helicopter rotor lag damping augmentation based …

The Pennsylvania State University

The Graduate School

Department of Aerospace Engineering

HELICOPTER ROTOR LAG DAMPING AUGMENTATION

BASED ON A RADIAL ABSORBER AND CORIOLIS COUPLING

A Thesis in

Aerospace Engineering

by

Lynn Karen Byers

© 2006 Lynn Karen Byers

Submitted in Partial Fulfillment of the Requirements

for the Degree of

Doctor of Philosophy

August 2006

The thesis of Lynn Karen Byers was reviewed and approved* by the following:

Farhan Gandhi Professor of Aerospace Engineering Thesis Advisor Chair of Committee

George A. Lesieutre Professor of Aerospace Engineering Head of the Department of Aerospace Engineering

Edward C. Smith Professor of Aerospace Engineering

Sean Brennan Assistant Professor of Mechanical Engineering

*Signatures are on file in the Graduate School

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ABSTRACT

A radial vibration absorber is proposed to augment rotor lag damping. Modeled

as a discrete mass restrained by a damped spring and moving along the spanwise

direction within the rotor blade, it introduces damping into the lag mode of the blade

through strong Coriolis coupling. A two-degree-of-freedom model is developed and used

to examine the effectiveness of the radial absorber in transferring damping to the rotor lag

mode. Results demonstrate that it is possible to introduce a significant amount of

damping in the lag mode with a relatively small absorber mass, and the corresponding

amplitudes of 1/rev periodic motions are not excessively large. The lag mode damping

and 1/rev motions are also compared with the results achieved for an embedded

chordwise inertial damper. A classical six-degree-of-freedom aeromechanical stability

analysis is augmented with two absorber cyclic degrees of freedom in the nonrotating

frame to examine the effect of the radial absorber on aeromechanical stability

characteristics. These results indicate that ground resonance instability is eliminated for

the range of absorber parameters considered, and in most cases, the stability margins are

significant. A rotor blade with a discrete radial vibration absorber is also analyzed to

examine the effect of the absorber on rotor blade and hub loads. The rotor blade is

modeled as an elastic beam undergoing flap and lag bending, with the absorber modeled

as a discrete mass restrained by a damped spring, moving in the spanwise direction

within the rotor blade. Results indicate that the addition of the absorber does not

detrimentally affect the blade spanwise and root loads, as well as steady and vibratory

hub loads. Finally, device concepts and implementation possibilities are considered for

the embedded radial vibration absorber.

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TABLE OF CONTENTS

LIST OF FIGURES .....................................................................................................viii

LIST OF TABLES.......................................................................................................xxi

LIST OF SYMBOLS ...................................................................................................xxiii

Chapter 1 Introduction ................................................................................................1

1.1 Background and Motivation ...........................................................................1 1.2 History of the Blade Lag Damper...................................................................2

1.2.1 Hydraulic Damper ................................................................................2 1.2.2 Elastomeric Damper .............................................................................3 1.2.3 Fluidlastic® Damper .............................................................................4

1.3 Literature Review ...........................................................................................4 1.3.1 Elastomeric Damper .............................................................................5 1.3.2 Fluid Elastic Damper............................................................................6 1.3.3 MR and ER Damper .............................................................................8 1.3.4 Constrained Layer Damping Treatment ...............................................8 1.3.5 Vibration Absorbers .............................................................................9 1.3.6 Aeromechanical Stability Research......................................................12

1.4 Focus of Present Research ..............................................................................14

Chapter 2 Fundamental Study of Blade Lag Damping with a Radial Vibration Absorber ...............................................................................................................22

2.1 Coordinate System..........................................................................................23 2.2 Position, Velocity, and Acceleration of Blade and Absorber.........................23

2.2.1 Blade.....................................................................................................24 2.2.2 Absorber ...............................................................................................24

2.3 Derivation of Blade and Absorber Equations of Motion................................25 2.3.1 Ordering Scheme ..................................................................................25 2.3.2 Newton’s Second Law.........................................................................26 2.3.3 Derivation of Blade and Absorber Equations of Motion using

Lagrange’s Equation ..............................................................................29 2.4 Complex Eigenvalue Analysis........................................................................31 2.5 Results.............................................................................................................32

2.5.1 Modal Frequencies and Damping Ratios .............................................32 2.5.2 Frequency Response.............................................................................35

2.6 Summary.........................................................................................................37

Chapter 3 Modeling and Aeromechanical Stability Analysis of a Rotor System with a Radial Vibration Absorber.........................................................................55

v

3.1 Coordinate System and Ordering Scheme......................................................56 3.1.1 Coordinate System................................................................................56 3.1.2 Ordering Scheme ..................................................................................57

3.2 Position, Velocity, and Acceleration of Blade and Absorber.........................57 3.2.1 Blade.....................................................................................................58 3.2.2 Absorber ...............................................................................................59

3.3 Derivation of Equations of Motion.................................................................59 3.3.1 Flap Equation of Motion ......................................................................60 3.3.2 Lag Equation of Motion .......................................................................63 3.3.3 Absorber Equation of Motion...............................................................67 3.3.4 Body Roll and Pitch Equations of Motion ...........................................69 3.3.5 Inertial Contributions to Perturbation Forces and Moments ................69 3.3.6 Aerodynamic Contributions to Perturbation Forces and Moments ......76

3.3.6.1 Perturbation Aerodynamic Flap Moment...................................77 3.3.6.2 Perturbation Aerodynamic Lag Moment....................................78 3.3.6.3 Perturbation Fuselage Aerodynamic Roll and Pitch Moments ..79

3.4 Complex Eigenvalue Analysis........................................................................91 3.5 Results.............................................................................................................92 3.6 Summary.........................................................................................................95

Chapter 4 Comparison with Chordwise Damped Vibration Absorber .......................116

4.1 Analysis ..........................................................................................................117 4.2 Blade Lag Damping and Absorber Response.................................................118 4.3 Advantages and Disadvantages of Both Systems...........................................121 4.4 Summary.........................................................................................................122

Chapter 5 Elastic Blade Analysis................................................................................143

5.1 Coordinate Systems ........................................................................................144 5.2 Ordering Scheme ............................................................................................144 5.3 Elastic Blade Model........................................................................................144 5.4 Absorber Model ..............................................................................................149 5.5 Blade and Absorber Responses ......................................................................153

5.5.1 Absorber Static Displacement ..............................................................154 5.5.2 Coupled Rotor-Absorber Response/Trim Calculation .........................155

5.6 Blade Loads ....................................................................................................157 5.7 Hub Loads.......................................................................................................161 5.8 Results.............................................................................................................162

5.8.1 Full-Scale BO-105 Rotor-Fuselage Model Results..............................162 5.8.1.1 Eigenvalue Analysis ...................................................................163 5.8.1.2 Blade and Absorber Responses ..................................................165 5.8.1.3 Blade Loads................................................................................167 5.8.1.4 Hub Loads ..................................................................................169

5.8.2 HART Rotor Results ............................................................................170

vi

5.8.2.1 Eigenvalue Analysis ...................................................................171 5.8.2.2 Response and Loads Results ......................................................172

5.9 Summary.........................................................................................................173

Chapter 6 Device Concepts.........................................................................................238

6.1 Fluid Elastic Absorber ....................................................................................238 6.2 Nonlinear Spring.............................................................................................240

6.2.1 Nonlinear Spring Simulation................................................................241 6.2.2 Buckling Beam .....................................................................................243 6.2.3 Aeromechanical Stability with Buckling Beam ...................................244

6.3 Summary.........................................................................................................246

Chapter 7 Conclusions and Recommendations...........................................................251

7.1 Conclusions.....................................................................................................251 7.1.1 Two-Degree-of-Freedom Model ..........................................................251 7.1.2 Aeromechanical Stability Analysis ......................................................252 7.1.3 Aeroelastic and Loads Analysis ...........................................................253 7.1.4 Implementation Concepts .....................................................................254

