symmetric Wing Mode Shapes ...... 19

2.1 Modular Model Development Approach ...... 22

2.2 Dymore X-57 Maxwell Semi-Span Model ...... 25

2.3 Dymore X-57 Maxwell Full-Span Model ...... 25

2.4 Dymore Full-Span Elastic Fuselage ...... 26

2.5 Dymore X-57 Maxwell Free-Flying Model ...... 27

2.6 Natural Frequency Variation with Rotor Speed ...... 30

2.7 X-57 Maxwell Propeller Dymore Model ...... 32

2.8 Propeller Pitch Change Under Windmilling with NACA 0012 ...... 32

2.9 Propeller Thrust with NACA 0012 ...... 33

2.10 Propeller Power with NACA 0012 ...... 33

xiii 2.11 Windmilling Trim Blade Pitch Change for NACA 0012 and the X-57 Air- foils on Rigid Blades ...... 34

2.12 Windmilling Trim Blade Pitch Change for the X-57 Airfoils on Rigid and Elastic Blades ...... 34

3.1 Whirl Flutter Stability Boundary for Diﬀerent Inﬂows [6] ...... 38

4.1 Goland Wing Dymore Model ...... 40

4.2 Goland Wing Frequency and Damping Ratio of Flutter Mode ...... 42

4.3 Dymore Isolated Propeller Model ...... 43

4.4 Propeller Hub Motion for Increasing Spring Stiffness at Constant Velocity: Top Has Lowest Stiffness, Bottom Has Highest Stiffness ...... 45

4.5 Propeller Hub Motion for Increasing Velocity at Constant Spring Stiﬀness: Top Has Lowest Velocity, Bottom Has Highest Velocity ...... 45

4.6 Damping Ratio of The Isolated Propeller for Diﬀerent Dynamic Pressures and Spring Stiﬀness ...... 46

5.1 Isolated X-57 Propeller with X-57 Airfoils: Contour Plots for Varying Py- lon Stiffness and Velocity; Pitch Stiffness Equal to Yaw Stiffness . . . . . 48

5.2 Isolated X-57 Propeller with X-57 Airfoils: Contour Plots for Varying Py- lon Stiﬀness and Velocity; Ratio of Pitch to Yaw Stiﬀness is 0.5405 . . . 50

5.3 Isolated X-57 Propeller with NACA 0012: Contour Plots for Varying Pylon Stiﬀness and Velocity; Ratio of Pitch to Yaw Stiﬀness is 0.5405 ...... 51

5.4 Contour of the Marginal Stability Location (Zero Damping) for Varying Pylon Stiﬀness and Velocity: Ratio of Pitch to Yaw Stiﬀness is 0.5405 with NACA 0012 and X-57 Airfoils ...... 52

5.5 Version 1: Full FEM and Equivalent Beam Model ...... 58

5.6 Version 2: Full FEM and Equivalent Beam Model ...... 59

5.7 Version 2: Full FEM Shapes: Mode 5 Detail ...... 60

5.8 Version 3: Full FEM and Equivalent Beam Model ...... 61

5.9 Frequency and Damping of Semi-Span X-57 Maxwell Model; Without Wing Aerodynamics; RPM 2250; Version 1 ...... 65

5.10 Dymore Frequency and Damping of Semi-Span X-57 Maxwell Model; With and Without Wing Aerodynamics; RPM 2250; Version 1 ...... 66

xiv 5.11 CAMRAD II Frequency and Damping of Semi-span X-57 Maxwell Model; Without Wing Aerodynamics; 2250 RPM; Version 2 ...... 67

5.12 Frequency and Damping of Semi-Span X-57 Maxwell Model; Without Wing Aerodynamics; RPM 2250; Version 3 ...... 68

5.13 Dymore Frequency and Damping of Semi-Span X-57 Maxwell Model; Com- parison of Adding Wing Aerodynamics; RPM 2250; Version 3 ...... 69

5.14 Dymore Frequency and Damping of Semi-Span X-57 Maxwell Model; Com- parison of Adding Wing Aerodynamics; RPM 2700; Version 3 ...... 70

5.15 Full-Span Frequencies and Damping Ratios for CAMRAD II and Dymore Symmetric Modes ...... 77

5.16 Full-Span Frequencies and Damping Ratios for CAMRAD II and Dymore Anti-Symmetric Modes ...... 78

5.17 Comparison of the Symmetric Modes for the Dymore Full-Span and Semi- Span Models ...... 79

5.18 Eﬀects of Applying Steady Aerodynamics to the Wing of the Full-Span Dymore Model Symmetric Modes ...... 80

5.19 Eﬀects of Applying Steady Aerodynamics to the Wing of the Full-Span Dymore Model Anti-Symmetric Modes ...... 81

5.20 Eﬀect of Blade Stiﬀness on Whirl Flutter for the Full-Span Dymore Model Symmetric Modes; X-57 Distribution ...... 82

5.21 Eﬀect of Blade Stiﬀness on Whirl Flutter for the Full-Span Dymore Model Anti-Symmetric Modes; X-57 Distribution ...... 83

5.22 Eﬀect of Wing Stiﬀness on Frequencies and Damping Ratios of the Sym- metric Out-of-Plane Bending Mode: Elastic X-57 Propellers, 2250 RPM and at 8000 Feet ...... 84

5.23 Eﬀect of Wing Stiﬀness on Frequencies and Damping Ratios of the Sym- metric In-Plane Bending Mode: Elastic X-57 Propellers, 2250 RPM and at 8000 Feet ...... 85

5.24 Eﬀect of Wing Stiﬀness on Frequencies and Damping Ratios of the Sym- metric Torsion Mode: Elastic X-57 Propellers, 2250 RPM and at 8000 Feet ...... 86

5.25 Eﬀect of Wing Stiﬀness on Frequencies and Damping Ratios of the Anti- Symmetric Out-of-Plane Bending Mode: Elastic X-57 Propellers, 2250 RPM and at 8000 Feet ...... 87

xv 5.26 Eﬀect of Wing Stiﬀness on Frequencies and Damping Ratios of the Anti- Symmetric In-Plane Bending Mode: Elastic X-57 Propellers, 2250 RPM and at 8000 Feet ...... 88

5.27 Comparison of Free-Flying and Full-Span X-57 Whirl Flutter Symmetric Frequencies and Damping Ratios Using Quasi-Steady Aerodynamics . . . 91

5.28 Comparison of Free-Flying and Full-Span X-57 Whirl Flutter Anti-Symmetric Frequencies and Damping Ratios Using Quasi-Steady Aerodynamics . . . 92

5.29 Free-Flying X-57 Whirl Flutter Symmetric Frequencies and Damping Ra- tios Using Unsteady and Quasi-Steady Aerodynamics ...... 94

5.30 Free-Flying X-57 Whirl Flutter Anti-Symmetric Frequencies and Damping Ratios Using Unsteady and Quasi-Steady Aerodynamics ...... 95

5.31 Aircraft Longitudinal Stability Derivatives Versus Velocity ...... 99

5.32 Frequencies and Damping Ratios of the Measured and Approximated Short Period Modes and Excited Elastic Wing Modes ...... 101

5.33 Frequencies and Damping Ratios of the Measured and Approximated Phugoid Modes ...... 102

5.34 Transient Pitch and Change in Altitude for All Measured Velocities . . . 104

5.35 Short Period States; Pitch and Angle of Attack ...... 106

5.36 Short Period States; Plunge and Velocity ...... 107

5.37 Phugoid Mode States; Pitch and Angle of Attack ...... 108

5.38 Phugoid Mode States; Plunge and Velocity ...... 109

5.39 X-57 With Damaged Pylon ...... 111

5.40 X-57 With Damaged Pylon Zoomed In ...... 112

5.41 X-57 Transient Results with Damaged Pylon: 214 Knots ...... 114

5.42 X-57 Transient Results with Damaged Pylon: 214 Knots ...... 115

5.43 X-57 Transient Results with Damaged Pylon: 233 Knots ...... 117

5.44 X-57 Transient Results with Damaged Pylon: 233 Knots ...... 118

5.45 X-57 Transient Results with Damaged Pylon: 292 Knots ...... 120

5.46 X-57 Transient Results with Damaged Pylon: 292 Knots ...... 121

5.47 Constrained X-57 With Damaged Pylon ...... 123

xvi 1 INTRODUCTION

As personal flight vehicles become a higher possibility, efficiency and emissions are going to be a significant driving force towards the design of this generation of aircraft. NASA is currently leading the way to fully electric vehicles with the development of the X-57, a NASA experimental aircraft [7]. Designed to be 500% more efficient than the Tecnam P2006T, the aircraft the X-57 is based on, where the wing and propulsion system are designed with drag reduction and efficiency in mind. The wing designed for the X-57 is a slender, high aspect ratio wing with large wing loading, as shown in Figure 1.1. The propulsion system chosen for the X-57 are propellers, including the distributed electric propulsion system involving multiple small inboard propellers. Propellers are chosen for the X-57 due to this propulsion method being quieter and more efficient than other conventional propulsion methods, as shown in

Figures 1.2 and 1.3 respectively. Further details on the X-57 and the program this experimental aircraft comes from are discussed in detail in the section on the X-57. For safety considerations, NASA designated that the X-57 had a potential for propeller whirl ﬂutter due to the aircrafts thin eﬃcient wings [8].

Propeller whirl ﬂutter is an aeroelastic instability where the propeller aerodynamic forces drive the propeller/pylon system to become unstable. Figure 1.4 shows a diagram of this instability, and it can be seen that there are two modes: a forward whirl mode and a backwards mode. Above a critical velocity, this whirling motion diverges and can

1 Figure 1.1: Comparison of the Wing Area of the X-57 to the Tecnam P2006T

Figure 1.2: Typical Propulsion Noise [1] cause large stresses in structures that lead to failures in those structures. The critical velocity for whirl flutter depends on the stiffness of the pitch and yaw degrees of freedom and the ratio of the pitch and yaw stiffness; a further explanation is given in the section titled Fundamental Theory of Whirl Flutter. Initially discovered by Taylor

2 Figure 1.3: Typical Propulsion Eﬃciencies [2] and Browne [9] in 1938, the speeds at which the instability occurs was deemed too high for aircraft of that era. This aeroelastic phenomenon was subsequently placed in the background of aerospace engineering for over twenty years until two fatal crashes of the

Electra in 1959 and 1960.

Figure 1.4: Diagram of Whirl Flutter [3]

The two fatal accidents of the Electra aircraft began a new-found interest in

propeller whirl ﬂutter when a model was put into the NASA Transonic Dynamics

3 Tunnel (TDT) at Langley, shown in Figure 1.5, and it was determined that a structural failure in the pylon reduced the stiffness of the mount and ultimately produced the conditions for propeller whirl flutter to occur [4]. This led to numerous studies to be conducted to develop an understanding of propeller whirl flutter at the TDT, including a state of the art review of the subject by Reed [3]. Whirl flutter subsequently became a requirement as a part of the dynamic evaluation of transport aircraft, such that no

ﬂutter shall occur as a result of a failure of any single element of an engine mount structure [10].

Despite this added regulation, another fatal accident involving propeller whirl

flutter occurred involving a Beechcraft 1900C in 1992. An undetected crack propagated and eventually reduced the stiffness enough such that whirl flutter conditions were encountered [5]. The propeller separated from the wing, causing the wing to detach, and then impacting the tail, Figure 1.6. The propeller location on both the Electra and the Beechcraft were near the root of the wing, where the stiffness associated with the wing is large.

In order to maximize efficiency with propeller location, the X-57 has the propeller located at the wingtips. Placing propellers, or any large mass, at the wingtips is sensitive aeroelastically and can lead to classical flutter issues. However, there is a substantial efficiency gain from placing the propellers at the wingtip that come in the form of reduced drag caused by the wingtip vortices and giving the aircraft a larger effective wing aspect ratio [11]. These efficiencies go towards the X-57 goal of being

500% more eﬃcient than the base plane it is based on. While the X-57 is derived from an already existing vehicle, the implementation of a new wing presents the possibility of new aeroelastic concerns. This dissertation looks into the whirl ﬂutter stability of the

4 (a) Electra Inside the TDT

(b) Electra After Destructive Testing

Figure 1.5: Electra Transonic Dynamics Tunnel Wind Tunnel Tests [4]

X-57 through the use of multibody dynamics analysis.

Multibody dynamics are increasingly popular to study the stability and loading on aircraft aeromechanics due to their generality and ability to model complex coupled systems [12,13]. Multibody dynamics analyses are the dominant tools for the study of tiltrotor whirl ﬂutter, where the blade elasticity must be taken into account as well.

Unlike tiltrotor whirl ﬂutter, where the wings on tiltrotors are stubby with large airfoil thickness to increase the torsional stiﬀness, the wings on the X-57 are long and thin

5 Figure 1.6: Beechcraft 1900C Whirl Flutter Accident [5] which can not only cause a destabilizing effect on propeller whirl flutter, but the low frequencies of the wing can interact with the rigid body flight dynamic modes.

The effect of a highly flexible wing on the longitudinal flight dynamics modes has been of increasingly more interest for the next generation of high-altitude, long-endurance aircraft [14]. In addition to the flight dynamics being affected by the elastic wings, there is the potential for the rigid body modes coupling with the wing modes. This possible coupling of the rigid flight dynamic modes with the elastic wing modes may have a destabilizing effect on the whirl flutter stability [15].

This dissertation focuses on the whirl flutter aeromechanics of the X-57. The physics behind whirl flutter are explored for an isolated propeller, and then a semi-span, full-span, and free-flying aircraft are developed and whirl flutter stability examined for various conditions and parametric studies of the X-57 performed.

1.1 Literature Review

Current texts on propeller whirl flutter are extensive as the physics behind the instability are relatively well understood. Taylor and Browne first discovered propeller whirl flutter in 1938 [9]. Their findings showed mathematically that an instability could

6 occur, but could only produce the instability in a specially designed model that had reduced friction. The instability could not be replicated in aircraft of the era and the expected pylon stiﬀness and damping values at the time ensured that propeller whirl

flutter would not occur. Following the fatal crashes of the Electra aircraft, an investigation into the cause of the crashes was performed. A model was developed and tested in the NASA Langley TDT [4]. It was determined that while the undamaged aircraft had an adequate safety margin, a reduction in pylon stiffness due to a damaged pylon mount could lead to the onset of whirl flutter. While the problem was found to be due to the outboard propeller, it was discovered that the inboard propeller had a greater sensitivity to pylon stiffness than the outboard propeller. Experimental results from the NASA Langley Research Center were correlated with theory, and a design space was tested to determine how the variation of stiffness and mass affected the stability of an isolated propeller [16].

The introduction of wing motion for whirl flutter stability was investigated via wind tunnel test by Bennett and Bland [17], where it was seen that wing motion is slightly stabilizing and that a wing in vacuum is too conservative as the wing aerodynamics dampen the wing motion. However, the propeller placement was near the root of the wing where the effects are limited. This study developed a mathematical model for a four degree-of-freedom system model: propeller pitch, propeller yaw, wing out-of-plane bending, and wing torsion. A wind tunnel model was also used to validate the mathematical model. The in-plane wing motion was not considered as the in-plane stiffness is typically much higher than the out-of-plane or the torsional stiffness, which is not a valid assumption for current and next-generation aircraft. In addition, the stiffness of the wing at the root was much higher than standard aircraft [18].

7 After the initial investigation into whirl ﬂutter, a full review was conducted by

Wilmer Reed III at NASA Langley [3,19]. Reed not only examined the stability behind rigid propeller blades but also looked into the effect of non-rigid blades and while blade elasticity had little effect on the backwards mode, the elasticity of the propeller blades can introduce flutter in the forward mode. Reed also showed that propeller thrust at high flight speeds have little effect on whirl flutter and the thrust can be ignored.

An energy balance equation for whirl ﬂutter was derived by Young and Lytwyn in

1967 [20], this equation showed that an instability is possible if αxα˙ y − αyα˙ x < 0.

Where αy and αx are the pitch and yaw degrees of freedom respectively. Replacing pitch and yaw with the propeller shaft angle such that αx = a cos θ and αy = a sin θ, gives a2θ˙ < 0. Therefore the instability occurs when the propeller shaft is rotating opposite of the propeller rotation.

Even with the current information on propeller whirl ﬂutter, there have still been fatal accidents. One such case is the Beechcraft 1900C [5]. It was determined that whirl

ﬂutter occurred on the starboard engine, causing it to become separated from the wing

(Fig.1.6). The separated engine crashed into the tail, damaging the horizontal stabilizer. The damaged wing became separated from the vehicle as well, and as a result of the loss of the starboard wing and horizontal tail, the aircraft pitched down, rolled right, and crashed. An acoustic analysis of the ﬂight recorder showed that frequencies were present that matched the structural dynamic and ﬂutter analysis of a damaged truss. It was also noted that the pilots’ voices showed no stress, indicating that the incident occurred suddenly and without warning.

Kunz [21] provided a more recent review of the analysis of proprotor whirl ﬂutter.

A propeller, proprotor, and helicopter rotor are examined for whirl stability and

8 compared to each other. The mathematical models for each are derived out for the three cases, the assumptions and parameters are identified and expounded. The stability boundary is identified for each system while the differences and similarities are compared among the different systems. Livne [22], in his review of active aircraft flutter suppression, emphasized that the dynamic analysis of whirl flutter must be included for any propeller-driven aircraft.

While mathematical models of whirl flutter do exist, they deal with limited degrees of freedom. To have higher fidelity in the whirl flutter stability, comprehensive computer analysis need to be used. Multibody dynamics have been shown to be able to accurately model complex coupled systems, which is why it is a preferred analysis tool in the rotorcraft community [12,13]. Three such multibody dynamic analyses are

CAMRAD II [23], RCAS [24], and Dymore [25,26]. These multibody dynamic codes have been used independently, in conjunction with one another, and compared with wind tunnel data to look at whirl ﬂutter stability of tiltrotors.

Tiltrotor whirl ﬂutter has been widely studied recently, driven by the need of developing future high-speed tiltrotor [27–31]. Shen [29] carried out a comparison study of tiltrotor whirl ﬂutter stability using multibody dynamics codes, RCAS [32] and

Dymore [33], and shows good comparison among the two analytical predictions and with wind tunnel test data. Kreshock [34] used CAMRAD II [35] and RCAS to predict the whirl ﬂutter boundary of a tiltrotor. CAMRAD II, Dymore, and RCAS were also shown to capture the whirl ﬂutter trends of a tiltrotor in cruise and the proprotor loads at various pylon conversion angles [36].

In addition to the effect of the whirl flutter stability, there can also be an influence of the rigid flight dynamics mode on the aeroelastic boundaries [37,38]. How this

9 interaction may affect whirl flutter was examined by Mattaboni et al. in 2012 [15] for a tiltrotor. While the short period mode was lower than the wing modes, due to the high required stiffness of the wings, there was a difference in whirl flutter stability when compared to a clamped wing rotor.

1.2 X-57 Maxwell

The X-57 Maxwell is the ﬁrst NASA X-Plane in over a decade and is a part of the

Scalable Convergent Electric Propulsion Technology Operations Research (SCEPTOR) program, which is a program that aims at designing, building and testing a demonstrator aircraft to showcase distributed electric propulsion technology. It is an all-electric propulsion vehicle, designed to be quieter, lighter, more eﬃcient and environmentally friendly than current turboprops [39]. The goal of SCEPTOR is to aid the development towards future electric aircraft with the X-57 being a general aircraft moving towards commuters, then to regional airliners and ﬁnally large aviation.

Based on the Italian Tecnam P2006T, the X-57 trades out the normal P2006T wing for a smaller, more eﬃcient wing. Replacing the two P2006T turboprops with 14 electric motors, two large outboard [40], and twelve smaller inboard, the X-57 aims for

500% higher high-speed cruise eﬃciency compared to the original Tecnam aircraft. The large outboard propellers are to be used in all ﬂight regimes while the small inboard propellers are used for lift augmentation, in a system called Distributed Electric

Propulsion (DEP), during slow speed areas such as takeoﬀ and landing. When the inboard propellers are not required, they fold in on themselves to reduce drag. Figure

1.7 shows a concept of the X-57 in ﬂight in an urban setting and Table 1.1 gives key parameters for this whirl ﬂutter study.

10 To achieve the 500% higher eﬃciency goal that the X-57 aims for, a redesigned wing is implemented. This high-eﬃciency wing is thinner and has a larger aspect ratio compared to the original Tecnam P2006T and can be seen in an overlay comparison in

Figure 1.1 [41]. In addition to the high-eﬃciency wing, the X-57 also considers the propeller placement, with two large cruise propellers placed at the wingtips to reduce the eﬀect of the wingtip vortices. An isometric view of the X-57 with all 14 propellers active can also be seen in Figure 1.8, while a CAD rendering of the Mod 4 X-57 and the

Tecnam P2006T that was delivered to NASA can be seen in Figures 1.9a and 1.9c respectively.

The X-57 has been divided up into four Mods of development, with this dissertation focusing on Mod 3. The ﬁrst Mod established up a baseline of the original Tecnam

P2006T and performed a ground validation test of the DEP high-lift system. Mod 2 swapped out the Tecnam P2006T propulsion system for the electric propulsion and examined the validity of electric motors, battery, and instrumentation. Ground and

flight tests were performed at this stage and established a baseline with the electric propulsion. Mod 3 moved the electric propulsion to the wing tips of the new high-efficiency DEP wing but does not include the individual DEP propellers. This stage includes the validation of the high-speed cruise efficiency and is where the whirl

flutter analysis is conducted. Mod 4 incorporates the propellers for the DEP system, performs flight tests, and examines acoustic testing as well as the low-speed control and certification of the DEP system. Figures 1.9b and 1.9c show the difference between Mod

3 and Mod 4 in CAD models.

11 Figure 1.7: Concept of X-57 Maxwell

Property Value Tip Prop Number of Blades 3 Tip Prop Diameter 5 ft Take-Oﬀ RPM 2700 Cruise RPM 2250 Cruise Speed 150 KTS Wing Semi-Span 14 ft Wing Norminal Chord 2.1 ft

Table 1.1: Key Aircraft Parameters

12 Figure 1.8: Multiview of the X-57

13 (a) NASA Tecnam P2006T CAD Model

(b) NASA X-57 Mod 3 CAD Model

Figure 1.9: Tecnam P2006T to X-57 (Courtesy of NASA)

14 1.3 Approach and Objectives

The primary goal for this dissertation is to assess the whirl ﬂutter stability of the X-57 and provide a design space for future electric aircraft. To meet this objective, the whirl

ﬂutter stability is determined for more than double the expected cruise speed of the

X-57. To accomplish this, a systematic approach is taken. Starting with the development of each component individually then put together after validation or comparison with other resources. The development and analysis of the X-57 is split into four parts:

1. Validate the use of multibody dynamics to use for the whirl ﬂutter stability of the

X-57 and perform a stability analysis on the standalone X-57 propeller

2. Develop semi-span wing and propeller models to analyze whirl ﬂutter stability for

symmetric modes

3. Expand to full-span with updated propeller blades and parametric study to

analyze whirl ﬂutter stability for symmetric and anti-symmetric modes

4. Analyze whirl ﬂutter stability on free-ﬂying aircraft with an emphasis on how the

coupling of the elastic modes with rigid ﬂight dynamic modes inﬂuences the

stability

As stated above, the wing elastic modes examined in this dissertation are the fundamental symmetric and anti-symmetric wing modes. These modes are the symmetric out-of-plane, in-plane and torsion while the anti-symmetric modes are the out-of-plane and the in-plane wing modes. The anti-symmetric torsion mode was not considered due to the frequency being considerably larger than the others fundamental

15 modes. In addition, the fundamental modes are only considered due to those modes being the most likely to go unstable ﬁrst [6]; therefore, the second and third wing bending modes are also not considered in this dissertation. The wing modes, via the

Dymore full-span model, are shown in Figures 1.10-1.12 along with the undeformed wing for reference. The frequency identiﬁcation is performed by using the Prony method to obtain the frequency and damping ratios from given signals [42].

16 (a) Top View

(b) Front View

Figure 1.10: X-57 Undeformed Wing

17 (a) Out-of-Plane

(b) In-Plane (c) Torsion

Figure 1.11: X-57 Symmetric Wing Mode Shapes

18 (a) Out-of-Plane

(b) In-Plane

Figure 1.12: X-57 Anti-Symmetric Wing Mode Shapes

19 2 ANALYTICAL MODELS

2.1 Analyses

This section gives a brief description of the analyses used in this dissertation.

2.1.1 Dymore

Dymore is the main analysis used in this Dissertation. Developed by Olivier Bauchau,

Dymore is a Finite Element (FE) based multibody dynamics code for the comprehensive modeling of nonlinear ﬂexible multibody systems [33]. The equilibrium equations are derived in a Cartesian inertial reference and constraints are modeled using the Lagrange multiplier technique. This leads to a system of diﬀerential-algebraic equations which is then solved using a robust time integration scheme. Dymore’s element library includes rigid and deformable bodies as well as joint elements.

Deformable bodies are modeled with the FE method and the formulations of beams and shells are geometrically exact, i.e., the ﬁnite rotation of the cross-section is treated exactly without small angle assumption [43]. This beam theory accounts for arbitrarily large displacements and ﬁnite rotations but is limited to small strains. The aerodynamic forces can be computed with the built-in lifting line theory or through coupling with an external aerodynamics code.

20 2.1.2 CAMRAD II

CAMRAD II is the main analysis used to compare with the Dymore results. Developed by Wayne Johnson of Johnson Aeronautics, CAMRAD II is an aeromechanical analysis tool for helicopters and rotorcraft that incorporates several features, including multibody dynamics, nonlinear ﬁnite elements, structural dynamics, and rotorcraft aerodynamics. CAMRAD II calculates the wing damping directly through eignenanalysis of the system equations of motion.

2.1.3 NASTRAN

NASA created a full FEM of the X-57 in NASTRAN in which the subsequent beam model of the wing and fuselage are derived from the mode shapes and frequencies given by NASTRAN. This process is explained in detail in the model portion of this dissertation.

2.2 Multibody Dynamics Model of the X-57

The multibody dynamics model of the X-57 is split up in the development of a semi-span model, then a full-span model with a simpliﬁed fuselage that incorporates the inertia and aerodynamic drag, and ﬁnally a full aircraft with an empennage. A modular approach was developed, and Figure 2.1 illustrates the modular procedure applied to the semi-span X-57 Maxwell model. The same modular approach was taken for the rest of the X-57 multibody dynamics model development.

2.2.1 Equivalent Beam Model of the Wing/Pylon System

To predict propeller whirl ﬂutter correctly, it is crucial to have an accurate model of the wing/pylon system. The full NASTRAN FE model uses quad-dominant shell elements

21 The Modular Modeling Validation Process

1 Clamped Wing Clamped Blade add pylon & multiple blades pylon springs hub connections

2 Wing/Pylon Isolated Propeller add propeller weight add pitch control

3 Wing/Pylon [Propeller Weight] Propeller with Control System

4 Add Aerodynamics Add Aerodynamics 5 Semi-Span Structural Model

6 Complete X-57 Wing and Propeller Model

Figure 2.1: Modular Model Development Approach to model the spar-rib wingbox and obtains the wind-off structural mode shapes and natural frequencies. CAMRAD II takes this modal representation of the wing/pylon system directly to facilitate its whirl flutter analysis. Dymore, however, needs a wing beam model, on which wing aerodynamics can be applied, and the effects of wing aerodynamics on propeller whirl flutter can be investigated. The full NASTRAN wing-box model is used to derive the equivalent beam model for the Dymore analysis.

The equivalent beam models of the wing/pylon systems were first constructed and verified in NASTRAN, and then re-constructed and re-verified in Dymore.

The following steps were undertaken to construct each equivalent beam model:

1. The elastic axis of the wingbox is located, by dividing the wingbox into several

spanwise sections. The root of each section is clamped, and vertical loads are

applied to the tip, in order to ﬁnd the force application location which produces a

pure bending response, without torsion. This process is repeated until the elastic

axis is mapped from wing root to wing tip.

22 2. Beam elements are laid-out along this elastic axis, and these beam elements are

connected to the wingbox with mass-less interpolation elements.

3. Forces and moments are applied to each beam node. Because the beam grids are

”spidered” to the wingbox (and because the beams do not yet have any stiﬀness

properties), these forces are counteracted by the wingbox alone. As the wingbox

deforms, the internal beam grids passively deform as well. Examining the relative

deformation and rotation of adjacent beam nodes produces estimates for the

bending and torsional stiﬀness of each beam element.

4. Mass properties of each beam element are computed from the known inertial

properties of the wingbox. Concentrated masses are created for each nacelle

location, and connected to the beam elements with rigid linkages.

5. The ﬁnal beam model is ”disconnected” from the wingbox, and the natural

frequencies and mode shapes can be computed for the equivalent beam, for

comparison with the eigen-data of the full wingbox model.

6. In order to further tune the modal response of the equivalent beam model to that

of the wingbox model, an optimization process is conducted. The elastic and

inertial properties of each beam element, as well as the pitch and yaw stiﬀness for

two torsional springs at the wing tip (connecting the wing to the tip propeller) are

optimized, in order to minimize a weighted objective function. This objective

includes the sum of the diﬀerences in the two sets of natural frequencies (beam

model and shell wingbox model), as well as the norm-diﬀerence between the

modal assurance criterion (MAC) matrix and the identity matrix. If the two sets

of mode shapes are identical, then the MAC matrix will become the identity

23 matrix. This optimization is conducted with a genetic algorithm.

The development of the beam models for Versions 2 and 3 utilize this complete

6-step procedure. The beam model for Version 1 only uses steps 1-5, however, neglecting the tuning optimization.

2.2.2 Semi-Span Dymore Model

The Dymore model of the X-57 Maxwell aircraft, shown in Figure 2.2, includes the wing, pylon, and the tip propeller. The wing and propeller blades are modeled as elastic beams. The wing beam properties are derived from full FEM models, and the propeller blades are modeled as very stiﬀ. The inboard propellers and the pylon and nacelle of the large tip propeller are modeled as rigid bodies with their appropriate inertial properties. The blade pitch bearing is modeled as a revolute joint. The pylon mount

ﬂexibility in yaw and pitch is captured with linear springs connecting the pylon to the wing. The aerodynamic forces acting on the propeller and wing are modeled with both quasi-steady and unsteady lifting line theories. The unsteady aerodynamics used in

Dymore are based on the unsteady aerodynamics theory developed in Ref. [44]. A table lookup format is used to define the propeller aerodynamic coefficients. Rotor inflow is calculated with a three-dimensional nonlinear dynamic inflow model using three inflow states [45]. The aerodynamic interaction between the propeller and wing is ignored.

The CAMRAD II X-57 Maxwell model includes a rigid propeller model and modal representations of the wing/pylon ﬁxed systems. Propeller aerodynamics are modeled with the lifting line theory coupled with a linear inﬂow model [31].

24 Figure 2.2: Dymore X-57 Maxwell Semi-Span Model

2.2.3 Full-Span Dymore Model

The Dymore full-span model, shown in Figure 2.3, is a continuation of the semi-span model. The semi-span wing was mirrored on the port side of the fuselage with the propeller spinning in the opposite direction to the starboard propeller. The structural properties for the port wing are the same as the structural properties for the starboard wing.

Figure 2.3: Dymore X-57 Maxwell Full-Span Model

Dymore Fuselage Model

The Dymore X-57 fuselage, shown in Figure 2.4, is modeled using beam elements with

ﬂexible joints at the wing roots and a rigid mass at the vehicle CG. The rigid mass is

used to account for the fuselage mass and moment of inertia while the beam elements

and ﬂexible joints are used to account for the elasticity of the fuselage. The fuselage

beam stiﬀness and ﬂexible joint properties are generated by a least squares optimization

using the ﬁrst four NASTRAN elastic mode frequencies as a reference. Table 2.4 shows

the Dymore fuselage for the X-57 along with a comparison to the NASTRAN

25 frequencies. The largest relative error is 8% in the symmetric out-of-plane bending mode; however, its absolute error is a 0.2Hz over-prediction of Dymore than the

NASTRAN prediction. The errors of all the other modes are below 3%. Since

CAMRAD II is using a modal representation, the NASTRAN fuselage mode shapes and frequencies are directly input into the analysis.

Figure 2.4: Dymore Full-Span Elastic Fuselage

Mode NASTRAN Dymore % Error Absolute Diﬀerence (Hz) (Hz) (Hz) Sym Out-of-Plane 2.39 2.58 7.73 0.19 Asym Out-of-Plane 5.05 4.93 2.47 0.12 Asym In-Plane 6.46 6.45 0.12 0.01 Sym In-Plane 7.64 7.58 0.78 0.06 Sym Torsional 17.43 16.99 2.48 0.44

Table 2.1: Dymore X-57 Fuselage Model

26 2.2.4 Free-Flying Aircraft Dymore Model

The Dymore model of the X-57 Maxwell aircraft, shown in Figure 2.5 includes the wing, fuselage, empennage, pylons, and the tip propellers. The wing, empennage, and propeller blades are modeled as elastic beams. The inboard propellers, the pylon and nacelle of the large tip propeller, and the fuselage are modeled as rigid bodies with their appropriate inertial properties. The blade pitch bearing is modeled as a revolute joint.

The pylon mount flexibility in yaw and pitch is captured with linear springs connecting the pylon to the wing. The aerodynamic forces acting on the propeller, wing, and empennage are modeled with both quasi-steady and unsteady lifting line theories developed by Peters [45]. A table lookup format is used to define the propeller aerodynamic coefficients on the propeller blades and the empennage while a zero lift angle is defined with a lift curve slope and the drag and moment coefficients are used for the wing. Rotor inflow is calculated with a three-dimensional nonlinear dynamic inflow model using three inflow states. The aerodynamic interaction between the propeller and wing is ignored, as is the downwash from the wing and empennage.