7.2 Recommendations for Future Work ...............................................................254 7.2.1 Articulated Rotor ..................................................................................254 7.2.2 Vibration Reduction .............................................................................255 7.2.3 Energy Harvesting ................................................................................256 7.2.4 Device Design and Experimental Investigation ..................................257 7.2.5 Other .....................................................................................................258

Bibliography ................................................................................................................260

Appendix A Mass, Stiffness, and Damping Terms in Ground Resonance Analysis..267

A.1 Mass Matrix ...................................................................................................267 A.2 Damping Matrix.............................................................................................268

A.2.1 Inertial Terms ......................................................................................268 A.2.2 Aerodynamic Terms ............................................................................270

A.3 Stiffness Matrix..............................................................................................273 A.3.1 Inertial and Elastic Terms....................................................................273 A.3.2 Aerodynamic Terms ............................................................................274

Appendix B Aerodynamic Formulation......................................................................276

B.1 Resultant Velocity..........................................................................................276 B.2 Inflow.............................................................................................................277 B.3 Blade Loads....................................................................................................278

Appendix C Rotor Data ..............................................................................................281

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C.1 AFDD Rotor Data ..........................................................................................281 C.2 Full Scale BO-105-Type Rotor Data .............................................................282 C.3 HART Rotor Data ..........................................................................................284

Appendix D Waterbed Effect......................................................................................287

Appendix E Elastic Blade Analysis – Lag Only.........................................................289

E.1 Coordinate Systems and Nondimensionalization...........................................290 E.1.1 Coordinate Systems..............................................................................290 E.1.2 Nondimensionalization ........................................................................290

E.2 Velocity and Acceleration of Blade and Absorber.........................................291 E.2.1 Blade ....................................................................................................291 E.2.2 Absorber...............................................................................................292

E.3 Derivation using Lagrange’s Equation...........................................................293 E.3.1 Strain Energy........................................................................................293 E.3.2 Kinetic Energy .....................................................................................294 E.3.3 Rayleigh Dissipation Function.............................................................294

E.4 Finite Element Discretization.........................................................................294 E.4.1 Blade Matrices .....................................................................................296 E.4.2 Absorber Terms....................................................................................297

E.5 Blade and Absorber Response Solution .........................................................298 E.6 Absorber Static Displacement........................................................................299 E.7 Blade Root and Hub Loads ............................................................................299

E.7.1 Blade Root Loads.................................................................................300 E.7.2 Rotor Hub Loads..................................................................................301

E.8 Shear Force and Moment Distributions Along the Blade Radius ..................302 E.9 Results ............................................................................................................305

E.9.1 Comparison with Rigid Blade Response .............................................306 E.9.2 Blade and Absorber Response .............................................................306 E.9.3 Blade Root Loads.................................................................................307 E.9.4 Radial Distribution of Blade Loads .....................................................308 E.9.5 Blade Loads at Absorber Location.......................................................309

E.10 Conclusions ..................................................................................................309

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LIST OF FIGURES

Figure 1-1: US Army TH-55 destroyed by ground resonance [51] .............................16

Figure 1-2: CH-47 hydraulic lag damper [52] .............................................................16

Figure 1-3: Schematic of elastomeric lag damper used on Boeing Model 360 [53] ...17

Figure 1-4: Elastomeric lag damper used on AH-64 Apache [52] ..............................17

Figure 1-5: Fluidlastic® lag damper used on NH-90 [52]............................................18

Figure 1-6: Elastomeric lag damper schematic............................................................18

Figure 1-7: Elastomer stiffness and damping dependence on amplitude [53].............19

Figure 1-8: Elastomeric and Fluidlastic® lag damper schematics used on RAH-66 Comanche [25]......................................................................................................20

Figure 1-9: Embedded chordwise inertial damper [43] ...............................................20

Figure 1-10: Radial vibration absorber schematic .......................................................21

Figure 1-11: Coriolis force on blade and absorber mass .............................................21

Figure 2-1: Coordinate systems used in two-degree-of-freedom model .....................38

Figure 2-2: Forces and moments acting on the blade contributing to moments about the lag hinge................................................................................................39

Figure 2-3: Forces acting on absorber .........................................................................39

Figure 2-4: Modal damping ratios (a) and frequencies (b) vs frequency ratio, fα ( 30a .= and 30a .=ζ ) ...........................................................................................40

Figure 2-5: Modal damping ratios (a) and frequencies (b) vs frequency ratio, fα ( 50a .= and 30a .=ζ )...........................................................................................41

Figure 2-6: Modal damping ratios (a) and frequencies (b) vs frequency ratio, fα ( 70a .= and 30a .=ζ ) ..........................................................................................42


ix


Figure 2-9: Modal damping ratios (a) and frequencies (b) vs frequency ratio, fα ( 70a .= and 50a .=ζ )..........................................................................................45



Figure 2-12: Modal damping ratios (a) and frequencies (b) vs frequency ratio, fα ( 70a .= and 70a .=ζ ) .........................................................................................48

Figure 2-13: FRF of the blade lag amplitude for undamped (no absorber) and damped (with absorber) cases ( 70a .= , 30a .=ζ , 030m .=α , 1f =α ) ............49

Figure 2-14: Frequency response function – absorber and blade lag amplitude ( 70a .= , 30a .=ζ , 030m .=α , 1f =α ) .............................................................49

Figure 2-15: Frequency response function - absorber amplitude in %R per degree of lag motion ( 70a .= , 30a .=ζ , 030m .=α , 1f =α ) ......................................50

Figure 2-16: 1/rev absorber amplitude per degree of lag motion vs frequency ratio, fα ( 30a .=ζ and 30a .= ) .........................................................................50

Figure 2-17: 1/rev absorber amplitude per degree of lag motion vs frequency ratio, fα ( 30a .=ζ and 50a .= ).........................................................................51

Figure 2-18: 1/rev absorber amplitude per degree of lag motion vs frequency ratio, fα ( 30a .=ζ and 70a .= ) ........................................................................51

Figure 2-19: 1/rev absorber amplitude per degree of lag motion vs frequency ratio, fα , ( 50a .=ζ and 30a .= ) ......................................................................52

Figure 2-20: 1/rev absorber amplitude per degree of lag motion vs frequency ratio, fα , ( 50a .=ζ and 50a .= ) .......................................................................52

x

Figure 2-21: 1/rev absorber amplitude per degree of lag motion vs frequency ratio, fα , ( 50a .=ζ and 70a .= ).......................................................................53

Figure 2-22: 1/rev absorber amplitude per degree of lag motion vs frequency ratio, fα , ( 70a .=ζ and 30a .= ) .......................................................................53

Figure 2-23: 1/rev absorber amplitude per degree of lag motion vs frequency ratio, fα , ( 70a .=ζ and 50a .= ).......................................................................54

Figure 2-24: 1/rev absorber amplitude per degree of lag motion vs frequency ratio, fα , ( 70a .=ζ and 70a .= ) ......................................................................54

Figure 3-1: Coordinate systems used in ground resonance analysis............................96

Figure 3-2: Blade root shear forces and moments .......................................................96

Figure 3-3: Comparison of baseline results with results from [55] .............................97

Figure 3-4: Modal frequencies and decay rates vs RPM ( 0103030a ma .,.,. === αζ ) ...............................................................................98





Figure 3-9: Modal frequencies and decay rates vs RPM ( 0503050a ma .,.,. === αζ )...............................................................................103



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Figure 3-12: Modal frequencies and decay rates vs RPM ( 0503070a ma .,.,. === αζ ) ..............................................................................106










Figure 4-1: Embedded chordwise damped vibration absorber [42].............................124

Figure 4-2: Embedded chordwise absorber schematic (redrawn from [44]) ...............124

Figure 4-3: Modal damping ratios vs frequency ratio, fα ( 30a .= and 30a .=ζ ) ..125

Figure 4-4: Modal damping ratios vs frequency ratio, fα ( 50a .= and 30a .=ζ ) ...125

Figure 4-5: Modal damping ratios vs frequency ratio, fα ( 70a .= and 30a .=ζ )...126


xii




Figure 4-10: Modal damping ratios vs frequency ratio, fα ( 50a .= and 70a .=ζ ) ..............................................................................................................128





Figure 4-15: 1/rev absorber amplitude per degree of lag motion ( 30a .= and 30a .=ζ )...............................................................................................................131

Figure 4-16: 1/rev absorber amplitude per degree of lag motion ( 50a .= and 30a .=ζ ) ..............................................................................................................132





xiii







Figure 5-1: Elastic blade coordinate system with absorber .........................................175

Figure 5-2 Spatial discretization of the rotor blade using the Finite Element Method (with a radial absorber located at the kth finite element node) ................175

Figure 5-3: Forces and moments exerted on a helicopter in level forward flight (figure redrawn from [57])....................................................................................176

Figure 5-4: Schematic of absorber mass and springs attached to rotor blade and equivalent radial forces on blade due to absorber – spring and inertial forces.....177

Figure 5-5: Fundamental lag mode shape – elastic and rigid blades ...........................177

Figure 5-6: Blade tip flap and lag responses ( 30a .= and 30a .=ζ ) .........................178


Figure 5-8: Blade tip flap and lag responses ( 70a .= and 30a .=ζ ).........................179


Figure 5-10: Blade tip flap and lag responses ( 50a .= and 50a .=ζ ) .......................180

Figure 5-11: Blade tip flap and lag responses ( 70a .= and 50a .=ζ ).......................180

xiv



Figure 5-14: Blade tip flap and lag responses ( 70a .= and 70a .=ζ ) ......................182

Figure 5-15: Absorber response ( 30a .= and 30a .=ζ ) ............................................182


Figure 5-17: Absorber response ( 70a .= and 30a .=ζ )............................................183






Figure 5-23: Absorber response ( 70a .= and 70a .=ζ ) ...........................................186

Figure 5-24: Blade root drag shear ( 30a .= and 30a .=ζ ) ........................................187

Figure 5-25: Blade root drag shear ( 50a .= and 30a .=ζ ) ........................................187

Figure 5-26: Blade root drag shear ( 70a .= and 30a .=ζ ) .......................................188

Figure 5-27: Blade root drag shear ( 30a .= and 50a .=ζ )........................................188

Figure 5-28: Blade root drag shear ( 50a .= and 50a .=ζ )........................................189




Figure 5-32: Blade root drag shear ( 70a .= and 70a .=ζ ).......................................191

xv

Figure 5-33: Blade root lag bending moment ( 30a .= and 30a .=ζ )........................191

Figure 5-34: Blade root lag bending moment ( 50a .= and 30a .=ζ ) .......................192


Figure 5-36: Blade root lag bending moment ( 30a .= and 50a .=ζ ) ......................193





Figure 5-41: Blade root lag bending moment ( 70a .= and 70a .=ζ ).......................195

Figure 5-42: Blade root vertical shear ( 30a .= and 30a .=ζ ) ...................................196

Figure 5-43: Blade root vertical shear ( 50a .= and 30a .=ζ )...................................196

Figure 5-44: Blade root vertical shear ( 70a .= and 30a .=ζ ) ..................................197







Figure 5-51: Blade root flap bending moment ( 30a .= and 30a .=ζ ) ......................200


Figure 5-53: Blade root flap bending moment ( 70a .= and 30a .=ζ )......................201

xvi






Figure 5-59: Blade root flap bending moment ( 70a .= and 70a .=ζ ) .....................204

Figure 5-60: Blade root radial shear ( 30a .= and 30a .=ζ ) ......................................205

Figure 5-61: Blade root radial shear ( 50a .= and 30a .=ζ )......................................205

Figure 5-62: Blade root radial shear ( 70a .= and 30a .=ζ ) .....................................206







Figure 5-69: Spanwise drag shear ( 30a .= and 30a .=ζ ) .........................................209


Figure 5-71: Spanwise drag shear ( 70a .= and 30a .=ζ ).........................................210




xvii



Figure 5-77: Spanwise drag shear ( 70a .= and 70a .=ζ ) ........................................213

Figure 5-78: Spanwise vertical shear ( 30a .= and 30a .=ζ ) ....................................214


Figure 5-80: Spanwise vertical shear ( 70a .= and 30a .=ζ )....................................215






Figure 5-86: Spanwise vertical shear ( 70a .= and 70a .=ζ ) ...................................218

Figure 5-87: Spanwise radial shear ( 30a .= and 30a .=ζ ) .......................................218


Figure 5-89: Spanwise radial shear ( 70a .= and 30a .=ζ ).......................................219






Figure 5-95: Spanwise radial shear ( 70a .= and 70a .=ζ ) ......................................222

xviii

Figure 5-96: Steady hub loads ( 30a .= and 30a .=ζ ) ...............................................223

Figure 5-97: Steady hub loads ( 50a .= and 30a .=ζ )...............................................223

Figure 5-98: Steady hub loads ( 70a .= and 30a .=ζ ) ..............................................224

Figure 5-99: Steady hub loads ( 30a .= and 50a .=ζ )...............................................224

Figure 5-100: Steady hub loads ( 50a .= and 50a .=ζ ).............................................225

Figure 5-101: Steady hub loads ( 70a .= and 50a .=ζ ) ............................................225



Figure 5-104: Steady hub loads ( 70a .= and 70a .=ζ )............................................227

Figure 5-105: 4/rev vibratory hub loads ( 30a .= and 30a .=ζ ) ................................227

Figure 5-106: 4/rev vibratory hub loads ( 50a .= and 30a .=ζ )................................228

Figure 5-107: 4/rev vibratory hub loads ( 70a .= and 30a .=ζ ) ...............................228






Figure 5-113: 4/rev vibratory hub loads ( 70a .= and 70a .=ζ )...............................231

Figure 5-114: Blade tip flap and lag responses (HART rotor) ...................................232

Figure 5-115: Absorber response (HART rotor) ........................................................232

Figure 5-116: Blade root drag shear (HART Rotor)....................................................233

xix

Figure 5-117: Blade root lag bending moment (HART rotor).....................................233

Figure 5-118: Blade root vertical shear (HART rotor) ................................................234

Figure 5-119: Blade root flap bending moment (HART rotor) ...................................234

Figure 5-120: Blade root radial shear (HART rotor) ...................................................235

Figure 5-121: Spanwise drag shear (HART rotor) ......................................................235

Figure 5-122: Spanwise vertical shear (HART rotor)..................................................236

Figure 5-123: Spanwise radial shear (HART rotor) ....................................................236

Figure 5-124: Steady hub loads (HART rotor) ............................................................237

Figure 5-125: 4/rev vibratory hub loads (HART rotor) ...............................................237

Figure 6-1: Conceptual design of a fluid elastic damper [redrawn from 44]...............247

Figure 6-2: Mechanical analogy of a fluid elastic damper [redrawn from 44] ............247

Figure 6-3: Nonlinear piecewise spring stiffness.........................................................248

Figure 6-4: Lag response with nonlinear spring ..........................................................248

Figure 6-5: Absorber response with nonlinear spring..................................................249

Figure 6-6: Force-displacement and frequency characteristics of ideal buckling [64]........................................................................................................................249

Figure 6-7: Nonlinear spring stiffness - critical buckling load changing as RPM increases................................................................................................................250

Figure 6-8: Schematic for changing effective length of Euler spring..........................250

Figure D-1: Frequency response function with and without absorber.........................288

Figure D-2: Frequency response function with and without absorber – fundamental lag and flap frequencies shown........................................................288

Figure E-1: Blade coordinate system...........................................................................311

Figure E-2: Global assembly of blade elemental matrices with absorber terms .........311

Figure E-3: Blade tip response comparison – rigid and elastic blade analyses ...........312

Figure E-4: First three lag mode shapes ......................................................................312

xx

Figure E-5: Blade tip response – with and without absorber.......................................313

Figure E-6: Blade tip and absorber responses .............................................................313

Figure E-7: Blade root drag shear force – with and without absorber.........................314

Figure E-8: Blade root lag moment – with and without absorber ...............................314