Figure 2.5: Dymore X-57 Maxwell Free-Flying Model

27 Fuselage Structure and Aerodynamic Model

The fuselage inertial properties are included by the use of a rigid body mass, with its

appropriate mass and moments of inertia. The drag of the fuselage, while not used for

the whirl flutter, is needed for the flight dynamics. The parasite drag coefficient for the

fuselage was chosen to be 0.04. This value was selected from using similar style aircraft

to the original Tecnam P2006T and using the aircraft’s total wetted area [46,47]. The

total drag from the fuselage is based on the wing area, and since the X-57 has reduced

wing area compared to the Tecnam, this drag value is an estimation. The drag due to

the folded in propellers was factored into the fuselage drag as well by multiplying the

fuselage drag by 1.025. This multiplier was determined by using a wing plus nacelle

drag value at low Mach number from Hoerner’s Fluid Dynamic Drag [48]. Therefore the

total fuselage drag coeﬃcient from the fuselage and the twelve inboard propellers is

0.041. The drag applied to the fuselage is given by the following equation:

DF use = q∞CDF S (2.1)

The drag is applied as a point load at the center of gravity.

Empennage Structure

The empennage geometry was taken from the original Tecnam P2006T, and the multibody dynamics model uses NACA0010 for the vertical tail and the horizontal tail.

The elevator is attached to the horizontal tail by rigid connections and uses revolute joints to allow elevator deﬂection. The tail boom and empennage are formed using massless and rigid beam elements with no aerodynamics applied on the tail boom.

Aerodynamics are applied to the vertical and horizontal tail using the aerodynamic

28 theory developed by Peters [45].

2.2.5 Propeller Dymore Model

The X-57 propeller, shown in Figure 2.7, uses revolute joints to model the hub and pitch bearings with linear springs that allow pitch and yaw ﬂexibility. Three propeller blades are modeled using beam elements. In addition, there are rigid masses along the propeller/pylon system that accounts for the inertial properties of the pylon, electric motor, and hub assembly. The hub and pylon masses are connected to the wingtip via a rigid connection. Table 2.2 gives the basic properties of the X-57 propeller model.

Parameter Value Number of Blades 3 Diameter, ft 5 Mean Chord, ft 0.308 Cruise RPM 2250

Table 2.2: X-57 Propeller Properties

Rigid and Elastic Blades

Two propeller blade stiffness values are used in this dissertation: a rigid propeller blade and an elastic propeller blade. The rigid propeller was modeled in Dymore using a beam element with high stiffness values. The stiffness values for the rigid propeller blade were selected such that the first modal frequency was significantly higher than the system, on the order of 500 Hertz. A beam model reflecting the real flexibility of the propeller blade is developed in the current study to investigate the elastic effects of the blade on whirl flutter stability. Figure 2.6 compares the CAMRAD II and Dymore predictions of the blade natural frequency in a fanplot, which is a plot of the variation of the blade natural frequencies against rotor speed. Also plotted are the propeller harmonics given in units of per revolutions (per rev). A fanplot is used in the design of

29 a propeller/rotor to ensure that the propeller/rotor does not operate at a speed such that the blade is in resonance. It is used here to show that the CAMRAD II and

Dymore blades are modeling the same propeller blade in the two analyses, as they have a good agreement in the fanplot. Table 2.3 shows that the diﬀerences for the ﬁrst four blade modes predicted by the two analyses are all below 2%.

500

450 9/rev

400 8/rev Dymore 350 CAMRAD II 7/rev

300 6/rev

250 5/rev

200 4/rev Frequency (Hz)Frequency 150 3/rev

100 2/rev

50 1/rev

0 0 500 1000 1500 2000 2500 3000 RPM Figure 2.6: Natural Frequency Variation with Rotor Speed

Mode CAMRAD II Dymore % Diﬀerence (Hz) (Hz) 1 79.46 80.67 1.5 2 177.54 178.42 0.5 3 267.55 263.54 1.5 4 448.67 453.97 1.2

Table 2.3: Blade Frequency at Cruise RPM (2250 RPM)

30 Airfoil Distribution and Blade Elasticity

Two different airfoil distributions are used in this dissertation. The first airfoil distribution is the NACA 0012 along the entire span of the blade. The next distribution is a series of eleven airfoils for an off-the-shelf propeller blade, designated as the X-57

Airfoils.

Figure 2.8 shows the variation of blade pitch change for windmilling with velocity for the NACA 0012 blades for CAMRAD II and Dymore. Both CAMRAD II and

Dymore are initialized at the same angles with the same geometry and trimmed such that the propellers are in windmilling conditions, that is that the propeller blade pitch angle is changed until zero torque is applied onto the propeller shaft. This ensures that the two analyses are modeling equivalent propeller blade aerodynamics and are trimming to the similar angles for windmilling. Figures 2.9 and 2.10 show the NACA

0012 propeller thrust and power respectively for CAMRAD II and Dymore. There is a slight diﬀerence between the two analyzes near the propeller stall conditions due to the diﬀerence in modeling stall, CAMRAD II adds a stall factor that Dymore does not.

Figure 2.11 shows that the NACA 0012 and the X-57 airfoils follow the same trend with a constant oﬀset from each other, with the NACA 0012 being a few degrees above the

X-57 airfoils. Figure 2.12 shows the trim angles for the X-57 airfoils distribution on rigid and elastic propellers. The elastic propeller has a similar oﬀset of a few degrees from the rigid propeller that grows as velocity increases.

31 Figure 2.7: X-57 Maxwell Propeller Dymore Model

CAMRAD 2250 RPM CAMRAD 2700 RPM 30 Dymore 2250 RPM Dymore 2700 RPM

0 Change in Pitch (degrees) Pitch in Change -10

-20 50 100 150 200 250 300 350 400 450 500 Velocity (knots)

Figure 2.8: Propeller Pitch Change Under Windmilling with NACA 0012

32 600

500

400

300

200

100

-100(lbs) Thrust

-200 CAMRAD II 2250 RPM -300 CAMRAD II 2700 RPM Dymore 2250 RPM Dymore 2700 RPM -400

-500 -30 -20 -10 0 10 20 Pitch (degrees)

Figure 2.9: Propeller Thrust with NACA 0012

300 CAMRAD II 2250 RPM CAMRAD II 2700 RPM Dymore 2250 RPM Dymore 2700 RPM 200

100

Power (hp) Power 0

-100

-200 -30 -20 -10 0 10 20 Pitch (degrees)

Figure 2.10: Propeller Power with NACA 0012

33 35

30 NACA 0012 25 X-57 Airfoil

-5 Change in Pitch (Degrees) Pitch in Change -10

-15

-20 50 100 150 200 250 300 350 400 450 Velocity (Knots)

Figure 2.11: Windmilling Trim Blade Pitch Change for NACA 0012 and the X-57 Airfoils on Rigid Blades

30 Rigid Propeller Elastic Propeller 25

-5 Change in Pitch (Degrees) Pitch in Change -10

-15

-20 50 100 150 200 250 300 350 400 450 Velocity (Knots)

Figure 2.12: Windmilling Trim Blade Pitch Change for the X-57 Airfoils on Rigid and Elastic Blades

34 3 FUNDAMENTAL THEORY OF PROPELLER WHIRL FLUTTER

Propeller whirl flutter is an aeroelastic instability where the propeller aerodynamics drive the airframe/pylon motion to become unstable. This section gives the fundamental equations of motion behind this instability and shows how stiffness and airspeed influence the stability.

3.1 Whirl Flutter Equations

The general equations behind whirl ﬂutter are given by the following from Johnson’s

Rotorcraft Aeromechanics [6]. The equations of motion for the tip-path-plane tilt and pylon angular motion are:

          ¨ ˙ β1c −1 0 α¨y −γM ˙ 2 β1c   +     +  β     ¨         ˙  β1s 0 1 α¨x −2 −γMβ˙ β1s         2 γM ˙ 2 + hγMµ˙ α˙ y ν − 1 −γM ˙ β1c +  β    +  β β   (3.1)      2    2 + hγMµ˙ −γMβ˙ α˙ x γMβ˙ νβ − 1 β1s     0 −γλM α  µ  y +     = 0 −γλMµ 0 αx

and

35         ¨ ˆ −1 0 β1c Iy + 1 0 α¨y     +        ¨   ˆ    0 1 β1s 0 Ix + 1 α¨x     ˙ γM ˙ − hγR ˙ −2 − hγH ˙ β1c +  β β β       ˙  −2 − hγHβ˙ −(γMβ˙ − hγRβ˙ ) β1s     ˆ 2 2 Cy − γM ˙ + hγR ˙ + h γ(Hµ + Rµ) −(2 + hγMµ + hγH ˙ − h γRr) α˙ y +  β β β     2 ˆ 2    2 + hγMµ + hγHβ˙ − h γRr Cx − γMβ˙ + hγRβ˙ + h γ(Hµ + Rµ) α˙ x     hγ(H ˙ − R ) γM ˙ − hγR ˙ β  β β β β   1c +     −(γMβ˙ − hγRβ˙ ) −hγ(Hβ˙ − Rβ) β1s     ˆ Ky − hγλ(Hµ + Rµ) γλMµ − hγλRr αy +     = 0  ˆ    −(γλMµ − hγλRr) Kx − hγλ(Hµ + Rµ) αx (3.2)

For a propeller mounted on an elastic pylon, where the blade frequencies are signiﬁcantly higher than pylon, the equations become:

            I 0 α¨ C −D α˙ K L α  y   y  y   y  y   y     +     +     = 0 (3.3) 0 Ix α¨x DCx α˙ x −LKx αx

ˆ ˆ ˆ Where Ix = Ix + 1, Cx = Cx + Ca, Kx = Kx + Ka, and

2 Ca = −γMβ˙ + hγRβ˙ + h γ(Hµ + Rµ) Z 1 4 Z 1 2 (3.4) ∼ r 2 λ ∼ γ 2 γ = γ dr + h γ dr = cos φe + h λ sin φe 0 2U 0 2U 8 2

Z 1 2 Z 1 2 2 ∼ λr γh ∼ D = 2 + hγMµ + hγHβ˙ − h γRr = 2 + hγ dr − hγ dr = 2 (3.5) 0 2U 0 2U

Z 1 3 ∼ λ ∼ γ 2 Ka = hγλ(Hµ + Rµ) = hγ dr = h λ sin φe (3.6) 0 2U 2 Z 1 2 2 ∼ λ r ∼ γ 2 L = γλMµ − hγλRr = γ dr = λ cos φe (3.7) 0 2U 4

36 −1 Approximating the inﬂow at re, φe = tan λ/re Therefore:

Z 1 Z 1 cl clM cdα Mµ + Hβ˙ = r(FzT + FxP )dr = rU 3 + M − dr (3.8) 0 0 2a 2a 2a

The characteristic equation for whirl ﬂutter is:

2 2 2 (Iys + Cys + ky)(Ixs + Cxs + Kx) + (−Ds + L) = 0 (3.9)

By substituting s = ωi into the characteristic equation, and taking the real component:

2 2 2 2 2 (Ky − Iyω )(Kx − Ixω ) − (D + CyCx)ω + L = 0 (3.10)

The imaginary part can be solved for the ﬂutter frequency:

K C + K C − 2DL ω2 = y x x y (3.11) IyCx + IxCy

When values are assigned and the whirl ﬂutter boundary plotted as a contour plot

with the pitch and yaw stiﬀness as the x-axis and y-axis respectively (Fig. 3.1). The

ˆ ˆ assumed system properties are h = 0.3, γ = 4, Ix = Iy = 3, and Cx = Cx = 0. It can be

seen in the figure that the stiffness required increases as the inflow increases. This

system stability depends wholly on aerodynamic damping, has zero structural damping,

and requires the highest stiffness when the pitch stiffness is equal to the yaw stiffness.

Overall, whirl ﬂutter is caused by the aerodynamic coupling spring

∼ ∼ γ 2 L = γλMµ = 4 λ cos φe. Pitch αy tilts the rotor with respect to the axial ﬂow γ, which causes an in-plane velocity λαy, which then gives a roll moment on the propeller hub through the aerodynamic coeﬃcient Mµ. L increases with forward speed, thus whirl

37 2.5 λ=1.5 STABLE

2.0

λ=1.0

1.5 y  I y  K

1.0 λ=0.5

0.5 UNSTABLE

0.0 0.0 0.5 1.0 1.5 2.0 2.5

 K x Ix

Figure 3.1: Whirl Flutter Stability Boundary for Diﬀerent Inﬂows [6]

ﬂutter for a rigid propeller on a pylon is a high inﬂow instability caused by the aerodynamic spring coupling Mµ [6].

38 4 VALIDATION OF MULTIBODY DYNAMICS ANALYSIS

This section is to validate the use of multibody dynamics for determining the whirl

flutter stability. Two cases are shown in this section, a cantilevered wing undergoing torsion-bending flutter and an isolated propeller undergoing propeller whirl flutter.

Both cases are compared with experimental and mathematical models. These two cases are chosen for the validation in addition to the cases presented in the literature review since the whirl ﬂutter cases examined in this dissertation is the combination of propeller and wing motion.

4.1 The Goland Wing Undergoing Torsion-Bending Flutter

This section is to validate the use of multibody dynamics for use of an elastic wing undergoing flutter. The Goland Wing is named after the aeroelastician that performed a flutter analysis of a wing undergoing torsion-bending flutter [49]. The wing is uniform, unswept, untapered, and untwisted. The wing properties are given below in

Table 4.1 and the Dymore model is shown in Figure 4.1.

Property Value slug Mass per Unit Length 0.746 ft slug·ft2 Moment of Inertia 1.943 ft Bending Stiﬀness 23.6 · 106 lb · ft2 Torsional Stiﬀness 2.39 · 106 lb · ft2 Chord 6 ft Span 20 ft Elastic Axis 33% of chord (from leading edge)

Table 4.1: Goland Wing Properties

39 Figure 4.1: Goland Wing Dymore Model

The properties are applied to a cantilevered wing in Dymore, the wing is perturbed and a Prony analysis run on the transient results [42]. The aerodynamics on the wing are modeled using an unsteady aerodynamic theory developed by Peters et al. [45] and use a table look-up for the aerodynamic coefficients of a NACA 0012 airfoil. Figure 4.2 shows the frequency and damping ratio of the wing undergoing flutter. The velocity range for Figure 4.2 was chosen as it encased the flutter speed and shows that Dymore can run past the flutter speed. Table 4.2 shows a comparison between the exact Goland

Wing solution, a solution using assumed modes [50], and the results from a Dymore analysis. The Dymore solution has a better comparison for the flutter speed than the method of assumed modes, but underpredicts the flutter frequency. However, this simple model verifies that multibody dynamics can identify the flutter speed and give an approximation of the flutter frequency.

40 Quantity Exact Assumed Mode Dymore ft ft ft Flutter Speed 576 s 565 s 570 s Flutter Speed Diﬀerence 0 % 1.9 % 1.0 % rad rad rad Flutter Frequency 66.2 s 67.4 s 61.0 s Flutter Frequency Diﬀerence 0 % 1.8 % 7.8 %

Table 4.2: Goland Wing Results Comparison

41 14

10 Frequency (Hz)Frequency

6 500 520 540 560 580 600 Velocity (ft/s)

(a) Frequency

Damping Ratio (%)Ratio Damping 0

-2

-4 500 520 540 560 580 600 Velocity (ft/s)

(b) Damping Ratio

Figure 4.2: Goland Wing Frequency and Damping Ratio of Flutter Mode

42 4.2 Isolated Propeller Undergoing Whirl Flutter

The section is to validate the use of multibody dynamics for a propeller undergoing whirl ﬂutter. This system is a propeller based on the properties from Ref. 51 that is used to validate the multibody dynamics model of a propeller undergoing whirl ﬂutter.

The propeller blades for this system are considered rigid and use the NACA 0012 for the blade airfoil. This system consists of three propeller blades, a propeller shaft, and linear springs that allow the propeller to pitch and yaw. Revolute joints are used for the hub, and pitch bearings and the propeller blades and shaft are modeled using beam elements. Figure 4.3 shows the Dymore model for the isolated propeller.

Figure 4.3: Dymore Isolated Propeller Model

The isolated propeller is using an unsteady aerodynamics theory developed by

Peters et al. [45] and use a table look-up for the aerodynamic coefficients. The propeller inflows are calculated using a three-dimensional nonlinear dynamic inflow model using

five inflow states. The propellers are trimmed to be windmilling by adjusting the pitch such that zero torque is applied to the propeller shaft. The whirl flutter stability is determined by introducing a gust and using the Prony method on the transient response to determine the frequency and damping ratio.