Figure E-9: Blade root axial force – with and without absorber .................................315

Figure E-10: Radial distribution of axial force – with and without absorber..............315

Figure E-11: Radial distribution of drag shear force – with and without absorber .....316

Figure E-12: Radial distribution of lag bending moment – with and without absorber.................................................................................................................316

Figure E-13: Drag shear force at absorber location – with and without absorber.......317

Figure E-14: Axial force at absorber location – with and without absorber ...............317

xxi

LIST OF TABLES

Table 2-1: Ordering scheme.........................................................................................25

Table 2-2: Forces and moments acting on the blade....................................................26

Table 2-3: Forces acting on absorber...........................................................................27

Table 2-4: Absorber parameter values used in simulation...........................................32

Table 3-1: Ordering scheme.........................................................................................57

Table 3-2: Forces and moments contributing to blade flapping ..................................60

Table 3-3: Forces and moments contributing to blade lead-lag motions....................64

Table 3-4: Forces contributing to absorber radial motion ..........................................67

Table 3-5: Absorber parameter values used in ground resonance simulations............92

Table 5-1: Absorber terms in ordering scheme...........................................................144

Table 5-2: Trim results (baseline BO-105-type rotor).................................................163

Table 5-3: Comparison of lag mode damping – elastic and rigid blades.....................165

Table 5-4: Trim results (baseline HART I rotor)........................................................170

Table 5-5: Comparison of baseline rotor modal frequencies with results from RCAS....................................................................................................................171

Table 5-6: Comparison of lag mode damping – HART I and rigid blades..................172

Table C-1: AFDD rotor data........................................................................................281

Table C-2: Main rotor data ..........................................................................................282

Table C-3: Tail and fuselage data................................................................................283

Table C-4: General rotor properties.............................................................................284

Table C-5: Structural properties - 1 .............................................................................285

Table C-6: Structural properties - 2 .............................................................................286

Table C-7: Property descriptions .................................................................................286

xxii

Table E-1: Comparison of lag mode frequencies with results generated by RCAS....307

xxiii

LIST OF SYMBOLS

C Global blade finite element damping matrix

C Global damping matrix

C~ Modal damping matrix

aaC Absorber terms in finite element damping matrix

baC Blade-absorber coupling term in finite element damping matrix

bbC Blade global damping matrix augmented with absorber terms

dC Airfoil drag coefficient

lC Airfoil lift coefficient

mC Airfoil moment coefficient

iC Elemental damping matrix

vvC , vwC Damping matrix associated with lag degree of freedom

wvC Damping matrix associated with flap degree of freedom

TC Rotor thrust coefficient

xC , yC Fuselage roll and pitch damping coefficients

[ ]C Damping matrix

D Blade section drag

ED Energy dissipated per cycle

fD Fuselage drag force

iF Elemental load vector

xF , zF Aerodynamic force components

xF , yF , zF Blade forces

p4xF , p4

yF , p4zF Vibratory hub forces

HxF , H

yF , HzF Hub forces

xxiv VxF , V

yF , VzF Vehicle equilibrium forces

βF , ζF Aerodynamic force in flap and lag directions

G Fluid elastic tuning port area ratio

H Rotor drag force

1H , 2H , 3H , 4H Finite element shape functions

bI Blade second moment of inertia

HHH KJI ˆ,ˆ,ˆ Unit vectors in hub-fixed nonrotating coordinate system

xI , yI Fuselage roll and pitch moments of inertia

βI , ζI Blade second moment of inertia about flap and lag hinges

K Global blade finite element stiffness matrix

K Global stiffness matrix

K~ Modal stiffness matrix

aaK Absorber terms in finite element stiffness matrix

baK Blade-absorber coupling term in finite element stiffness matrix

bbK Blade global stiffness matrix augmented with absorber terms

iK Elemental stiffness matrix

vvK , vvK Stiffness matrix associated with lag degree of freedom

wvK , wwK Stiffness matrix associated with flap degree of freedom

xK , yK Fuselage roll and pitch stiffness

[ ]K Stiffness matrix

L Lagrangian; Length of buckling beam; Blade section lift AL Aerodynamic loads used in force summation method IL Inertial loads used in force summation method IaL Inertial loads due to absorber

htL Horizontal tail lift AvL , A

wL Distributed airloads in chordwise and vertical directions

xxv

M Global finite element mass matrix

M Global mass matrix augmented with absorber terms

M~ Modal mass matrix

aaM Absorber terms in finite element mass matrix

bbM Blade global mass matrix augmented with absorber terms

bM Total blade mass ∗

bM Blade mass (nondimensionalized)

iM Elemental mass matrix

vvM Mass matrix associated with lag degree of freedom

wwM Mass matrix associated with flap degree of freedom

xM , yM Rotor roll and pitch moments

xM , yM , zM Blade moments

p4xM , p4

yM , p4zM Vibratory hub moments

HxM , H

yM , HzM Hub moments

VxM , V

yM , VzM Vehicle equilibrium moments

xfM , yfM , zfM Fuselage moments

βM , φM Blade flap and pitching moments

ζM , AeroMζ Nondimensional aerodynamic lag moment

[ ]M Mass matrix

bN Number of blades

crP Critical buckling load

iQ Generalized force

R Rotor radius; Finite element constraint equation reaction forces

S Blade nodal load vector

rS , xS , zS Blade radial, drag, and vertical shear forces

xxvi

βS , ζS Blade first moment of inertia about flap and lag hinges

T Kinetic energy

aT Absorber kinetic energy

DUT Rotation matrix

trT Tail rotor thrust

U Strain energy

aU Absorber strain energy

V Potential energy; Blade section velocity

W Virtual work; Vehicle gross weight

aW Absorber virtual work

HHH ZYX ,, Hub-fixed nonrotating coordinate system

Y Rotor side force

fY Fuselage side force

a Absorber offset from rotor hub

a Lift curve slope

a Generic acceleration vector

aa Absorber acceleration vector

ba Blade acceleration vector

ca Chordwise inertial damper position relative to blade feathering axis

c Chord

0c Constant lift coefficient

1c Linear lift coefficient

ac Absorber damping coefficient

ζc Lag damping coefficient

0d Constant drag coefficient

2d Nonlinear drag coefficient

xxvii

e Hinge offset

ge Offset of blade elastic axis and center of gravity

h Vertical distance from helicopter cg to hub

kji ˆ,ˆ,ˆ Unit vectors in blade coordinate system

RRR kji ˆ,ˆ,ˆ Unit vectors in hub-fixed rotating coordinate system

ζηξ kji ˆ,ˆ,ˆ Unit vectors in deformed coordinate system

ak Absorber spring stiffness

βk Flap hinge spring stiffness

ζk Lag hinge spring stiffness

m , 0m Blade mass per unit length

am Absorber mass

apm Absorber primary mass

atm Absorber tuning mass

p Modal coordinate

iq Generalized coordinate

r Radial distance to arbitrary point along the blade

r Generic position vector

ar Absorber position vector

br Position of arbitrary point along the blade

cr Radial location of the chordwise inertial damper

dr Rayleigh dissipation function

u Elastic blade axial displacement

Tu , Pu , Ru Tangential, normal, and radial airflow components

v Generic velocity vector

av Absorber velocity vector

xxviii

bv Blade velocity vector

v Elastic blade lag displacement

w Elastic blade flap displacement

x Nondimensional position of arbitrary point along the blade

cx Chordwise displacement of inertial damper mass

cgx , cgy Hub offset from center of gravity in x and y directions

rx Radial displacement of absorber mass

tx Radial displacement of absorber tuning mass

zyx ,, Blade coordinate system

RRR zyx ,, Hub-fixed rotating coordinate system

trz Tail rotor offset from center of gravity in z direction

Λ Radial yaw angle

Φ Eigenvectors ψ Rotor azimuth angle

Ω Rotor speed

α Blade section angle of attack

fα Frequency ratio (ratio of the absorber rotating natural frequency to

the rotating lag frequency)

fα Mass ratio (ratio of absorber mass to blade mass)