43 4.2.1 Transient Results

Figures 4.4 and 4.5 show the isolated propeller hub displacements for different spring stiffness values at the same velocity and for different velocities at the same spring stiffness respectively. A gust is applied at two seconds and the spring stiffness are the same for the pitch and yaw directions. It can be seen that there is an additional frequency at the beginning of the perturbation. This is due to there being two whirl

flutter modes, a stable forward procession, and an unstable backward procession. The stable forward mode dampens out quickly and only the backward mode remains. There is a 90◦ phase difference between the pitch and yaw motion. Figure 4.4 shows that as the spring stiffness increases while keeping the velocity constant that the frequency of the whirl flutter mode also increases and the amplitude decreases. The top plot in

Figure 4.4 is dynamically unstable, the middle plot is near the marginally stable stiffness, and the bottom plot is dynamically stable. Figure 4.5 shows that increasing the velocity while keeping the spring stiffness constant does not affect the whirl flutter frequency but causes the amplitude to increase. The top plot in Figure 4.5 is dynamically stable, the middle plot is near the marginally stable dynamic pressure, and the bottom plot is dynamically unstable.

44 )

h 0.02 c n i (

t n

e 0 m e c a l p

s -0.02 i

D 0 0.5 1 1.5 2 2.5 3 3.5 4 Time (sec) )

h 0.02 c

n pitch i (

t yaw n

e 0 m e c a l p

s -0.02 i

D 0 0.5 1 1.5 2 2.5 3 3.5 4 Time (sec) )

h 0.02 c n i (

t n

e 0 m e c a l p

s -0.02 i

D 0 0.5 1 1.5 2 2.5 3 3.5 4 Time (sec)

Figure 4.4: Propeller Hub Motion for Increasing Spring Stiffness at Constant Velocity: Top Has Lowest Stiffness, Bottom Has Highest Stiffness )

h 0.02 c n i (

t n

e 0 m e c a l

p -0.02 s i 0 0.5 1 1.5 2 2.5 3 3.5 4 D Time (sec) )

h 0.02 c n

i pitch (

t yaw n

e 0 m e c a l

p -0.02 s i 0 0.5 1 1.5 2 2.5 3 3.5 4 D Time (sec) )

h 0.02 c n i (

t n

e 0 m e c a l

p -0.02 s i 0 0.5 1 1.5 2 2.5 3 3.5 4 D Time (sec)

Figure 4.5: Propeller Hub Motion for Increasing Velocity at Constant Spring Stiﬀness: Top Has Lowest Velocity, Bottom Has Highest Velocity

45 4.2.2 Comparison to Mathematical Model and Experimental Data

Figure 4.6 shows are how the damping ratio contours of the multibody dynamics

representation of the isolated propeller compare to both the mathematical model and

experimental results presented in Ref. 51 as a function of both dynamic pressure and

spring stiﬀness. The results from Ref. 51 are the experimental results of the propeller in

whirl ﬂutter conditions near marginal stability and the negative of the artiﬁcial

structural damping, g, required for marginal stability. The negative of the required structural damping is shown to compare to the Dymore damping ratio taken from the

Prony analysis on the transient results. There is good agreement between all three results shown with the multibody dynamic representation contours lining up well with the mathematical model and experimental data.

20 0 -g=-1 5 18 . -1 0

16 -2 14 -g=-3

) 12 d -3 a 0

r 1 / - b l 10 2 - - 4 n - i (

-5 θ 8 -3 k -2 6 -3 0 -4 4 Dymore Damping Ratio % -g=-1% 2 -g=-3% Experimental 0 0 0.5 1 1.5 2 2.5 3 3.5 4 Dynamic Pressure (lb/ft2)

Figure 4.6: Damping Ratio of The Isolated Propeller for Diﬀerent Dynamic Pressures and Spring Stiﬀness

46 5 ANALYTICAL RESULTS

5.1 X-57 Standalone Propeller

This section shows the whirl ﬂutter stability of the isolated X-57 propeller. The RPM is kept constant at 2250 RPM and the propeller is windmilling for all cases in this section.

Two different airfoil distributions are examined and two different pitch and yaw stiffness ratios are used. The same approach was taken for the isolated propeller validation, a gust was applied at two seconds and the response analyzed.

5.1.1 Equal Pitch and Yaw Stiﬀness

Figures 5.1a and 5.1b show the frequency and damping ratio contours for varying the stiﬀness of the pylon mount and speed. Note that the y-axis is a logarithmic scale.

Figure 5.1a shows that the frequency is relatively constant with velocity for constant stiffness values of the pylon system with the lower stiffness values showing a slight change in frequency as the propeller goes unstable. Figure 5.1b shows that the gradient of stability with respect to stiffness is small on the stable side of marginal stability and has a large increase on the unstable side. This means that a reduction in pylon stiffness can cause large displacements very suddenly near the marginal stability point. However, since the usual stiffness of the X-57 is at the top of the contour plot, the isolated propeller should not undergo whirl flutter unless damage is sustained to the pylon.

When the contour ends in the middle of the ﬁgure, it is due to the simulation diverging

47 before the Prony method can be fully implemented on the response.

) 35 35 35 35 35 35 35 35 d 30 30 30 30 30 30 30 30 a

r 25 25 25 25 25 25 25 25 /

b 20 20 20 20 20 20 20 20 l 5 -

t 10 15 15 f 15 15 15 15 15 15 (

s 10 10 s 10 10 10 10 10 10 e n f f i

t 4 5 5 5 5 5 5 5 5

S 10

4 4 4 4 4 n 3 o 3 3 3 l y

P 2 2

3 10 100 150 200 250 300 350 400 Velocity (Knots) (a) Whirl Flutter Frequency (Hz) ) d a r / 1 1 1 1 b l

- 5 1 t

f 10 1 1 1 (

1 s

s 0 e 0

n 0 f -1

f 0 -1 i -2

t -1 -2 4 1 0 -1 -2 S -2 -4 -4 10 0 -1 -2

n -4 -6 o -1 -2 l -4 -8 y 0

P -2 -4 -1 -6 -8 -4-6 -8 3 -10 10 100 150 200 250 300 350 400 Velocity (Knots) (b) Whirl Flutter Damping Ratio (%)

Figure 5.1: Isolated X-57 Propeller with X-57 Airfoils: Contour Plots for Varying Pylon Stiffness and Velocity; Pitch Stiffness Equal to Yaw Stiffness

48 5.1.2 Diﬀerent Pitch and Yaw Stiﬀness

The X-57 pylon stiffness ratio between the pitch and yaw stiffness is 0.5405, with the pitch having the larger stiffness. Keeping this ratio, the pylon stiffness is again varied and the stability examined for the isolated X-57 propeller. Figures 5.2a and 5.2b show the contours of the isolated X-57 propeller frequency and damping ratio. Note that the y-axis is a logarithmic scale. Compared to 5.1b, the contours are closer together for each of the constant damping lines indicating a larger gradient with respect to pylon stiffness. Figure 5.2a also shows that the contours of constant frequencies at higher stiffness values are closer together and are not constant for constant stiffness values compared to 5.1a. This indicates that there is a dominant spring stiffness between the pitch and yaw springs that dictate the whirl flutter frequency.

5.1.3 Eﬀect of Diﬀerent Airfoil

Figure 5.3 shows the frequency and damping ratio contours for the X-57 propeller with the NACA 0012 airfoil along the span of the propeller blades. Compared to Fig. 5.2b,

Fig. 5.3b shows that the damping ratio contours are closer together, indicating that the gradient of the damping ratio is larger for the NACA 0012 propeller.

Figure 5.4 shows the contour for the marginal stability of both the X-57 and NACA

0012 airfoil distributions. It can be seen that the NACA 0012 distribution requires a higher pylon stiﬀness in order to remain stable compared to the X-57 distribution.

49 35 35 35 35 35 35 35 ) 30 30 30 30 30 30 30 3035 25 25 25 25 25 25 25 d 25

a 20 20 20 20 r 20 20 20 / 20 b 15 15 15 15 15 l 15 15 15 - 5 t

f 10 ( 10 10 10 10 10 10 10 10 s s e n

f 5 5 5 5 f 5 5 5 5 i 4 4 4 t 4 4 4 4 4 4 3 S

3 10 3 3 3 n

o 2 l 2 2

y 2 P

3 10 1 100 150 200 250 300 350 400 Velocity (Knots) (a) Whirl Flutter Frequency (Hz) ) d a r / b l

- 5 t

f 10 (

s s

e 1 1

n 1 1 f 1 0 f 0 i 1 -1 t 0 -1 4 0 -1 -2 -2 S -2 -4 1 0 -1 -2 10 0 -1 -2

n -4 -6 o -1 -2 l -4 -8 -10

y -6 0 -2 -4

P -1 -6-8 -4 -2 -6 3 -8 10 100 150 200 250 300 350 400 Velocity (Knots) (b) Whirl Flutter Damping Ratio (%)

Figure 5.2: Isolated X-57 Propeller with X-57 Airfoils: Contour Plots for Varying Pylon Stiﬀness and Velocity; Ratio of Pitch to Yaw Stiﬀness is 0.5405

50 3 1 0 52 5 2030 5 5 2 1 33 2 3 5 05 0 2105 35 35 35 35 35 35 5 25 30 30 30 30 30 30 303 ) 25 25 25 25 25 25 25 25 d 20 20 20 20 20 a 20 20 20 r / 15 15 15 15 15 15 15 15 b 5 l -

t 10

f 10 10 10 ( 10 10 10 10 10

s s

e 5 5

n 5 5 4 f 5 5 5 5 4 f 4 4 3 i 4 3

t 4 4 4 3 4 3 2 2

S 10 3

3 3 3 2

n 2

o 2 l 2 2 y 1 P 1 3 1 10 100 150 200 250 300 350 400 Velocity (Knots) (a) Whirl Flutter Frequency (Hz) ) d a r / b l 5 - t

f 10 (

s s

e 1

n 1 1 1 0 f 1 -1 f 0 0 -2 i 1 0 -1-2 -4

t -1 0 -1 -2 -1-40 -6 4 -2 -4 -6 -8-14 -8 S -1 10 1 0 -2-4 -4 -1-2 -6 -6 n -10 -8 0 -8 o -4 -14

l -6 -1-2 -8 -10 -14 y -14 1 -4

P -6 -10 -20 -8 -14 -2 -4 -10 -20 3 -6 10 100 150 200 250 300 350 400 Velocity (Knots) (b) Whirl Flutter Damping Ratio (%)

Figure 5.3: Isolated X-57 Propeller with NACA 0012: Contour Plots for Varying Pylon Stiﬀness and Velocity; Ratio of Pitch to Yaw Stiﬀness is 0.5405

51 NACA0012

) X-57 d a r / b l

- 5 t f

( 10

s s e n f f i t 4 S 10 n o l y P

3 10 100 150 200 250 300 350 400 Velocity (Knots)

Figure 5.4: Contour of the Marginal Stability Location (Zero Damping) for Varying Pylon Stiﬀness and Velocity: Ratio of Pitch to Yaw Stiﬀness is 0.5405 with NACA 0012 and X-57 Airfoils

52 5.1.4 Summary and Conclusions for Isolated Propelller

This section has been to show the propeller whirl ﬂutter stability for a standalone X-57 propeller. The propeller blades are the elastic blades with either the X-57 airfoils or

NACA 0012. While there was little change in the location of marginal stablity, the gradient of the damping ratio was much greater for the X-57 airfoils. In addition, both equal pitch and yaw stiffness and the actual pitch and yaw stifness ratio have been exmained. For the nominal stiffness values for pitch and yaw, whirl flutter does not occur. Only a severe reduction in stiffness, by several orders of magnitude, will cause the isolated propeller to go unstable.

5.2 Propeller Whirl Flutter Study with a Semi-Span X-57 Propeller/Wing/Pylon Model

This section looks at the development of the semi-span X-57 Dymore model. Starting from the full NASTRAN FEM model to an equivalent beam for Dymore input. Since

CAMRAD II uses modal inputs for the wing, only the development of the Dymore wing is looked at in detail.

5.2.1 Three Wing/Pylon Design Versions

This study covers three versions of the wing/pylon system, and the eﬀects of various design decisions on whirl ﬂutter is studied. The three versions will be referred to as

Versions 1-3.

Version 1

Version 1 occurred during the initial wing design of the X-57. The wing design and aeroelastic analyses were performed in concert, so that one would impact the other and

53 presumably keep the design in a region of the design space with robust aeroelastic safety margins. This process necessitates that the structural model be analyzed at various levels of completion. The initial FE structural model of the X-57 wing included only primary components of the wing; the cruise propulsion system components (tip nacelle, pylon, propeller and motor) were represented solely by their mass properties.

This included a full 3D FE model of the wing. For whirl ﬂutter analysis in Dymore, an equivalent beam model was derived.

Version 2

Between Version 1 and Version 2, two substantial changes were made in the structural design. The close spacing observed for the first two modal frequencies in the Version 1 design was an aeroelastic concern, more from a wing flutter perspective than from a whirl flutter perspective. This led to incorporation of unidirectional fibers in the spar caps.

The second important development was that the design of the tip nacelle geometry, structure and propulsion system had advanced so that details could be included in the analytical model. The design of the tip nacelle included a firewall that served as a faceplate for mounting the tip propeller motor. This faceplate provides much of the structural stiffness between the propeller system and the nacelle, strongly influencing the in-plane behavior. Due to this being a local phenomenon, it is difficult for the equivalent beam model required by the Dymore analysis to capture this local deformation.

54 Version 3

The ﬁrewall was stiﬀened in Version 3, substantially reducing the faceplate deformations. Additional design and modeling developments were incorporated, including more detailed representations of the control surfaces, control systems and cabling. The mode shapes and frequencies for Version 3 were generated using a FE model where the control surfaces and systems were removed, as was also done in

Version 2. This was necessitated by the equivalent beam model process. The mass and inertial properties of the control surfaces were not incorporated into the beam model for

Version 3.

The analytical results include the verification of the X-57 Maxwell analytical models and the whirl flutter stability predictions. The important sub-components of the semi-span X-57 Maxwell analytical model, the fixed subsystem (wing/pylon) and the rotating subsystem (isolated propeller), are developed and verified separately by correlating the predictions among different analyses (Dymore, CAMRAD II,

NASTRAN).

5.2.2 X-57 Maxwell Analytical Model Veriﬁcation

The analytical model development is carried out component by component, through a modular approach with correlation among the analytical models at each step. The comparison of predictions of characteristics of these key components among CAMRAD

II, Dymore, and NASTRAN are presented below. The NASTRAN analysis is used to validate the structural model of the wing/pylon system, while the aerodynamic model of the propeller is correlated between CAMRAD II and Dymore.

55 5.2.3 Equivalent Beam Model of Three Fixed System Versions

This section compares the modal vibration results of the full FE wingbox model and the derived equivalent beam model.

Version 1

In Version 1, the first vertical bending and first in-plane bending modes were much closer in frequencies, 1.8 Hz and 2.8 Hz respectively. Since the frequencies were so close together, design changes were made in Versions 2 and 3 to further separate the two in order to help with potential classical flutter issues. The design change implemented was the addition of unidirectional fibers in the spar caps that raised the first in-plane bending mode to its current frequency. As the model developed complexity, a modal analysis of the firewall revealed significant local deformation to the faceplate, particularly in the second in-plane bending mode. A tuned tip yaw spring used in the equivalent beam model is able to partially account for this relative deformation as well.

The mode shapes and frequencies for this FE model are shown in Figure 5.5. The

ﬁrst two mode shapes can be described qualitatively as wing vertical ﬁrst bending mode

(1.8 Hz) and wing inplane ﬁrst bending mode (2.8 Hz). As noted above, these ﬁrst two frequencies are relatively close to each other. Due to concerns about aeroelastic wing

ﬂutter problems, design changes were made in the next version to separate these frequencies. It can also be seen in Figure 5.5 that the equivalent beam model is unable to correctly predict the torsional vibration shape of the sixth mode. This is due to the fact that Step 6 in Section 2.2.1 (additional tuning of the beam model) was not performed for this version.

56 Version 2

The unidirectional fibers in the spar caps raised the modal frequency of the first in-plane mode to 6.8 Hz. It can also be seen that all six modes in Figure 5.6 agree well between the equivalent beam and the full FE model, both in terms of shape and frequency. Modal analysis of the firewall also revealed significant local deformation of the faceplate, shown in Figure 5.7, particularly in the second in-plane bending mode.The tuned tip yaw spring used in the equivalent beam model is able to partially account for this relative deformation as well.

Version 3

Figure 5.8 shows the mode shapes and frequencies for Version 3, which were generated using a FE model where the control surfaces and systems were removed. This was necessitated by the equivalent beam model process. It can also be seen that the local deformation of the ﬁrewall faceplate seen in Figure 5.6 is less prevalent here.

Wing/Pylon System

The properties of the NASTRAN equivalent beam model was used in Dymore to develop the Dymore wing/pylon system and the predicted natural frequencies were compared to the NASTRAN calculations for the Version 3 conﬁguration. Table 5.1 lists the ﬁrst ten natural frequencies, with the maximum error between Dymore and the

NASTRAN beam model predictions less than 2.0%. Figure 5.8 compares the mode shapes of the ﬁrst four modes of the wing/pylon model between the NASTRAN and

Dymore predictions. These four modes are respectively: the ﬁrst wing bending, in-plane bending (knife-edge), second bending (out-of-plane), and torsion mode. These modes play critical roles in the dynamics of propeller whirl ﬂutter stability, and the good

57 (a) Full FEM

(b) Beam Model

Figure 5.5: Version 1: Full FEM and Equivalent Beam Model agreements between the analytical predictions of the frequencies and mode shapes ensure accurate structural representation of the wing/pylon system.

Mode NASTRAN Dymore N/D Hz Hz (% Error) 1 2.18 2.19 0.5 2 7.09 7.11 0.3 3 13.91 13.87 -0.3 4 16.77 16.89 0.7 5 34.70 34.11 -1.7 6 34.82 35.35 1.5 7 47.78 47.89 0.2 8 54.86 54.70 -0.3 9 59.55 59.56 0.0 10 87.21 86.74 -0.5

Table 5.1: Frequencies of Wing/Pylon Model; Version 3

58 (a) Full FEM

(b) Beam Model

Figure 5.6: Version 2: Full FEM and Equivalent Beam Model

5.2.4 Whirl Flutter Stability

The whirl ﬂutter stability of three wing design versions are presented. The whirl ﬂutter stability of the X-57 Maxwell aircraft is studied with two semi-span analytical models.

Although modeling the same phenomenon, CAMRAD II and Dymore use diﬀerent methods to calculate the wing damping. CAMRAD calculates the wing damping directly through eigenanalysis of the system equations of motion. Dymore, instead, uses the Prony method to identify the wing damping based on the wing transient response [42]. In addition, all the calculations of whirl ﬂutter stability in CAMRAD II are carried out without wing aerodynamics since it uses a modal representation of the wing.

59 Figure 5.7: Version 2: Full FEM Shapes: Mode 5 Detail

Whirl Flutter Stability: Version 1

The frequencies and damping ratios of the wing bending and torsional modes are predicted for a range of airspeeds, as shown in Figure 5.9. The potential critical mode, that is the mode with the possible onset of whirl flutter, is identified as the torsional mode by both analyses. While CAMRAD II tends to predict a higher damping ratio for the first bending and first inplane bending (knife-edge), both CAMRAD II and Dymore nearly overlay for the first torsional bending mode. The flutter speed is found to be around 200 knots, however, the damping is calculated without wing aerodynamics.

The whirl flutter stability of the X-57 Maxwell is recalculated using Dymore with a quasi-steady wing aerodynamics model. Figure 5.10 shows the wing frequency and damping variation with airspeed for the bending and torsional modes. The wing aerodynamics has a stabilizing effect on the torsional mode as the torsional flutter is seen to be eliminated when wing aerodynamics is included. The damping of the first wing bending mode is also shown to increase substantially when including the wing aerodynamics. In contrast, the wing aerodynamics has a negligible effect on the

60 (a) Full FEM

(b) Beam Model

Figure 5.8: Version 3: Full FEM and Equivalent Beam Model damping of the inplane (knife-edge) bending mode.

In addition, the wing aerodynamics has a small effect on the frequencies of the wing bending (out-of-plane and inplane) modes. However, the wing torsional frequency is affected by the wing aerodynamics noticeably, showing a different trend or variation with airspeed for with and without wing aerodynamics. This large difference in frequency is due to the aerodynamic forces being dependent on the pitch frequency and airspeed [45].

61 Whirl Flutter Stability: Version 2

Due to the equivalent beam not being able to properly represent the flexible faceplate, only a CAMRAD II analysis was performed (Figure 5.11). The flexibility of the faceplate caused large rotations in both mode 3 and mode 4 in the pitch direction, and the yaw flexibility was introduced through mode 5. For this version, the critical mode is the second beam bending mode and is also around 200 knots.

Whirl Flutter Stability: Version 3

Figure 5.12 shows the Dymore and CAMRAD II frequencies and damping ratios of

Version 3 in cruise, at a density altitude of 8000 feet and a propeller speed of 2250

RPM. The frequencies of the ﬁrst (ﬁrst bending), second (knife-edge), and fourth

(torsion) modes show good agreement with CAMRAD II predicting a slightly lower frequency for the fourth mode at higher free stream velocities. The damping ratios agree well for the second and fourth modes with Dymore predicting a slight lower damping ratio for the second mode at higher free stream velocities. While the ﬁrst predicted mode (ﬁrst bending) agrees between Dymore and CAMRAD II, Dymore predicted a lower damping ratio as velocity increases.

Figure 5.13 shows the eﬀects of three diﬀerent wing aerodynamics models on

Version 3 of the X-57: no wing aerodynamics, quasi-steady wing aerodynamics, and unsteady wing aerodynamics. While the frequencies of the lower two modes are similar among the three cases, the fourth mode (torsional) frequency increases drastically from the case with no wing aerodynamics to the cases that include wing aerodynamics as the velocity increases. The damping ratio of the ﬁrst mode is also very diﬀerent between the case of no wing aerodynamics and those including wing aerodynamics, with the

62 wing aerodynamics increasing the damping ratio of the mode. The damping on the fourth mode is the only mode where the quasi-steady and unsteady aerodynamics diﬀer signiﬁcantly, with the unsteady aerodynamics contributing to a higher damping ratio.

Figure 5.14 shows the similar results of whirl ﬂutter with diﬀerent aerodynamics models but at sea level conditions and at 2700 RPM. The modes in Figure 5.13 at high altitude present an overall higher damping than those at sea level (Figure 5.14). The damping of the fourth mode has a similar trend as those at higher altitude conditions

(Figure 5.13) for quasi-steady and unsteady wing aerodynamics.

5.2.5 Semi-Span Summary and Conclusions

Two multibody dynamics codes, Dymore and CAMRAD II, are used to study the whirl

ﬂutter stability of the experimental NASA aircraft designated the X-57 Maxwell. Three design versions of the wing and pylon have been studied. The loads and power of an isolated propeller are predicted by the multibody dynamics codes and compared to each other. The thrust and power of the propeller agreed well within the stall angles. The natural frequencies and mode shapes of the semi-span model are compared between

Dymore and NASTRAN predictions and show good correlations. For versions 2 and 3, the ﬁrst four modes of the semi-span model, which are ﬁrst bending, in-plane

(knife-edge) bending, second bending, and torsion, are examined for whirl ﬂutter stability. The whirl ﬂutter analysis from both CAMRAD II and Dymore showed similar frequencies and damping ratios for each mode when no wing aerodynamics were used although Dymore predicted slightly lower damping ratios than CAMRAD II. CAMRAD

II results captured a whirl ﬂutter case for the wing design version 2, which prompted a redesign of the pylon mount system to eliminate the issue. When quasi-steady and unsteady aerodynamics were included in the Dymore analysis, the damping of the

63 fundamental bending mode increased dramatically with velocity and became the most stable among the four modes. This also corresponds to a decrease in the damped frequency compared to the no wing aerodynamics case, as expected. The inplane bending mode was not affected significantly by including wing aerodynamics. The torsion frequencies increased with velocities when wing aerodynamics were applied. The torsion damping ratio varied depending on the type of aerodynamics, showing higher damping ratio with unsteady aerodynamics than quasi-steady aerodynamics. As altitude increases, density decreases, and the damping ratios decrease. Overall the damping ratios of all the semi-span model modes are positive for the latest wing version, indicating that the design is clear of whirl flutter for the symmetric modes up to 400 knots, well above the operating range of the X-57.

64 25

Out-of-Plane; CAMRAD II 20 In-Plane; CAMRAD II Torsion; CAMRAD II Out-of-Plane; Dymore In-Plane; Dymore Torsion; Dymore 15

10 Frequency (Hz)Frequency

0 50 100 150 200 250 300 350 400 450 500 Velocity (knots)

(a) Frequency

10 Out-of-Plane; CAMRAD II In-Plane; CAMRAD II 8 Torsion; CAMRAD II Out-of-Plane; Dymore In-Plane; Dymore Torsion; Dymore 6

Damping Ratio (%)Ratio Damping 0

-2

-4 50 100 150 200 250 300 350 400 450 500 Velocity (knots)

(b) Damping

Figure 5.9: Frequency and Damping of Semi-Span X-57 Maxwell Model; Without Wing Aerodynamics; RPM 2250; Version 1

65 25

Out-of-Plane; No Wing Aero In-of-Plane; No Wing Aero 20 Torsion; No Wing Aero Out-of-Plane; Quasi-Steady Wing Aero In-of-Plane; Quasi-Steady Wing Aero Torsion; Quasi-Steady Wing Aero

10 Frequency (Hz)Frequency

0 50 100 150 200 250 300 350 400 450 500 Velocity (knots)

(a) Frequency

Out-of-Plane; No Wing Aero 30 In-Plane; No Wing Aero Torsion; No Wing Aero Out-of-Plane; Quasi-Steady Wing Aero 25 In-Plane; Quasi-Steady Wing Aero Torsion; Quasi-Steady Wing Aero

10 Damping Ratio (%)Ratio Damping 5

-5 50 100 150 200 250 300 350 400 450 500 Velocity (knots)

(b) Damping

Figure 5.10: Dymore Frequency and Damping of Semi-Span X-57 Maxwell Model; With and Without Wing Aerodynamics; RPM 2250; Version 1

66 25

Mode 1 20 Mode 2 Mode 3 Mode 4 Mode 5 15

10 Frequency (Hz)Frequency

0 100 200 300 400 Velocity (knots)

(a) Frequency

6 Mode 1 Mode 2 Mode 3 Mode 4 4 Mode 5

2 Damping (% Crit)(% Damping 0

•2 100 200 300 400 Velocity (knots)

(b) Damping

Figure 5.11: CAMRAD II Frequency and Damping of Semi-span X-57 Maxwell Model; Without Wing Aerodynamics; 2250 RPM; Version 2

67 25 Out-of-Plane; CAMRAD II In-Plane; CAMRAD II Torsion; CAMRAD II Out-of-Plane; Dymore 20 In-Plane; Dymore Torsion; Dymore

10 Frequency (Hz)Frequency

0 50 100 150 200 250 300 350 400 450 500 Velocity (knots)

(a) Frequency

Out-of-Plane; CAMRAD II In-Plane; CAMRAD II 8 Torsion; CAMRAD II Out-of-Plane; Dymore In-Plane; Dymore 6 Torsion; Dymore

Damping Ratio (%)Ratio Damping 0

-2

-4 50 100 150 200 250 300 350 400 450 500 Velocity (knots)

(b) Damping Ratio

Figure 5.12: Frequency and Damping of Semi-Span X-57 Maxwell Model; Without Wing Aerodynamics; RPM 2250; Version 3

68 25

15 Out-of-Plane; No Wing Aero In-of-Plane; No Wing Aero Torsion; No Wing Aero Out-of-Plane; Quasi-Steady In-of-Plane; Quasi-Steady Torsion; Quasi-Steady 10 Out-of-Plane; Unsteady In-of-Plane; Unsteady

Frequency (Hz)Frequency Torsion; Unsteady

0 100 150 200 250 300 350 400 450 500 Velocity (knots)

(a) Frequency

35 Out-of-Plane; No Wing Aero In-of-Plane; No Wing Aero 30 Torsion; No Wing Aero Out-of-Plane; Quasi-Steady In-of-Plane; Quasi-Steady 25 Torsion; Quasi-Steady Out-of-Plane; Unsteady In-of-Plane; Unsteady 20 Torsion; Unsteady

10 Damping Ratio (%)Ratio Damping 5

-5 50 100 150 200 250 300 350 400 450 500 Velocity (knots)

(b) Damping Ratio

Figure 5.13: Dymore Frequency and Damping of Semi-Span X-57 Maxwell Model; Com- parison of Adding Wing Aerodynamics; RPM 2250; Version 3

69 25

15 Out-of-Plane; No Wing Aero In-Plane; No Wing Aero Torsion; No Wing Aero Out-of-Plane; Quasi-Steady In-Plane; Quasi-Steady Torsion; Quasi-Steady 10 Out-of-Plane; Unsteady In-Plane; Unsteady Torsion; Unsteady

Frequency (Hz)Frequency 5

-5 50 100 150 200 250 300 350 400 450 500 Velocity (knots)

(a) Frequency

35 Out-of-Plane; No Wing Aero In-Plane; No Wing Aero 30 Torsion; No Wing Aero Out-of-Plane; Quasi-Steady In-Plane; Quasi-Steady Torsion; Quasi-Steady 25 Out-of-Plane; Unsteady In-Plane; Unsteady Torsion; Unsteady 20

10 Damping Ratio (%)Ratio Damping 5

-5 50 100 150 200 250 300 350 400 450 500 Velocity (knots)

(b) Damping Ratio

Figure 5.14: Dymore Frequency and Damping of Semi-Span X-57 Maxwell Model; Com- parison of Adding Wing Aerodynamics; RPM 2700; Version 3

70 5.3 Propeller Whirl Flutter Study with a Full-Span X-57 Propeller/Wing/Pylon Model

This section looks at the development of the full-span X-57 Dymore model and a brief parametric study in a few design parameters. First, a transition from the assumed rigid blades in part one to blades with elasticity values that match the oﬀ-the-shelf blades used on the X-57. Next, an elastic fuselage model is developed in Dymore and compared with frequencies in NASTRAN. Finally, the whirl ﬂutter stability is looked at for the same range of speeds from Part 1.

The analytical results include the whirl ﬂutter stability of the baseline the X-57 full-span models and a parametric study of the eﬀects of key design variables on whirl

flutter stability. The full-span X-57 analytical model, with its structural and aerodynamics modeling of key sub-components, are developed and verified separately by correlating the predictions among different analyses (Dymore and CAMRAD II). The results include a look at the fundamental modes: symmetric out-of-plane bending, anti-symmetric out-of-plane bending, symmetric in-plane bending, anti-symmetric in-plane bending, and symmetric torsional. The parametric study looks into changes in propeller blade elasticity, propeller blade airfoils, and wing stiffness.

5.3.1 Whirl Flutter Stability of the Full-Span X-57

Figures 5.15 and 5.16 show the comparison of the frequencies and damping ratios between CAMRAD II and Dymore for the full-span X-57. The properties of the blades are the rigid propeller and have the X-57 airfoil distribution.

Figures 5.15a and 5.16a show the symmetric and anti-symmetric mode frequencies as velocity increases. Figure 5.15a shows the symmetric out-of-plane frequencies have a

71 slight oﬀset with the Dymore symmetric out-of-plane mode having a higher frequency than CAMRAD II. This is also shown for the torsion mode at lower velocities, but as velocity increases the two analyses have similar frequencies. The symmetric in-plane mode for both Dymore and CAMRAD II show good agreement for the entire velocity range. Figure 5.16a shows the anti-symmetric out-of-plane and in-plane frequencies are in good agreement for both CAMRAD II and Dymore.

Figures 5.15b and 5.16b show the damping ratios of the symmetric and anti-symmetric modes as velocity increases. Figure 5.15b shows the symmetric modes begin at similar damping ratios but diverge as velocity increases. CAMRAD II predicts a lower damping ratio compared to Dymore for both the symmetric out-of-plane and symmetric in-plane modes. CAMRAD II and Dymore show good agreement on the damping ratio for the symmetric torsion mode with CAMRAD II having a slight increase in damping ratio around 400 knots. Figure 5.16b shows the anti-symmetric out-of-plane mode for both CAMRAD II and Dymore begin at similar damping ratios but Dymore showing a higher damping ratio after 200 knots. The anti-symmetric in-plane damping ratio for both analyses show a near constant damping ratio that slightly dips down at higher velocities, but with CAMRAD II showing a lower damping ratio overall.

5.3.2 Comparison of the Full-Span to the Semi-Span X-57 Model

Figure 5.17 shows the frequencies and damping ratios of the symmetric modes for the

Dymore full-span and semi-span X-57 model. Both models have elastic propellers with the X-57 airfoil distribution.

Figure 5.17a shows the frequencies of the symmetric modes. There is a slight oﬀset in the frequencies between the symmetric out-of-plane and in-plane modes, with the

72 semi-span model having lower frequencies for those two modes. The frequency of the symmetric torsion mode for the semi-span and full-span models show good agreement.

Figure 5.17b shows the damping ratios of the symmetric modes. The full-span and semi-span out-of-plane and torsion mode start off at similar damping ratios. The torsion mode for the two models show good agreement as velocity increases while the out-of-plane modes diverge as velocity increases, with the semi-span having a higher damping ratio. The semi-span and full-span in-plane modes show similar trends with and offset showing the semi-span mode having a higher damping ratio. The difference between the semi-span and full-span symmetric modes could be due to the difference in boundary conditions. The semi-span model is clamped at the root whereas the full-span model is free-free and can have marginal displacements at the wing root from the elastic fuselage.