sα Longitudinal shaft tilt

xα Hub roll angle

yα Hub pitch angle

β Blade flap angle

0β Steady blade coning angle

γ Lock number ( )ε Indication of order for ordering scheme

ζ Blade lag angle

xxix

aζ Absorber damping ratio

Lζ Blade damping ratio

nζ Modal damping ratio

rη Offset between elastic axis and blade quarter chord

ζηξ ,, Deformed coordinate system

θ Rotor pitch angle

75θ Collective pitch at the 75% span location

c1θ , s1θ Lateral and longitudinal cyclic pitch controls settings

trθ Tail rotor collective pitch

twθ Blade pre-twist angle referenced to the 75% span location

xκ , yκ Drees inflow parameters

λ Rotor inflow μ Rotor advance ratio

βν Nondimensional rotating flap frequency

ζν Nondimensional rotating lag frequency

ρ Air density

sφ Lateral shaft tilt

ω Frequency vector

aω Absorber rotating frequency

nω Modal frequency

ζω Rotating lag frequency

0ζω Nonrotating lag frequency

( )c , ( )s Cyclic modes

( ) c1 , ( ) s1 First harmonic cyclic modes

( )δ Perturbation term; Variation term

xxx

( )Aero , ( )A Aerodynamic term

( )I Inertial term

( )•

Derivative with respect to time

( )∗

Derivative with respect to azimuth

( )′ Derivative with respect to space

( ) Nondimensional quantity

Chapter 1

Introduction

1.1 Background and Motivation

Helicopters with articulated and soft in-plane hingeless rotors are known to be

susceptible to aeromechanical instabilities such as ground and air resonance, which arise

due to the coupling of the poorly damped rotor cyclic lag modes with the fuselage modes.

Adding damping to the lag mode is one of the most common ways to overcome these

instabilities. Unlike the flap mode, the lag mode has very little contribution from

aerodynamic damping. Rotor lag damping must come almost entirely from mechanical

dampers or structural damping.

Aeromechanical instabilities are characterized by a coupling of the low frequency

cyclic lag modes with a fuselage natural frequency. An instability is possible if the

nondimensional rotating lag frequency, νζ, is less than 1/rev as is the case for articulated

(νζ on the order of 0.2/rev to 0.4/rev) and soft in-plane hingeless rotors (νζ on the order of

0.7/rev). These instabilities can result in the complete destruction of the aircraft [1].

Figure 1-1 shows a Hughes/Schweizer 269 (the Army TH-55 version shown) destroyed

by ground resonance.

Ground resonance is most common in articulated rotors and can occur when the

fuselage oscillates on its landing gear. In addition to lag damping, the landing gear on

helicopters with an articulated rotor system will contain some sort of mechanical dampers

or structural damping. Air resonance is most common in hingeless or bearingless rotors

2

and can occur when a lag mode is coupled with a rigid fuselage mode. The conventional

approach to alleviating these instabilities has been to ensure an adequate amount of

damping in the lag mode through the provision of auxiliary lag dampers at the rotor blade

root, although other methods, such as aeroelastic coupling and composite tailoring, have

been investigated. However, associated with the use of auxiliary lag dampers are issues

such as increased hub complexity, aerodynamic drag, and high maintenance

requirements. Lag dampers must provide sufficient damping and stiffness for a wide

range of flight conditions. Conditions such as temperature, frequency, and displacement

amplitude can all affect damper performance. It is also important that lag dampers not

apply excessive loads to the blade root or generate excessive heat during their operation.

1.2 History of the Blade Lag Damper

The lag damper was incorporated into the rotor system as part of the earliest

helicopter designs due to work done previously on autogiros [2]. Friction dampers were

first used on helicopters to provide lag damping, but the damping provided by the friction

damper was not predictable or reliable. The friction damper was also complex and

required constant maintenance.

1.2.1 Hydraulic Damper

In the 1960s, friction dampers were replaced with hydraulic orifice dampers,

which are still in use on helicopters today. The hydraulic damper provides more reliable

3

damping than a friction damper. The hydraulic fluid in the damper is forced to flow

through restricted outlets and valve systems, thus generating hydraulic resistance. The

damping provided by the hydraulic damper is proportional to the square of the lag speed

[3]. Like all lag dampers, the hydraulic damper operates in a dual frequency

environment. It undergoes transient motion at the lag natural frequency and steady state

motion at the 1/rev frequency (and higher harmonics). This dual frequency loading

environment results in operating loads that are much higher than those required to

provide stability, and relief valves must be added to the damper design to minimize the

damper force experienced at 1/rev. This all adds to the complexity of the hydraulic

damper. Hydraulic dampers are also subject to oil leakage from the numerous valves and

seals. While hydraulic dampers have served their purpose well, they have several

disadvantages described above that have led to research in several areas to find suitable

replacements. In addition, the fluid used in hydraulic lag dampers can be hazardous. The

material safety data sheet for one of the common hydraulic fluids in use today (MIL-L-

83282) lists several precautions and hazards to humans [4]. Hydraulic dampers are

currently in use on the Sikorsky UH-60 Blackhawk and the Boeing CH-47 Chinook. The

hydraulic damper used on the Chinook is shown in Figure 1-2.

1.2.2 Elastomeric Damper

In recent years, elastomeric dampers have emerged as a popular choice for use in

prevention of aeromechanical instabilities. Elastomeric dampers have several significant

advantages over the traditional hydraulic dampers. The dampers are simpler and have no

4

moving parts, they are not affected by sand and dust, and they have no seals that can wear

out. They have also proven to be extremely reliable. The gradual deterioration of the

elastomers allows for simple visual inspection of the components and replacement “on

condition” rather than at specified intervals [5]. Elastomeric lag dampers are currently

in use on several helicopters, including the Boeing AH-64 Apache and Model 360, as

well as the Bell 412. A schematic of the elastomeric lag damper used on the Model 360

is shown in Figure 1-3, and the elastomeric lag damper used on the Apache is shown in

Figure 1-4.

1.2.3 Fluidlastic® Damper

Fluidlastic® dampers were introduced late in the last century to overcome some of

the difficulties associated with elastomeric dampers. Fluidlastic® dampers can be used in

smaller space envelopes, have higher damping performance, and have potentially higher

service life than pure elastomeric dampers. Fluidlastic® dampers are currently in use on

the NHI NH-90, and prototype applications are under development for the UH-60, Bell

H-1, and the Agusta A-109 [6]. The Fluidlastic® lag damper installed on the NH-90 is

shown in Figure 1-5.

1.3 Literature Review

The literature review describes relevant research in the areas of auxiliary blade lag

dampers, which includes elastomeric dampers and fluid elastic dampers, as well as other

5

possibilities for lag damping which have been investigated: magnetorheological (MR)

and electrorheological (ER) dampers, constrained layer damping treatments, and

embedded vibration absorbers. The literature review will also describe applicable

aeromechanical stability research, as well as other approaches under investigation which

would allow for aeromechanical stability augmentation without the use of auxiliary lag

dampers.

1.3.1 Elastomeric Damper

Elastomeric lag dampers are used in helicopter rotor systems to prevent

instabilities such as air and ground resonance by dissipating mechanical energy.

Figure 1-6 shows a schematic of an elastomeric damper. The in-plane lag motion of the

rotor blade causes the elastomer to shear. The energy associated with this shearing action

is dissipated as internal heat by material hysteresis. This energy dissipation provides the

damping required for aeromechanical stability [7]. Designing, analyzing, and modeling

elastomeric dampers for helicopters has proven to be challenging. Elastomers are

viscoelastic materials; the elastomer provides both stiffness and damping to the system.

The stiffness and damping characteristics of elastomers are nonlinear functions of the

amplitude and frequency of the blade lag motion, as well as the temperature of the

damper. As a result, several dynamic characteristics of the rotor system are also

nonlinear functions of the amplitude and frequency of the lag motion [8]. These

nonlinearities make modeling the elastomer a difficult process. The modeling of

elastomeric materials has been the subject of numerous studies [7-22], with the need for

6

more accurate, nonlinear models driven by two primary objectives: the accurate

prediction of damper dissipation energy and blade loading [24].

While elastomeric dampers have been successfully used on a number of

helicopters, they do not always provide the desired performance, primarily due to the

nonlinearities associated with elastomeric materials. For example, in the development

and testing of the RAH-66 Comanche, a limit cycle instability in hover was discovered.

After additional wind tunnel tests and analytical studies, it was determined that the

nonlinear stiffness and damping characteristics of the elastomeric lag damper, combined

with the low roll inertia of the aircraft was the fundamental cause of the instability. The

limit cycle instability occurred at very small damper motions. At these small motions,

the elastomer stiffness was high, and the loss factor was low; thus, the available lag mode

damping was low. Figure 1-7 shows the dependence of the stiffness and damping of an

elastomer on displacement amplitude. These conclusions led to a design change for the

Comanche, and the elastomeric dampers were replaced with Fluidlastic® lag dampers

[25]. The elastomeric and Fluidlastic® damper schematics for the Comanche are shown

in Figure 1-8.

1.3.2 Fluid Elastic Damper

Fluid elastic lag dampers have been studied as an alternative to hydraulic and

elastomeric dampers and have been primarily developed under the trade name

Fluidlastic® at Lord Corporation, although the concept of a fluid elastic vibration isolator

was investigated as early as 1972. In 1980, a Liquid Inertia Vibration Eliminator (LIVE)

7

was proposed as an alternative to isolate the vibrations of the main rotor from the

fuselage [26]. In a fluid elastic damper/isolator, the applied oscillatory forces cause the

elastomer to oscillate along with the fluid vessel. As the fluid vessel oscillates, the fluid

is pumped through the inner chamber of the fluid vessel in the opposite direction of the

elastomer. Pumping fluid from one chamber into another through a restriction generates

damping. Fluid elastic dampers combine fluid and bonded elastomeric elements to

provide the damping required in the lag mode [6]. The energy dissipation in Fluidlastic®

dampers is often shared between the fluid and the elastomer. Since the fluid can provide

the majority of the damping associated with the Fluidlastic® damper, the elastomeric

element can be selected based on properties other than its damping value, for example,

strength or shear fatigue properties [3]. In general, Fluidlastic® dampers can provide

higher loss factors than elastomeric dampers. They also exhibit a more linear dynamic

performance than elastomeric dampers as shown in Figure 1-7. Fluidlastic® lag dampers

can be used for articulated and hingeless rotors. Although elastomeric dampers have

been used successfully on articulated rotor systems, such as the AH-64, some articulated

rotors require more damping than can be achieved with current elastomers, particularly

rotors that were originally designed for hydraulic dampers [6]. One problem encountered

with elastomeric dampers on hingeless rotors has been low amplitude limit cycle lead-lag

motions as exhibited with the RAH-66 Comanche described previously. This can be

disturbing to a pilot and can be detrimental to the handling qualities of the helicopter.

Tests have shown that the limit-cycle instabilities associated with elastomeric dampers

were eliminated with Fluidlastic® dampers [25].

8

1.3.3 MR and ER Damper

Magnetorheological (MR) and electrorheological (ER) dampers have been

investigated for use as helicopter lag dampers. MR and ER fluids are colloidal

suspensions that exhibit dramatic reversible changes in properties when a magnetic or

electric field is applied to the fluid. MR and ER fluids consist of micron-sized

polarizable particles in a fluid such as mineral oil. In the presence of a magnetic or

electric field, the fluid undergoes a pseudo-phase change from a liquid to a solid. When

the magnetic or electric field passes through the fluid, the polarizable particles form

chain-like microstructures almost immediately, and as the microstructures form, the

viscosity and dynamic yield stress change dramatically. While similar in principle, the

MR and ER dampers have significantly different properties. The yield stress for MR

fluids is an order of magnitude greater than the yield stress for ER fluids. Additionally,

MR fluids have a wider operating temperature range than ER fluids [27]. While ER fluid

applications have historically outnumbered those of MR fluids primarily due to the wider

commercial availability of ER fluids, MR fluids have recently begun to gain researchers’

attention due to the higher dynamic yield stresses [28]. Numerous studies have been

conducted on the modeling and application of MR/ER dampers [27-34].

1.3.4 Constrained Layer Damping Treatment

A constrained layer damping treatment consists of a thin viscoelastic material and

a stiff constraining layer applied to the surface of the base structure. This concept has

been studied for use on rotorcraft flex beams as a way to increase the damping in the lag

9

mode of hingeless rotor systems [35-36,38]. The purpose of the constraining layer is to

induce shear strain in the highly dissipative damping layer. The cyclic shear deformation

is the mechanism by which energy is dissipated and damping occurs [37]. Both passive

and active constrained layer damping treatments have been studied. A passive

constrained layer damping treatment consists of embedding some distributed damping

material such as an elastomeric material in the flex beam of the hingeless rotor. Since

elastomeric materials have a significant frequency and temperature dependence, to

achieve adequate performance over the required operating temperature range, different

viscoelastic materials would have to be used [38]. Like the passive constrained layer

damping treatments, active constrained layer damping treatments consist of a high

damping viscoelastic material and a stiff constraining layer. However, in an active

constrained layer damping (ACLD) treatment, the stiff constraining layer is a piezo-

crystal or piezo-electric material. In this case, the dimensions of the constraining layer

can be actively altered by the application of an electric field to further augment the shear

in the viscoelastic layer.

1.3.5 Vibration Absorbers

An alternative concept to adding damping to the lag mode through the use of

auxiliary root-end dampers is through the use of embedded vibration absorbers. A

damped vibration absorber can be constructed in several different ways, including a

linear spring with viscous damping, a viscoelastic spring, a viscoelastically damped

resonant beam, or a tuned viscoelastic link joining several elements of a complex

10

structure [37]. Regardless of construction, a damped vibration absorber contributes to the

damping of the system through the dissipation of energy. Since these types of absorbers

dissipate energy, they can be considered broadband systems and are effective over a

wider range of frequencies than an undamped vibration absorber. Vibration absorbers

have been used on helicopters for vibration reduction in several areas for years, but only

recently have vibration absorbers been considered for lag mode damping.

The concept of applying vibration absorbers for lag damping was first analyzed

using highly distributed tuned vibration absorbers [39]. This approach consists of

multiple individual vibration absorbers which are distributed both in space and in

frequency to provide broadband energy dissipation. The absorbers are embedded in the

rotor blade, with the mass of the absorbers coming from the leading edge weights

installed in the blade. By tuning the absorbers over a range of frequencies, several

natural vibration frequencies can be included, as well as the frequencies involved in

ground or air resonance. The investigation showed that for as little as 3% of the total

blade mass, with the appropriate absorber frequency band, the distributed absorbers could

achieve damping levels required to maintain aeromechanical stability in the lag mode.

Recent work on the application of vibration absorber-type devices for lag

damping has focused on using a single embedded chordwise inertial damper. A

schematic of this type of damper is shown in Figure 1-9. The use of an elastomeric

inertial damper was studied and determined that a device of this nature could produce

adequate blade lag damping. It was also determined that this type of damper had the

potential to maintain aeromechanical stability. The amount of damping an embedded

chordwise inertial damper could achieve ranged from 0.3% to 15% critical damping [40].

11

However, an embedded elastomeric damper designed with a specific dynamic stiffness

experienced significant static displacement due to the centrifugal force acting on the mass

of the damper [41].

Later research focused on using an embedded chordwise fluid elastic inertial

damper. The fluid elastic damper could be conceivably designed with a large enough

static stiffness to withstand the high centrifugal force acting on the absorber mass, yet a

low enough dynamic stiffness to achieve the desired tuning frequency [42-45]. This

damper was also shown to achieve sufficient damping in the lag mode.