5.3.3 Full-Span X-57 With Wing Aerodynamics

Figures 5.18 and 5.19 show the frequencies and damping ratios of the Dymore full-span model with and without steady wing aerodynamics. The frequencies of the modes are the same with the exception of the ﬁrst symmetric out-of-plane bending and the symmetric torsional modes. The frequency of the symmetric out-of-plane mode decreases as velocity increases when wing aerodynamics are applied whereas the torsion mode frequency increases as velocity increases. The damping ratios for all the modes, both symmetric and anti-symmetric, are higher when wing aerodynamics are applied compared to no wing aerodynamics, with the exception of the symmetric in-plane mode. The symmetric in-plane mode with wing aerodynamics has similar damping ratios compared to the symmetric in-plane mode without wing aerodynamics. The anti-symmetric in-plane mode also shows similar damping ratios but there is an increase

73 in damping ratio as wing aerodynamics are applied.

5.3.4 Eﬀect of Change in Blade Stiﬀness on the Full-Span X-57

Figures 5.20 and 5.20 show the frequencies and damping ratios for the full-span X-57 with the X-57 airfoil distribution and for two different blade stiffness properties: a blade with large stiffness values, and a blade with stiffness properties matching the blade being used on the X-57. Figures 5.20a and 5.21a show the comparison of the wing frequencies for both propellers. The reduction in blade stiffness from the rigid propeller to the elastic propeller does not affect the frequencies of the wing modes with the exception of the symmetric torsional mode. The wing torsional frequency having a slightly lower frequency compared to the wing with the rigid propeller.

Figures 5.20b and 5.21b show the comparison of the wing damping ratios for both propellers. The damping ratios for each of the modes, excluding the torsional mode, show similar damping values at low velocities and diverge as velocity increases. The damping ratio of the torsional mode shows the opposite trend, there is a large oﬀset between the rigid propeller and elastic propeller with the elastic propeller having a larger damping ratio. As velocity increases, the damping ratio of the torsional mode for the elastic propeller starts to drop to match more closely with the rigid propeller. For each mode considered, the damping ratio at high velocities was lower for the elastic propeller with the exception of the symmetric out-of-plane mode.

5.3.5 Eﬀect of Wing Stiﬀness on Whirl Flutter Stability of the Full-Span X-57 Model

Figures 5.22-5.24 show the variations of frequencies and damping ratios of the symmetric modes while Figures 5.25-5.26 show the frequencies and damping ratios for

74 the anti-symmetric modes for three cases of wing flexibility: the baseline, 50% of baseline, and 200% of baseline. Overall the frequencies increase with wing stiffness for all the presented modes while the damping ratios decrease with wing stiffness. The only exception are the damping ratios of the anti-symmetric in-plane mode and the symmetric torsion mode. The anti-symmetric damping ratios of the 50% and 200% stiff wing have similar starting points and are both below the baseline wing stiffness. As velocity increases the damping ratio of the anti-symmetric in-plane 50% stiff wing rises above the baseline while the baseline converges with the 200% stiff wing until around

300 knots when all three converge to a similar damping ratio. The damping ratios of the symmetric torsion mode show that increasing the stiffness of the wing does not alter the damping ratio significantly while decreasing the stiffness does affect the damping ratio. The 50% wing stiffness for the symmetric torsion mode has the highest damping ratio of the three wing stiffness values.

5.3.6 Full-Span Summary and Conclusions

A whirl flutter stability study is carried out on the full-span X-57 using the multibody dynamics codes Dymore and CAMRAD II. An additional parametric study is carried out on key design parameters using Dymore. The whirl flutter stability analysis showed that for the full-span vehicle, the whirl flutter boundary is not crossed for any of the symmetric or asymmetric fundamental modes. CAMRAD II and Dymore compared well with the frequencies of the full-span modes and observed similar trends for the damping ratios. The parametric study showed that changing the blade stiffness or the airfoil distribution on the blade causes a significant change in the damping ratio of the vehicle, but does not drive it to become unstable for the range of velocities considered.

The parametric study also showed that changing the wing stiﬀness aﬀects the

75 frequencies and damping ratios for each mode, but does not cause the vehicle to become unstable in the expected flight envelope. There is a difference between the full-span and semi-span frequencies and damping ratios, but this is likely because of the difference in boundary conditions for the two models. The semi-span model is clamped at the wing root while the full-span model is a free-free model that additionally has some small displacements at the wing root due to the elastic fuselage. Overall the damping ratios for all the cases studied, both symmetric and asymmetric modes, have positive damping ratios and are stable well beyond the expected flight envelope of the X-57.

76 18

14 Out-of-Plane; CAMRAD II In-Plane; CAMRAD II Torsion; CAMRAD II 12 Out-of-Plane; Dymore In-Plane; Dymore Torsion; Dymore 10

Frequency (Hz)Frequency 6

0 50 100 150 200 250 300 350 400 450 Velocity (Knots)

(a) Frequencies of the Symmetric Modes

Out-of-Plane; CAMRAD II In-Plane; CAMRAD II Torsion; CAMRAD II Out-of-Plane; Dymore 1.5 In-Plane; Dymore Torsion; Dymore

1 Damping Ratio (%)Ratio Damping 0.5

0 50 100 150 200 250 300 350 400 450 Velocity (Knots)

(b) Damping Ratio of Symmetric Modes

Figure 5.15: Full-Span Frequencies and Damping Ratios for CAMRAD II and Dymore Symmetric Modes

77 18

14 Out-of-Plane; CAMRAD II In-Plane; CAMRAD II 12 Out-of-Plane; Dymore In-Plane; Dymore

Frequency (Hz)Frequency 6

0 50 100 150 200 250 300 350 400 450 Velocity (Knots)

(a) Frequencies of the Anti-Symmetric Modes

1.5

Out-of-Plane; CAMRAD II In-Plane; CAMRAD II Out-of-Plane; Dymore 1 In-Plane; Dymore Damping Ratio (%)Ratio Damping 0.5

0 50 100 150 200 250 300 350 400 450 Velocity (Knots)

(b) Damping Ratios of Anti-Symmetric Modes

Figure 5.16: Full-Span Frequencies and Damping Ratios for CAMRAD II and Dymore Anti-Symmetric Modes

78 18

Out-of-Plane; Semi-Span 14 In-Plane; Semi-Span Torsion; Semi-Span Out-of-Plane; Full-Span 12 In-Plane; Full-Span Torsion; Full-Span 10

Frequency (Hz)Frequency 6

0 50 100 150 200 250 300 350 400 450 Velocity (Knots)

(a) Frequencies of Symmetric Modes

2 Out-of-Plane; Semi-Span 1.8 In-Plane; Semi-Span Torsion; Semi-Span Out-of-Plane; Full-Span 1.6 In-Plane; Full-Span Torsion; Full-Span 1.4

1.2

0.8

Damping Ratio (%)Ratio Damping 0.6

0.4

0.2

0 50 100 150 200 250 300 350 400 450 Velocity (Knots)

(b) Damping Ratios of Symmetric Modes

Figure 5.17: Comparison of the Symmetric Modes for the Dymore Full-Span and Semi- Span Models

79 22

14 Out-of-Plane; No Aero In-Plane; No Aero 12 Torsion; No Aero Out-of-Plane; With Aero In-Plane; With Aero 10 Torsion; With Aero

8 Frequency (Hz)Frequency

0 50 100 150 200 250 300 350 400 450 Velocity (Knots)

(a) Frequencies of the Symmetric Modes

18 Out-of-Plane; No Aero In-Plane; No Aero Torsion; No Aero 16 Out-of-Plane; With Aero In-Plane; With Aero Torsion; With Aero 14

Damping Ratio (%)Ratio Damping 6

0 50 100 150 200 250 300 350 400 450 Velocity (Knots)

(b) Damping Ratio of Symmetric Modes

Figure 5.18: Eﬀects of Applying Steady Aerodynamics to the Wing of the Full-Span Dymore Model Symmetric Modes

80 22

18 Out-of-Plane; No Aero In-Plane; No Aero Out-of-Plane; With Aero 16 In-Plane; With Aero

8 Frequency (Hz)Frequency

0 50 100 150 200 250 300 350 400 450 Velocity (Knots)

(a) Frequencies of the Anti-Symmetric Modes

16 Out-of-Plane; No Aero In-Plane; No Aero 14 Out-of-Plane; With Aero In-Plane; With Aero 12

Damping Ratio (%)Ratio Damping 6

0 50 100 150 200 250 300 350 400 450 Velocity (Knots)

(b) Damping Ratios of Anti-Symmetric Modes

Figure 5.19: Eﬀects of Applying Steady Aerodynamics to the Wing of the Full-Span Dymore Model Anti-Symmetric Modes

81 18

14 Out-of-Plane; Elastic In-Plane; Elastic Torsion; Elastic 12 Out-of-Plane; Rigid In-Plane; Rigid Torsion; Rigid 10

Frequency (Hz)Frequency 6

0 50 100 150 200 250 300 350 400 450 Velocity (Knots)

(a) Frequencies of the Symmetric Modes

2 Out-of-Plane; Elastic In-Plane; Elastic Torsion; Elastic Out-of-Plane; Rigid In-Plane; Rigid 1.5 Torsion; Rigid

1 Damping Ratio (%)Ratio Damping 0.5

0 50 100 150 200 250 300 350 400 450 Velocity (Knots)

(b) Damping Ratio of Symmetric Mode

Figure 5.20: Eﬀect of Blade Stiﬀness on Whirl Flutter for the Full-Span Dymore Model Symmetric Modes; X-57 Distribution

82 18

14 Out-of-Plane; Elastic 12 In-Plane; Elastic Out-of-Plane; Rigid In-Plane; Rigid 10

Frequency (Hz)Frequency 6

0 50 100 150 200 250 300 350 400 450 Velocity (Knots)

(a) Frequencies of the Anti-Symmetric Modes

1.5 Out-of-Plane; Elastic In-Plane; Elastic Out-of-Plane; Rigid In-Plane; Rigid 1 Damping Ratio (%)Ratio Damping 0.5

0 50 100 150 200 250 300 350 400 450 Velocity (Knots)

(b) Damping Ratios of Anti-Symmetric Mode

Figure 5.21: Eﬀect of Blade Stiﬀness on Whirl Flutter for the Full-Span Dymore Model Anti-Symmetric Modes; X-57 Distribution

83 4

3.5

2.5

1.5 Frequency (Hz)Frequency

Baseline 1 50% Stiffness 200% Stiffness

0.5

0 50 100 150 200 250 300 350 400 450 Velocity (Knots)

(a) Frequencies

Baseline 50% Stiffness 200% Stiffness 1.5

1 Damping Ratio (%)Ratio Damping 0.5

0 50 100 150 200 250 300 350 400 450 Velocity (Knots)

(b) Damping Ratios

Figure 5.22: Eﬀect of Wing Stiﬀness on Frequencies and Damping Ratios of the Sym- metric Out-of-Plane Bending Mode: Elastic X-57 Propellers, 2250 RPM and at 8000 Feet

84 12

Frequency (Hz)Frequency 4 Baseline 50% Stiffness 200% Stiffness 2

0 50 100 150 200 250 300 350 400 450 Velocity (Knots)

(a) Frequencies

Baseline 1.5 50% Stiffness 200% Stiffness

1 Damping Ratio (%)Ratio Damping 0.5

0 50 100 150 200 250 300 350 400 450 Velocity (Knots)

(b) Damping Ratios

Figure 5.23: Eﬀect of Wing Stiﬀness on Frequencies and Damping Ratios of the Sym- metric In-Plane Bending Mode: Elastic X-57 Propellers, 2250 RPM and at 8000 Feet

85 24

Frequency (Hz)Frequency 8

6 Baseline 50% Stiffness 4 200% Stiffness

0 50 100 150 200 250 300 350 400 450 Velocity (Knots)

(a) Frequencies

1.5

Damping Ratio (%)Ratio Damping Baseline 0.5 50% Stiffness 200% Stiffness

0 50 100 150 200 250 300 350 400 450 Velocity (Knots)

(b) Damping Ratios

Figure 5.24: Eﬀect of Wing Stiﬀness on Frequencies and Damping Ratios of the Sym- metric Torsion Mode: Elastic X-57 Propellers, 2250 RPM and at 8000 Feet

86 10

9 Baseline 50% Stiffness 200% Stiffness 8

4 Frequency (Hz)Frequency 3

0 50 100 150 200 250 300 350 400 450 Velocity (Knots)

(a) Frequencies

1.5

Baseline 50% Stiffness 200% Stiffness

1 Damping Ratio (%)Ratio Damping 0.5

0 50 100 150 200 250 300 350 400 450 Velocity (Knots)

(b) Damping Ratios

Figure 5.25: Eﬀect of Wing Stiﬀness on Frequencies and Damping Ratios of the Anti- Symmetric Out-of-Plane Bending Mode: Elastic X-57 Propellers, 2250 RPM and at 8000 Feet

87 8

Baseline 4 50% Stiffness 200% Stiffness 3 Frequency (Hz)Frequency

0 50 100 150 200 250 300 350 400 450 Velocity (Knots)

(a) Frequencies

1.5

Baseline 1 50% Stiffness 200% Stiffness Damping Ratio (%)Ratio Damping 0.5

0 50 100 150 200 250 300 350 400 450 Velocity (Knots)

(b) Damping Ratios

Figure 5.26: Eﬀect of Wing Stiﬀness on Frequencies and Damping Ratios of the Anti- Symmetric In-Plane Bending Mode: Elastic X-57 Propellers, 2250 RPM and at 8000 Feet

88 5.4 Whirl Flutter Analysis of a Free-Flying Electric Driven Propeller Aircraft

This section looks at the results of the free-ﬂying X-57 Dymore model. Elastic blades are used for this study with the X-57 airfoil distribution. A fuselage drag model is implemented and an empennage is attached to the fuselage. The whirl ﬂutter stability is looked at for a reduced range of speeds as in Part 2, this is due to the propellers not able to produce the thrust required to overcome drag at very high speeds.

5.4.1 Whirl Flutter of the Free-Flying X-57

Figures 5.27 and 5.28 show the symmetric and anti-symmetric frequencies and damping ratios of the full-span Dymore X-57 multibody dynamics model compared to the free-flying Dymore model. There are slight differences in boundary conditions for the two models, with the free-flying model fully released in all six degrees of freedom whereas the full-span has to be constrained in the pitch direction. The full-span has to be constrained due to there being a pitch moment that the tail in the free-flying model can provide a reactionary moment for.

Figures 5.27a and 5.28a show the frequencies for the symmetric and anti-symmetric wing mode respectively. It can be seen that for the symmetric modes, there are is a slight offset in frequencies for the torsion and out-of-plane bending modes, with the free-flying torsion mode have a lower frequency and a higher frequency for the out-of-plane bending. The symmetric in-plane bending mode has little variation between the full-span and free-flying models. The anti-symmetric modes show no difference between the full-span and the free-flying X-57 models at low velocities and a slight rise in frequency for the anti-symmetric in-plane wing mode at high velocities.

Figure 5.27b and 5.28b show the damping ratios for the symmetric and

89 anti-symmetric wing modes respectively. For the symmetric modes, the out-of-plane and torsion modes have higher damping for the full-span than the free-flying X-57 model but the in-plane wing mode has a larger damping ratio for the free-flying compared to the full-span. The anti-symmetric out-of-plane and in-plane modes both have a higher damping ratio for the free-flying X-57 model compared to the full-span. The full-span has a dip the in the damping ratio at higher velocities where the free-flying model continues to increase as velocity increases. The difference in damping ratios is due to the different boundary conditions that the two models have, the full-span in constrained in the pitch direction and the free-flying is completely unconstrained.

5.4.2 Eﬀect of Unsteady Aerodynamics on Whirl Flutter Stability

Figures 5.29 and 5.30 show the eﬀect of applying unsteady aerodynamics and quasi-steady on the whirl ﬂutter stability. The propellers are windmilling, the drag on the fuselage is ignored, and the aircraft is trimmed but allowed to move in all six degrees of freedom.

Figure 5.29a shows the frequencies of the symmetric wing modes with both quasi-steady and unsteady aerodynamics applied to the wing. The symmetric out-of-plane and in-plane modes show no difference between unsteady and quasi-steady aerodynamics and the torsion mode shows a slight offset with the unsteady torsion mode having a lower frequency. As in the previous studies, the torsion mode increases in frequency as velocity also increases and the symmetric out-of-plane mode slightly decreases in frequency. Figure 5.30a shows the frequencies of the anti-symmetric out-of-plane and in-plane modes. There is no difference in the frequencies when unsteady aerodynamics are applied compared to quasi-steady aerodynamics and the modes maintain a constant frequency as velocity increases.