There are certain disadvantages to an embedded chordwise inertial damper. First,

there is a significant stroke restriction; the damper’s displacement must be limited to a

small fraction of the chord length. Kang also investigated the aeroelastic stability of a

blade with an embedded chordwise damper, focusing on flap-lag flutter, pitch-flap flutter,

and pitch divergence [40]. He found coupling existed between the blade pitch rotation

and the motion of the damper. This resulted in the pitch mode of the blade becoming

more unstable as the thrust increased. The pitch mode also became more unstable as the

damper tuning frequency decreased, and the blade became more pitch divergent as the

mass of the damper increased.

While the disadvantages described above are significant, vibration absorbers

embedded in the rotor blade have several distinct advantages over root end lag dampers.

It is possible for the embedded absorber to be part of the existing leading edge mass or tip

mass of the rotor blade; therefore, there would be little increase in blade weight with the

addition of the absorber. Additionally, since the absorber is embedded in the rotor blade,

12

it reduces the rotor hub complexity and drag that is typically associated with a root end

lag damper.

1.3.6 Aeromechanical Stability Research

Blade lag damping is a vital addition to articulated and soft in-plane hingeless

rotor systems to prevent aeromechanical instabilities such as air and ground resonance.

Research efforts in this area have focused on understanding, predicting, and preventing

aeromechanical instabilities.

A classic analysis of ground resonance was first published in 1957 by Coleman

and Feingold [46]. This type of analysis considers four degrees of freedom: longitudinal

and lateral in-plane motion of the hub and two cyclic lag degrees of freedom of the rotor.

In the early analyses when computing power was low or when the computations were

done by hand, the aerodynamic terms were neglected, and often the analysis was reduced

to three degrees of freedom by considering longitudinal or lateral hub motion by itself.

While the classical analysis is simple, it adequately predicts the ground resonance

characteristics of an articulated rotor system [47].

The simplified ground resonance model, however, is not sufficient to predict

ground and air instabilities for soft in-plane hingeless rotors. These rotors have

considerable structural couplings between blade flap and lag modes, as well as between

rotor flapping and fuselage angular motion. As a result, for soft in-plane hingeless rotors,

additional rotor and fuselage degrees of freedom, as well as rotor aerodynamics, are

required. Soft in-plane hingeless rotors have a higher lag stiffness than articulated rotors,

13

and the ground resonance instability, while a possibility, is generally not as critical for

hingeless rotors.

Air resonance is similar to ground resonance in that the lag mode coalesces with a

fuselage mode. Air resonance occurs in a helicopter in flight and involves a coupling of

the low frequency lag mode and rigid body airframe modes. A basic air resonance

analysis consists of six degrees of freedom: two cyclic flap modes, two cyclic lag modes,

and two body pitch and roll modes. In a more advanced analysis, the rotor blade is

modeled as an elastic beam undergoing flap and lag bending, as well as elastic twist.

Additionally, the airframe is generally modeled with five degrees of freedom: three

translation and two rotation (pitch and roll). Aerodynamic forces must be included in an

air resonance stability analysis. Air resonance is primarily a problem associated with

hingeless and bearingless rotors and is generally not associated with articulated rotors

[47].

While adding an auxiliary lag damper is the conventional means to introduce lag

damping into the rotor system, there are other possible approaches to increase the

aeromechanical stability of the rotor. The first is through the use of active controls. In-

plane blade forces, as well as fuselage moments can be generated by actively changing

the blade pitch, either by inputs through a conventional swashplate or even trailing edge

flaps. These forces and moments can be used to reduce the original unstable blade lag

and fuselage motions. The concept of active controls was first proposed in the 1970s

[48], with experimental investigations beginning in the late 1990s [49]. While

implementation is potentially complicated, the use of active controls can augment

aeromechanical stability [50].

14

Another alternative to auxiliary root end dampers is the use of aeroelastic

couplings, such as pitch-lag coupling, pitch-flap coupling, and flap-lag coupling. These

couplings can be achieved in a number of ways, including elastically tailored composites,

skewed flexures, hub and control system geometry, and distribution of flap and lag

stiffness relative to the torsion bearing. Most studies indicate that it is possible to

improve aeromechanical stability through a combination of couplings, but not necessarily

eliminate the instability over a wide range of operating conditions and rotor

configurations [50].

1.4 Focus of Present Research

The objective of the present research is to investigate the feasibility of

helicopter blade lag damping using an embedded radial vibration absorber. A schematic

of the proposed concept is shown in Figure 1-10. As depicted in the schematic, the

absorber mass (restrained by a spring and damper mechanism) oscillates along the

spanwise direction (potentially along a track within the spar). In the process, it exerts a

tangential Coriolis force on the rotating blade, in the lead-lag direction. The lead-lag

motion of the rotating blade (and the mass), in turn, exerts a radial Coriolis force on the

absorber mass (refer to Figure 1-11). Thus, there exists a Coriolis coupling between the

lead-lag motion of the blade and the radial motion of the absorber mass. For the damped

absorber under consideration, a significant amount of damping can be introduced into the

rotor lag mode through this strong Coriolis coupling.

15

The radial absorber concept proposed here offers several potential advantages

over the chordwise absorber in References [40-44]. First, the restrictions on stroke-

length of the absorber mass are not as stringent (due to availability of space for motion in

the spanwise direction). Second, motion of the absorber mass along the radial direction

does not result in movement of the blade center of gravity in the chordwise direction.

Hence it is unlikely to have any negative impact on blade aeroelastic stability. Finally,

with the proposed configuration, a much smaller absorber mass can introduce a

significant amount of damping into the lag mode. A more detailed comparison between

the radial absorber and chordwise inertial damper can be found in Chapter 4.

This investigation consists of four major parts. First, a two-degree-of-freedom

model representing the coupled dynamics of the lag motion of an isolated rotor blade and

the embedded radial absorber is developed to investigate the damping that can be

introduced into the lag mode through the Coriolis coupling between the radial motion of

the absorber and the lag motion of the rotor blade. This model is also used to analyze the

dynamic amplitude the absorber undergoes to achieve the required damping levels.

Second, a rotor-fuselage aeromechanical stability analysis is developed to evaluate the

impact of the radial absorber in improving the aeromechanical stability behavior. Third,

an elastic blade analysis is developed to evaluate the effect of the radial absorber on the

vibration characteristics of the rotor blade and its effect on the rotor hub loads. Finally,

device concepts are examined for their implementation potential.

16

Figure 1-1: US Army TH-55 destroyed by ground resonance [51]

Figure 1-2: CH-47 hydraulic lag damper [52]

17

Figure 1-3: Schematic of elastomeric lag damper used on Boeing Model 360 [53]

Figure 1-4: Elastomeric lag damper used on AH-64 Apache [52]

18

Figure 1-5: Fluidlastic® lag damper used on NH-90 [52]

Metal Shims

Elastomer

Damper Motion

Figure 1-6: Elastomeric lag damper schematic

19

Figure 1-7: Elastomer stiffness and damping dependence on amplitude [53]

20

Figure 1-8: Elastomeric and Fluidlastic® lag damper schematics used on RAH-66 Comanche [25]

Figure 1-9: Embedded chordwise inertial damper [43]

21

xr

Ω

ζ

ma

a

eka

ca

xr

Ω

ζ

ma

a

eka

ca

Figure 1-10: Radial vibration absorber schematic

Coriolis force on blade in the chordwise (lag) direction due to motion of mass

Ωζ

ΩΩζζ

x&

ζζΩΩ

ζ&aζ&a

Coriolis force on the mass due to blade lag motion

Discrete mass moving in the spanwise direction

Figure 1-11: Coriolis force on blade and absorber mass

Chapter 2

Fundamental Study of Blade Lag Damping with a Radial Vibration Absorber

A simple two-degree-of-freedom model is first developed to gain insight into the

coupled dynamics of the isolated blade lead-lag motion, ζ , in the rotating frame of

reference, and the radial motion, rx , of the absorber mass within the blade (non-

dimensionalized by the rotor radius). The blade is modeled as a rigid body undergoing

lead-lag rotations about a spring-restrained hinge near the root with no hinge offset and

no elastic bending deformations considered. The absorber is assumed to be embedded

within the rotor blade at a distance a from the hub. The embedded absorber is modeled

as a simple spring-mass-damper system and moves radially within the blade.