90 20

Out-of-Plane; Full-Span 14 In-Plane; Full-Span Torsion; Full-Span Out-of-Plane; Free-Flying 12 In-Plane; Free-Flying Torsion; Free-Flying 10

8 Frequency (Hz)Frequency 6

0 50 100 150 200 250 300 350 Velocity (Knots)

(a) Frequencies

Out-of-Plane; Full-Span 14 In-Plane; Full-Span Torsion; Full-Span Out-of-Plane; Free-Flying 12 In-Plane; Free-Flying Torsion; Free-Flying

6 Damping Ratio (%)Ratio Damping 4

0 50 100 150 200 250 300 350 Velocity (Knots)

(b) Damping Ratios

Figure 5.27: Comparison of Free-Flying and Full-Span X-57 Whirl Flutter Symmetric Frequencies and Damping Ratios Using Quasi-Steady Aerodynamics

91 20

12 Out-of-Plane; Full-Span In-Plane; Full-Span Out-of-Plane; Free-Flying 10 In-Plane; Free-Flying

8 Frequency (Hz)Frequency 6

0 50 100 150 200 250 300 350 Velocity (Knots)

(a) Frequencies

10 Out-of-Plane; Full-Span In-Plane; Full-Span Out-of-Plane; Free-Flying 8 In-Plane; Free-Flying

6 Damping Ratio (%)Ratio Damping 4

0 50 100 150 200 250 300 350 Velocity (Knots)

(b) Damping Ratios

Figure 5.28: Comparison of Free-Flying and Full-Span X-57 Whirl Flutter Anti- Symmetric Frequencies and Damping Ratios Using Quasi-Steady Aerodynamics

92 Figure 5.29b shows the damping ratio for the symmetric wing modes for both quasi-steady and unsteady wing aerodynamics. The symmetric in-plane wing mode has no variation when unsteady aerodynamics are applied instead of quasi-steady and the symmetric out-of-plane bending mode develops a slight oﬀset that grows as velocity increases. The torsion mode has a large oﬀset between aerodynamic models that increases with velocity with the unsteady aerodynamics mode having a larger damping ratio compared to the quasi-steady. Both the symmetric out-of-plane and torsion mode are beginning to peak or have already begun to decrease in the velocity range examined.

For the symmetric modes the critical mode, the ﬁrst mode to go unstable, will most likely be the torsion mode with quasi-steady aerodynamics applied since it has already begun to have its damping ratio drop.

Figure 5.30b shows the damping ratio for the anti-symmetric out-of-plane and in-plane wing modes. The out-of-plane mode has no difference when quasi-steady or unsteady aerodynamics are applied, but the in-plane mode has a slight offset that increases with velocity with the unsteady mode having a larger damping ratio. The unsteady aerodynamics have a stabilizing effect or have no effect on all the fundamental modes considered.

93 20

14 Out-of-Plane; Quasi-Steady In-Plane; Quasi-Steady Torsion; Quasi-Steady Out-of-Plane; Unsteady 12 In-Plane; Unsteady Torsion; Unsteady 10

8 Frequency (Hz)Frequency 6

0 50 100 150 200 250 300 350 Velocity (Knots)

(a) Frequencies

Out-of-Plane; Quasi-Steady In-Plane; Quasi-Steady 12 Torsion; Quasi-Steady Out-of-Plane; Unsteady In-Plane; Unsteady Torsion; Unsteady 10

Damping Ratio (%)Ratio Damping 4

0 50 100 150 200 250 300 350 Velocity (Knots)

(b) Damping Ratios

Figure 5.29: Free-Flying X-57 Whirl Flutter Symmetric Frequencies and Damping Ratios Using Unsteady and Quasi-Steady Aerodynamics

94 20

14 Out-of-Plane; Quasi-Steady In-Plane; Quasi-Steady Out-of-Plane; Unsteady 12 In-Plane; Unsteady

8 Frequency (Hz)Frequency 6

0 50 100 150 200 250 300 350 Velocity (Knots)

(a) Frequencies

Out-of-Plane; Quasi-Steady In-Plane; Quasi-Steady 12 Out-of-Plane; Unsteady In-Plane; Unsteady

Damping Ratio (%)Ratio Damping 4

0 50 100 150 200 250 300 350 Velocity (Knots)

(b) Damping Ratios

Figure 5.30: Free-Flying X-57 Whirl Flutter Anti-Symmetric Frequencies and Damping Ratios Using Unsteady and Quasi-Steady Aerodynamics

95 5.4.3 Longitudinal Flight Dynamics

The longitudinal dynamics were isolated by constraining the motion of the aircraft in

yaw, roll, and sideslip. The remaining degrees of freedom, pitch, plunge, and surge are

kept unconstrained. The sections below include the mathematical approximations of the

longitudinal ﬂight dynamics as well as the Prony analysis of the modes present in the

transient results. A sample of the transient results is included at a ﬂight speed of 156

knots (180 mph), which is around the cruise speed of the X-57.

Approximated Short Period and Phugoid Modes

The short period mode can be approximated by the following:

      ∆α ˙ Zα 1 ∆α   =  u0    (5.1)    Zα    ∆q ˙ Mα + Mα˙ Mq + Mα˙ ∆q u0

with the frequency and damping ratio given by,

r Zα Mq + Mα˙ + ZαMq U0 ωnsp = − Mα , ζsp = − (5.2) u0 2ωnsp

With the phugoid mode being approximated by,

      ∆u ˙ X −g ∆u    u      =     (5.3) ∆θ˙ −Zu 0 ∆θ u0

and the frequency and damping ratio given by,

r Zug −Xu ωnρ = , ζρ = − (5.4) u0 2ωnρ

By ignoring compressibility, the phugoid approximation further simpliﬁes to,

96 √ g 1 1 ωnρ = 2 , ζρ = √ (5.5) u0 2 L/D

Where the longitudinal stability derivatives for the aircraft deﬁned in Nelson [52] and

are shown with their respective units in Table 5.2.

Table 5.2: Longitudinal Stability Derivatives with Appropriate Units

Stability Derivative Units

Zu 1/sec 2 Zα ft/sec Xu 1/sec Mq 1/sec 2 Mα 1/sec Mα˙ 1/sec

Stability Derivatives

Figure 5.31 shows the stability derivatives calculated using Dymore, obtained by

perturbing an aircraft state while ﬁxing the other states. While in most cases the values

were linear throughout the entire velocity range, the derivative at the trim condition

was used for cases where the derivative was nonlinear. Figure 5.31a shows the stability

derivatives with Zα having the largest magnitude. This is due to Zα being equal to

Zwu0, so the high velocities being factored in with the magnitude of Zw being around 2.

Mα has the next largest magnitude and is negative, which is required for a stable

aircraft. Figure 5.31b shows Figure 5.31a zoomed in to show the diﬀerences between the

remaining stability derivatives which are small comparatively. There are small

variations in the axial force and normal force due to changes in velocity,Xu and Zu respectively, that decrease as the trimmed velocity increases. The change in moment due to pitch rate, Mq, decreases with velocity as does Mα˙ . Since Mα˙ was calculated by slowly rotating the vehicle, the unsteadiness was not able to be captured and is Mq.

97 Also shown in Figure 5.31 are the NASA stability derivatives for a single velocity.

Since the tail location and airfoils are approximated and assumed respectively, there will be some diﬀerences between the two sets of stability derivatives. However, since the derivatives are all in the same order of magnitude, the stability derivatives computed through Dymore are considered acceptable. A comparison of the short period and phugoid modes using the stability derivatives and the actual short period and phugoid modes calculated in the Dymore transient results are shown below in the next section.

Stability Analysis of the Flight Dynamic Modes

Figures 5.32 and 5.33 show the frequencies and damping ratios of the modes identified by the Prony analysis. Two separate analyses were performed on the transient results, one immediately after the aircraft was perturbed by a gust for five seconds and the other two seconds after the gust for 170 seconds. The first analysis is to capture the short period mode and any excited elastic modes and the second is to capture the phugoid mode.

Figures 5.32a and 5.32b show the frequency and damping ratio for the short period analysis. There are five modes identified by the Prony analysis: four elastic wing modes and one rigid mode. Additionally, the approximated short period frequency and damping ratio are also shown for comparison. The elastic wing modes identified are the symmetric out-of-plane, symmetric in-plane, second symmetric out-of-plane, and the symmetric torsion mode. The damping ratios for all the elastic modes are positive; therefore the system is stable. The second symmetric out-of-plane bending mode appears, which has not been specifically targeted for excitation. However, this second bending mode is still stable and follows a damping ratio similar to the symmetric in-plane bending mode. The short period mode identified by Dymore increases with

98 1

-1

-2

-3

Mw; Dymore Mdw/dt; Dymore -4 Mq; Dymore Zw; Dymore Zu; Dymore Xu; Dymore Stability Derivative Stability Values -5 Mw; NASA Mdw/dt; NASA Mq; NASA Zw; NASA Zu; NASA -6 Xu; NASA

50 100 150 200 250 300 350 Velocity

(a) Stability Derivatives

1 Mw; Dymore Mdw/dt; Dymore 0.8 Mq; Dymore Zw; Dymore Zu; Dymore Xu; Dymore 0.6 Mw; NASA Mdw/dt; NASA Mq; NASA 0.4 Zw; NASA Zu; NASA Xu; NASA 0.2

-0.2

-0.4

Stability Derivative Stability Values -0.6

-0.8

-1 50 100 150 200 250 300 350 Velocity

(b) Stability Derivatives Zoomed In

Figure 5.31: Aircraft Longitudinal Stability Derivatives Versus Velocity

99 velocity and compares well with the approximated mode for lower velocity range considered. Around the velocity the two modes cross, there is a negative interaction and both modes have a reduction in damping ratio that rebounds as the distance between the frequencies of the two modes grows after crossing. The coupling of the short period mode and the out-of-plane wing mode affects the frequency and damping ratio of the rigid body mode and cannot be predicted by the approximation. The approximated damping ratio, however, does not compare well with the identified mode. This is due to the approximated mode depending on the unsteady angle of attack stability derivative and the change due to unsteadiness that was not able to be identified by the steady pitching done in calculating the stability derivatives. The identified mode does have the expected damping ratio for the short period mode and is highly damped.

Figures 5.33a and 5.33b show the frequency and damping ratio for the phugoid analysis. Only one mode was identified by the multibody dynamics analysis and is compared to both the approximated phugoid mode and incompressible approximated phugoid mode. The frequency of the identified mode compares well to both the approximated and incompressible approximated phugoid mode for low velocities. After about 200 knots, Mach 0.31, the incompressible approximated diverges from the identified and approximated phugoid mode. The damping ratio for the identified mode compares well with the approximated phugoid mode with a slight offset that grows as velocity increases. The incompressible approximation does not compare well to either the identified mode or the approximated mode. Since the frequency of the identified phugoid mode is significantly lower than any of the elastic modes, no interaction was expected or identified by the Prony analysis.

100 20

Short Period Short Period Approx Symmetric Out-of-Plane Symmetric In-Plane Second Symmetric Out-of-Plane 10 Symmetric Torison Frequency (Hz)Frequency

0 50 100 150 200 250 300 350 Velocity (Knots)

(a) Frequencies

Short Period 20 Short Period Approx Symmetric Out-of-Plane Symmetric In-Plane Second Symmetric Out-of-Plane 15 Symmetric Torison Damping Ratio (%)Ratio Damping 10

0 50 100 150 200 250 300 350 Velocity (Knots)

(b) Damping Ratios

Figure 5.32: Frequencies and Damping Ratios of the Measured and Approximated Short Period Modes and Excited Elastic Wing Modes

101 0.05

0.045

0.04 MBD Incompressible Approx. 0.035 Approx.

0.03

0.025

0.02

0.015 Frequency (Hz)Frequency

0.01

0.005

-0.005 50 100 150 200 250 300 350 Velocity (Knots)

(a) Frequencies

100

MBD Incompressible Approx. Approx. 60

40 Damping Ratio (%)Ratio Damping

0 50 100 150 200 250 300 350 Velocity (Knots)

(b) Damping Ratios

Figure 5.33: Frequencies and Damping Ratios of the Measured and Approximated Phugoid Modes

102 5.4.4 Transient Results of the Longitudinal Flight Dynamics

Figure 5.34 shows the pitch and change in altitude for all the velocities considered. The aircraft was constrained by a pseudo-spring for the initialization of the ﬂight condition and released after ﬁve seconds. A gust was then applied at nine seconds and the response recorded. Figure 5.34a shows that the gust immediately causes the vehicle to pitch up and the short period mode can be seen as small oscillations after the gust

ﬁnishes. Figure 5.34b shows that the gust does not give as dramatic a change in altitude as it does for pitch. Then the phugoid mode can be seen as time progresses with the higher velocity cases damping out quicker than the slower cases. Shown in both the pitch and change in altitude is that the last two velocities considered are unstable, with the 330 knot case quickly diverging and the 311 knot case slowly growing. It can also be seen that as velocity increases, the further from the initial starting position the aircraft settles at.

Short Period Transient Results at 156 Knots

Figures 5.35 and 5.36 show the states for the longitudinal modes for the first five seconds directly after the gust perturbation has ended. The four states shown are the aircraft pitch from which the pitch rate can also be seen, the angle of attack, the change in altitude from which the vertical velocity can be seen, and the velocity. Additionally, the Prony analysis of the states are also shown in each of the figures. The pitch of the aircraft starts at a high angle and quickly comes back down, signifying the beginning of the phugoid mode but the variations before one second are the short period mode being quickly damped out. The angle of attack illustrates the largest variation from the short period mode, with the two elastic modes also being displayed after the short period

103 3 97 2.5 117 136 156 2 172 194 214 1.5 233 253 272 1 292 311 330 0.5

Pitch (Deg) Pitch 0

-0.5

-1

-1.5

-2 0 20 40 60 80 100 120 140 160 180 Time (Sec) (a) Pitch Angle Versus Time 40 97 117 30 136 156 172 194 214 20 233 253 272 292 10 311 330

Change in Height (Feet) Height in Change -10

-20

-30 0 20 40 60 80 100 120 140 160 180 Time (Sec) (b) Change in Altitude Versus Time

Figure 5.34: Transient Pitch and Change in Altitude for All Measured Velocities

104 mode has damped out. The altitude of the aircraft rises for the ﬁve seconds analyzed here and begins to rise more slowly as time progresses. The velocity drops less than one knot during the ﬁve seconds, which is expected as the short period mode occurs at nearly constant speed.

Phugoid Transient Results at 156 Knots

Figures 5.37 and 5.38 show the transient results two seconds after the gust has ended for

170 seconds. The states shown are the same as the ones in the above section. The pitch of the aircraft slowly damps out as is expected for the phugoid mode. The angle of attack ranges from -0.01 and 0.01, which is an order of magnitude lower than the short period mode. The small change in angle of attack is expected as the phugoid mode occurs at near constant angle of attack. The change in altitude and velocity inversely correlate to each other, as the aircraft climbs the velocity decreases and the velocity increases as the aircraft descends. Compared to the short period mode of less than one knot diﬀerence, the phugoid mode has almost a four knot diﬀerence. The overall change in altitude for the aircraft is small and nearly returns to its original position.

105 1.8

1.6

MBD Prony 1.4

1.2

1 Pitch (Degrees) Pitch 0.8

0.6

0.4 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time (Seconds)

(a) Aircraft Pitch Versus Time

0.1

0.08 MBD Prony

0.06

0.04

0.02

0 Angle of Attack of (Degrees) Angle

-0.02

-0.04 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time (Seconds)

(b) Angle of Attack Versus Time

Figure 5.35: Short Period States; Pitch and Angle of Attack

106 40

20 MBD Prony

15 Change in Altitude (Feet) Altitude in Change

5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time (Seconds)

(a) Change in Altitude Versus Time

154

153.8

MBD Prony 153.6

153.4

Velocity (Knots) Velocity 153.2

153

152.8 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time (Seconds)

(b) Aircraft Velocity Versus Time

Figure 5.36: Short Period States; Plunge and Velocity

107 2

1.5 MBD Prony 1

0.5

0 Pitch (Degrees) Pitch -0.5

-1

-1.5 0 20 40 60 80 100 120 140 160 180 Time (Seconds)

(a) Aircraft Pitch Versus Time

0.025

0.02

0.015 MBD Prony 0.01

0.005

-0.005

Angle of Attack of (Degrees) Angle -0.01

-0.015

-0.02 0 20 40 60 80 100 120 140 160 180 Time (Seconds)

(b) Aircraft Angle of Attack Versus Time

Figure 5.37: Phugoid Mode States; Pitch and Angle of Attack

108 40

30 MBD Prony 25

0 Change in Altitude (Feet) Altitude in Change -5

-10

-15 0 20 40 60 80 100 120 140 160 180 Time (Seconds)

(a) Change in Altitude Versus Time

157

156.5

MBD 156 Prony

155.5

155

154.5

Velocity (Knots) Velocity 154

153.5

153

152.5 0 20 40 60 80 100 120 140 160 180 Time (Seconds)

(b) Aircraft Velocity Versus Time

Figure 5.38: Phugoid Mode States; Plunge and Velocity

109 5.4.5 Severely Damaged Pylon

The left pylon mount that holds the propeller to the wing is reduced to 0.65% of the nominal stiﬀness values in both the pitch and yaw directions to simulate a severely damaged pylon. When this occurs, there is coupling with the rigid ﬂight dynamic short period mode and this causes the pylon mode to couple with both the short period and the elastic wing mode to create an unstable system. In Figure 5.39a it can be seen that the short period mode crosses the wing out-of-plane bending mode and the pylon mode drops down as well. At the velocities the frequencies cross, there is a reduction in damping that goes negative as shown in Figure 5.39b. When the frequencies separate, however, the damping ratio goes back to being positive and the system is stable again.

This is conﬁrmed by Figures 5.41-5.46 that show the angle of attack of the aircraft, the pylon pitch motion, and the wing tip displacements and rotations for three diﬀerent velocities. The velocities shown are before the frequencies cross, during the frequency cross, and after the frequencies have separated.

Figure 5.39a also shows that other frequencies appear in the transient results. The anti-symmetric out-of-plane and in-plane wing modes appear with the anti-symmetric out-of-plane mode going unstable along with the symmetric out-of-plane mode (5.39b).

The symmetric in-plane wing mode appears in the transient results as well, but this mode does not go unstable but may for higher velocities.

Figures 5.40a and 5.40b show the unstable locations zoomed in, so there is a clearer image of which modes go unstable. It can be seen that the symmetric out-of-plane bending mode is unstable for three diﬀerent velocities tested, but the anti-symmetric bending mode is only unstable for 272 knots. The phugoid mode is also present in

Figures 5.39a and 5.39b, but the frequency is considerably lower than the other modes

110 that there is no interaction that could cause the system to become unstable.

Phugoid Short Period Symmetric Out-of-Plane 6 Pylon Anti-Symmetric Out-of-Plane Anti-Symmetric In-Plane Symmetric In-Plane

4 Frequency (Hz)Frequency

0 50 100 150 200 250 300 350 Velocity (Knots)

(a) Frequencies

20 Phugoid Short Period Symmetric Out-of-Plane 15 Pylon Anti-Symmetric Out-of-Plane Anti-Symmetric In-Plane Damping Ratio (%)Ratio Damping 10 Symmetric In-Plane

-5 50 100 150 200 250 300 350 Velocity (Knots)

(b) Damping Ratios

Figure 5.39: X-57 With Damaged Pylon

111 4.5

3.5

2.5

Phugoid 1.5 Short Period

Frequency (Hz)Frequency Symmetric Out-of-Plane 1 Pylon Anti-Symmetric Out-of-Plane Anti-Symmetric In-Plane 0.5 Symmetric In-Plane

-0.5 50 100 150 200 250 300 350 Velocity (Knots)

(a) Frequencies

Damping Ratio (%)Ratio Damping 0 Phugoid Short Period Symmetric Out-of-Plane -1 Pylon Anti-Symmetric Out-of-Plane Anti-Symmetric In-Plane Symmetric In-Plane -2 50 100 150 200 250 300 350 Velocity (Knots)

(b) Damping Ratios

Figure 5.40: X-57 With Damaged Pylon Zoomed In

112 Damaged Pylon - Transient Response at 214 Knots

Figures 5.41 and 5.42 show the transient response at 214 knots for the normalized angle of attack, pylon pitch motion, and the wing tip displacement and rotations. These transient responses are what the previous section shows in the frequency and damping

figures. The normalized angle of attack shows two curves, the original Dymore response and a curve fit that is from the summation of the frequencies that the Prony analysis gives. Since the two curves lie on top of each other, the Prony analysis has identified the frequencies and damping ratios in the original signal. This velocity is before the short period, pylon, and symmetric out-of-plane bending modes have crossed and are not interacting yet. It can be seen for the normalized angle of attack, the pylon motion, and the wing motion that the system is stable at this velocity, as is shown in the above

ﬁgures (Fig. 5.39 and 5.40).

113 Angle of Attack Fit 1 Original 0.8 Curve ﬁt 0.6 0.4 0.2 0 Normalized Signal Normalized -0.2 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time [sec] (a) Normalized Angle of Attack, Prony Analysis Curve Fit

Pylon Pitch Motion -1

-1.5

-2

-2.5 Pylon Pitch [deg] Pylon Pitch -3 10 15 20 25 30 35 Time [sec] (b) Pylon Pitch Motion

Figure 5.41: X-57 Transient Results with Damaged Pylon: 214 Knots

114 Wing Tip Displacement 0.4

0.3

0.2 X-Disp 0.1 Y-Disp Z-Disp 0 Displacements [ft] -0.1 10 15 20 25 30 35 Time [sec] (a) Wing Displacements

Wing Tip Rotation 0

-0.5 X-Rot Y-Rot -1 Z-Rot -1.5

-2 Rotations [deg] Rotations

-2.5 10 15 20 25 30 35 Time [sec] (b) Wing Rotations

Figure 5.42: X-57 Transient Results with Damaged Pylon: 214 Knots

115 Damaged Pylon - Transient Response at 233 Knots

Figures 5.43 and 5.44 show the transient response at 233 knots for the normalized angle of attack, pylon pitch motion, and the wing tip displacement and rotations. These transient responses are what the previous section shows in the frequency and damping

figures. The normalized angle of attack shows two curves, the original Dymore response and a curve fit that is from the summation of the frequencies that the Prony analysis gives. Since the two curves lie on top of each other, the Prony analysis has identified the frequencies and damping ratios in the original signal. This velocity is during the short period, pylon, and symmetric out-of-plane bending mode crossing. It can be seen for the normalized angle of attack, the pylon motion, and the wing motion that the system is unstable at this velocity, as is shown in the above figures (Fig. 5.39 and 5.40).

Compared to the transient responses at 214 knots, the pylon and wing motion grow exponentially and will eventually cause the system to destroy itself.

116 Angle of Attack Fit 1 0.8 Original Curve ﬁt 0.6 0.4 0.2 0 -0.2 Normalized Signal Normalized -0.4 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time [sec] (a) Normalized Angle of Attack, Prony Analysis Curve Fit

Pylon Pitch Motion -1

-1.5

-2

-2.5 Pylon Pitch [deg] Pylon Pitch -3 10 15 20 25 30 35 Time [sec] (b) Pylon Pitch Motion

Figure 5.43: X-57 Transient Results with Damaged Pylon: 233 Knots

117 Wing Tip Displacement 0.4

0.3

0.2 X-Disp 0.1 Y-Disp Z-Disp 0 Displacements [ft] -0.1 10 15 20 25 30 35 Time [sec] (a) Wing Displacements

Wing Tip Rotation 0

-0.5 X-Rot -1 Y-Rot Z-Rot -1.5 -2

Rotations [deg] Rotations -2.5 -3 10 15 20 25 30 35 Time [sec] (b) Wing Rotations

Figure 5.44: X-57 Transient Results with Damaged Pylon: 233 Knots

118 Damaged Pylon - Transient Response at 292 Knots

Figures 5.45 and 5.46 show the transient response at 292 knots for the normalized angle of attack, pylon pitch motion, and the wing tip displacement and rotations. These transient responses are what the previous section shows in the frequency and damping

figures. The normalized angle of attack shows two curves, the original Dymore response and a curve fit that is from the summation of the frequencies that the Prony analysis gives. Since the two curves lie on top of each other, the Prony analysis has identified the frequencies and damping ratios in the original signal. This velocity is after the short period, pylon, and symmetric out-of-plane bending modes have separated enough not to interact. It can be seen for the normalized angle of attack, the pylon motion, and the wing motion that the system is stable at this velocity, as is shown in the above figures

(Fig. 5.39 and 5.40). Compared to the transient responses at 233 knots, the pylon and wing motion are decaying and are stable at this velocity.

119 Angle of Attack Fit 0.8 0.6 Original 0.4 Curve ﬁt 0.2 0 -0.2 -0.4 -0.6

Normalized Signal Normalized -0.8 -1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time [sec] (a) Normalized Angle of Attack, Prony Analysis Curve Fit

Pylon Pitch Motion -1

-1.5

-2

-2.5 Pylon Pitch [deg] Pylon Pitch -3 10 15 20 25 30 35 Time [sec] (b) Pylon Pitch Motion

Figure 5.45: X-57 Transient Results with Damaged Pylon: 292 Knots

120 Wing Tip Displacement 0.4

0.3

0.2 X-Disp 0.1 Y-Disp Z-Disp 0 Displacements [ft] -0.1 10 15 20 25 30 35 Time [sec] (a) Wing Displacements

Wing Tip Rotation 0 -0.5 -1 X-Rot Y-Rot -1.5 Z-Rot -2 -2.5 -3 Rotations [deg] Rotations -3.5 -4 10 15 20 25 30 35 Time [sec] (b) Wing Rotations

Figure 5.46: X-57 Transient Results with Damaged Pylon: 292 Knots

121 5.4.6 Severely Damaged Pylon With Constrained Rigid Body Motion

A heavy spring is placed to restrict the motion of the aircraft in the longitudinal motion. This heavy spring is in addition to a spring already in place to constrain in the lateral-directional motion. The pylon is still simulated as damaged as the previous section with a reduction in pylon stiffness to 0.65%, the only difference is the heavy spring that restricts motion. Figure 5.47 shows the frequencies and damping ratios of the modes identified by the prony analysis of the transient Dymore signal. It can be seen that there still exists a short period mode, but the damping ratio of the short period is lower compared to Figure 5.39, therefore there is still some rigid body motion but not enough to drive the system unstable as seen in the damping ratios figure (Fig. 5.47b).

The frequencies identiﬁed by the prony analysis are the short period, pylon mode, symmetric out-of-plane bending, and anti-symmetric in-plane bending. These frequencies are all stable even through the short period, pylon mode, and symmetric out-of-plane bending mode all cross. Compared with the previous case of the unconstrained aircraft in longitudinal motion, none of these modes go unstable.

Therefore, the motion of the short period mode causes enough perturbation of the wing and pylon to become unstable that when constrained does not occur.

122 8

Short Period Symmetric Out-of-Plane Pylon Anti-Symmetric In-Plane 6

4 Frequency (Hz)Frequency

0 50 100 150 200 250 300 350 Velocity (Knots)

(a) Frequencies

18 Short Period 16 Symmetric Out-of-Plane Pylon 14 Anti-Symmetric In-Plane 12

8 Damping Ratio (%)Ratio Damping 6

0 50 100 150 200 250 300 350 Velocity (Knots)

(b) Damping Ratios

Figure 5.47: Constrained X-57 With Damaged Pylon

123 5.4.7 Free-Flying Summary and Conclusions

This section studies the whirl flutter stability of a free-flying X-57 multibody dynamics model. It was shown that the fundamental wing bending modes are stable well beyond the expected flight envelope of the X-57 and the effect of quasi-steady and unsteady aerodynamics have on the stability. In addition, the longitudinal flight dynamics modes are analyzed using the Prony method and the phugoid and short period modes identified and their respective frequencies and damping ratios plotted. It was shown that the short period mode frequency interacts with the symmetric out-of-plane bending mode, decreasing the stability of both modes. While the modes never became unstable, their interaction was viewed and captured by the multibody dynamics analysis and should be studied further. When the pylon stiffness was reduced significantly as if it was severely damaged, the coupling between the short period mode and the pylon and symmetric out-of-plane bending modes caused the system to go unstable. When the longitudinal motion is constrained with a heavy spring, but still having a damaged pylon, the system does not go unstable showing that the motion of the short period mode causes the system to go unstable. The longitudinal modes were also compared with the approximations, with the stability derivatives also calculated using results from Dymore. It was especially promising that the phugoid frequency agreed well with the incompressible approximation, at low mach numbers, as it uses no stability derivatives and is only a function of the flight speed.

124 6 CONCLUSIONS

The development and analysis of the X-57 for whirl ﬂutter stability is performed in stages, beginning with the standalone propeller and concluding with the coupling of the elastic modes with the longitudinal ﬂight dynamic modes.

Part 1 considers the standalone X-57 propeller and a validation study on the ability to capture the stability margin of propeller whirl ﬂutter using multibody dynamics. The validation study was performed using a propeller wind tunnel model from Houbolt and

Reed [16]. The multibody dynamics results showed a very good comparison to the wind tunnel and mathematical results given by Houbolt and Reed, thus indicating that multibody dynamics can accurately predict the whirl ﬂutter stability of whirl ﬂutter.

Next, the standalone X-57 propeller system was analyzed for whirl flutter stability. The pylon stiffness in both the pitch and yaw were swept for equal stiffness and a stiffness ratio equivalent to that seen in the X-57 wing and propeller model. The stiffness of the pitch and yaw springs have to be severely reduced, on the order of two magnitudes, to get to marginal stability. A look into different airfoils and pylon stiffness combinations show that the location of marginal stability does not move by a significant degree, but the gradient of damping ratio changes significantly based on the system parameters.

In Part 2, an equivalent beam was created from the full FEM model in NASTRAN and used as an input in Dymore. From that equivalent beam and the development of a propeller model, the whirl ﬂutter stability boundary was determined for the

125 fundamental wing modes. The whirl ﬂutter boundary is crossed in Version 1 of the X-57 wing, and therefore structural design changes were made to alter the frequencies of individual modes. The frequencies and damping ratios of the fundamental modes were compared with CAMRAD II and showed good comparison for only structural damping.

When aerodynamic loads were applied to the wing in Dymore, the damping ratios for the out-of-plane mode and the torsional mode increased when compared to only structural damping. The in-plane bending mode showed little variation when aerodynamics loads on the wing were applied. Overall, in Part 2 the whirl ﬂutter boundary is not crossed in Version 3 for velocities well beyond the ﬂight envelope for the X-57.

In Part 3, an elastic fuselage was added to the NASTRAN model and had to be implemented in Dymore as well. An equivalent beam approach was used towards the development of the Dymore fuselage and tuned to the NASTRAN model using a least squared optimization. The full-span Dymore model compared well in the symmetric and anti-symmetric in-plane bending modes but Dymore underpredicted the symmetric and anti-symmetric out-of-plane frequency. When the frequencies and damping ratios were compared to CAMRAD II for whirl flutter, while Dymore underpredicted the frequencies, the damping ratios followed the same trend and compared relatively well in magnitudes. A parametric study was also performed on the Dymore model for wing stiffness, blade stiffness, and airfoil distribution. While the change in stiffness altered the damping ratios and frequencies of the fundamental modes, the whirl flutter boundary was not crossed for any of the conditions considered. The change in airfoil distribution also affected the damping ratios of the X-57 modes but did not drive the system unstable.

126 Finally, Part 4 has the development of the free-flying X-57 model from the addition to the tail to adding drag to the fuselage. Whirl flutter stability is analyzed by exciting each of the fundamental modes at their specific frequency and by applying a gust to see how the elastic modes couple with the rigid flight dynamic modes. In addition, the stability derivatives are calculated using multibody dynamics to perturb the vehicle.

The stability derivatives are used to calculate the short period and phugoid mode approximations to show that the frequencies identified in the Prony analysis are the short period and phugoid modes of the aircraft. For the whirl flutter analysis, each of the symmetric and anti-symmetric modes are shown to be stable. When the longitudinal flight dynamics are coupled with the elastic wing modes, there is an interaction between the short period and the symmetric out-of-plane wing bending mode that negatively affects the stability but does not cause the system to go unstable.

However, when a pylon mount is simulated to be severely damaged, the coupling of the rigid ﬂight dynamic modes and the symmetric out-of-plane and pylon modes do cause the system to go unstable where the frequencies cross. When the aircraft motion is constrained by a heavy spring, with the damaged pylon mount, the system does not go unstable indicating that the rigid body motion is what perturbs the aircraft enough to drive the system unstable.

Future work to be considered is how the lateral-directional ﬂight dynamics modes can also couple with the elastic wing modes, mainly the dutch roll mode since that is the oscillating mode that could have its frequency near the symmetric out-of-plane wing bending mode.

127 References

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[3] W. H. Reed III, “Propeller-rotor whirl ﬂutter: A state-of-the-art review,” in Sound and Vibration, vol. 4, pp. 526–544, November 1966.

[4] F. T. Abbott, H. Kelly, and K. D. Hampton, “Investigation of propeller-power-plant autoprecession boundaries for a dynamic-aeroelastic model of a four-engine transport airplane,” NASA tech. Note D-1806, 1963.

[5] R. O. Stearman, G. H. Schulze, and S. M. Rohre, “Aircraft damage detection from acoustic and noise impressed signals found by a cockpit voice recorder,” The Journal of the Acoustical Society of America, vol. 101, no. 5, pp. 3085–3085, 1997.

[6] W. Johnson, Rotorcraft Aeromechanics. Cambridge University Press, 2013.

[7] C. Bolden, “Speech at aiaa aviation and aeronautics forum and exposition,” 2017. Remarks by NASA Administrator Charles Bolden during his keynote speech in Washington.

[8] J. V. Papathakis, A. M. Sessions, P. A. Burkhardt, and D. W. Ehmann, “A nasa approach to safety considerations for electric propulsion aircraft testbeds,” AIAA Propulsion and Energy Forum, (Atlanta, Georgia), July 10-12 2017.

[9] E. S. Taylor and K. A. Browne, “Vibration isolation of aircraft power plants,” Aero Sci 6, no. 2, 43, 1938.

[10] Anon., “Airplane airworthiness; transport categories - ﬂutter, deformation, and vibration requirements. civil air regulations amendment 4b-16,” August 1964.

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