The governing differential equations of motion are derived in two ways: with

Newton’s second law and Lagrange’s equation. The equations of motion are then

linearized and non-dimensionalized. The forcing terms in the equations of motion are set

to zero, and an eigenvalue analysis is performed. From the eigenvalue analysis, the

modal frequencies and damping characteristics are determined.

The steady state displacement amplitude of the absorber is evaluated by applying

a harmonic excitation force to the lag equation of motion and determining the frequency

response functions of the blade lag displacement, ζ , and the absorber displacement, rx .

The displacement amplitude of the absorber at an excitation frequency of 1/rev is

examined, as 1/rev is the dominant excitation frequency in flight.

23

2.1 Coordinate System

The coordinate systems used in this analysis are shown in Figure 2-1. The inertial

frame of reference is defined as the hub-fixed nonrotating coordinate system

( HHH ZYX ,, ), with unit vectors HHH KJI ˆ,ˆ,ˆ . The HX axis points to the rear of the

rotor, the HY axis points to the advancing side of the rotor, and the HZ axis points

upward. The hub-fixed rotating coordinate system ( RRR zyx ,, ) with unit vectors,

RRR kji ˆ,ˆ,ˆ , is attached to the hub and rotates with the blades at an angular velocity of

Rk̂Ω , relative to the hub-fixed nonrotating coordinate system. The blade coordinate

system ( zyx ,, ), with unit vectors, kji ˆ,ˆ,ˆ , is also attached to the hub. The x axis is

coincident with the blade, and the y axis is in the plane of rotation. The transformation

between the blade coordinate system and the hub-fixed rotating coordinate system is as

follows:

where ζ is the blade lag angle.

2.2 Position, Velocity, and Acceleration of Blade and Absorber

The blade and absorber velocities are determined for use in Lagrange’s equation,

and the accelerations are determined for use in the derivation of the equations of motion

using Newton’s second law.

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −=

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

R

R

R

kji

kji

ˆˆˆ

1000cossin0sincos

ˆˆˆ

ζζζζ

2.1

24

2.2.1 Blade

The velocity and acceleration of the blade are calculated by first determining the

position of an arbitrary point along the blade:

The velocity can be calculated by taking the first time derivative of the position vector.

Since the position vector is defined in a rotating reference frame, its first time derivative

is given by Eq. 2.3

and its second time derivative is given by Eq. 2.4

The rotating reference frame is rotating at an angular velocity, k̂Ω , and the rotor’s

angular acceleration is considered to be zero for this analysis. The velocity and

acceleration of an arbitrary point on the blade are then

2.2.2 Absorber

The acceleration of the absorber can be similarly determined, starting with its

position vector:

irr bˆ= 2.2

rrv r ×+= ω& 2.3

rr2rra r ×+×+××+= ωωωω &&&& )( 2.4

jrvbˆ)( ζ&−Ω= 2.5

jrirr2ra 22b

ˆˆ)( ζζζ &&&& −Ω−Ω+−= 2.6

25

where a is the radial offset of the absorber from the hub, and rx is the displacement of

the absorber. The velocity and acceleration of the absorber are then

2.3 Derivation of Blade and Absorber Equations of Motion

2.3.1 Ordering Scheme

In order to reduce the complexity of the equations of motion, an ordering scheme is

applied. This provides a method for systematically neglecting terms based on their

relative magnitude. The larger terms are kept, while the higher order terms are neglected.

The order of magnitude of the quantities used in the equations of motion (and defined in

section 2.3.2) is shown in Table 2-1.

ixar raˆ)( += 2.7

jxaixv rraˆ))((ˆ ζ&& −Ω++= 2.8

[ ][ ] jxax2x2

ixaxa2xaxa

rrr

2rr

2rra

ˆ)(

ˆ)()()(

ζζ

ζζ&&&&&

&&&&

+−−Ω+

Ω+−Ω+++−= 2.9

Table 2-1: Ordering scheme

Variable Symbol Order

Blade Properties bI O(1) Absorber Properties am , a , aζ O(1) Absorber/Blade Ratios fα , mα O(1) Lag DOF

ζ , ∗

ζ , ∗∗

ζ O(ε)

Absorber DOF rx , rx

∗

, rx∗∗

O(ε)

26

The equations of motion are simplified by eliminating terms of order 3ε or higher.

2.3.2 Newton’s Second Law

The differential equations of motion for the blade and absorber system are first

derived using Newtonian mechanics. The forces acting on the blade and absorber can be

determined from the accelerations of the blade and absorber, after the application of the

ordering scheme. The forces and moments acting on the blade are listed in Table 2-2 and

graphically depicted in Figure 2-2. While there is a centrifugal force present, it will not

appear in the blade equation of motion. This is due to the fact that there is no hinge offset

modeled, thus there is no component of the centrifugal force that will cause a moment

about the lag hinge.

The radial forces acting on the absorber are listed in Table 2-3 and graphically depicted

in Figure 2-3.

Table 2-2: Forces and moments acting on the blade

Force/Moment Magnitude Moment Arm about Lag Hinge

Inertial Force ζ&&rmdr)( ζ&&ama

r )( rxa +

Coriolis Force ra xm2 &Ω )( rxa + Aerodynamic Force ζF r Spring Moment ζζk -- Damping Moment ζζ &c --

27

To obtain the blade equation of motion, moments are summed about the lag

hinge, resulting in the following equation:

Using the following:

and dividing throughout by 23mR31

Ω , yields the non-dimensional form of the lag

equation:

The absorber equation of motion is obtained by summing forces on the absorber

in the radial direction:

Table 2-3: Forces acting on absorber

Force Magnitude

Inertial Force ra xm && Coriolis Force ζ&Ωam2 a Centrifugal Force 2

ra xam Ω+ )( Spring Force ra xk Damping Force ra xc &

drrFkcxam2amdrmrR

0raa

R

0

2 ∫∫ =++Ω−+ ζζζ ζζζ &&&&)( 2.10

b3R

0

2 ImR31drmr ==∫ (for a uniform blade)

( ) ( )∗•

Ω= and ( ) ( )∗∗••

Ω= 2 , where ( ) ( )ψd

d=

∗

2.11

ζζζ γζνζνζαζ M2xa6am31 2Lrma =++−+

∗∗∗∗

)( 2.12

2ar

2aaraara amxmkxcam2xm Ω=Ω−++Ω+ )(&&&& ζ 2.13

28

Like the lag equation, the absorber equation of motion can be non-dimensionalized by

using ( ) ( )∗•

Ω= and ( ) ( )∗∗••

Ω= 2 , where ( ) ( )ψd

d=

∗

and dividing throughout by

Rm 2aΩ , resulting in the following equation:

In Eq. 2.12 and Eq. 2.14 , the following definitions were used:

Two ratios are introduced. fα is the ratio of the absorber natural frequency to the

rotating lag frequency: ζω

ωα a

f = . mα is the ratio of the absorber mass to the blade mass:

b

am M

m=α .

The blade and absorber equations of motion are coupled by the Coriolis forces,

ζ&Ωam2 a and ra xam2 &Ω . In matrix form, the linearized, nondimensional, coupled

equations of motion can be written as:

axx2a2x r22

frfar =+++∗∗∗∗

ζζ ναναζζ 2.14

Rx

x rr =

Raa =

2

a

a2a m

kΩ−=ω

2b

2

IkΩ

= ζζν (for this case of a rigid blade with no hinge offset)

2.15

29

In the above equation, a is the absorber offset from hub (non-dimensionalized by the

rotor radius, R), and aζ is the absorber damping ratio. The rotor blade parameters in the

above equation are ζν (the non-dimensional rotating lag frequency) and Lζ (the blade

lag damping ratio).

The forcing term on the right hand side of Eq. 2.13 is due to the static centrifugal

force acting on the absorber mass. With a constant spring stiffness, ak , the static

displacement of the absorber is very large due to this centrifugal force. This issue is

described in detail in Chapter 5. The results presented in this chapter include only the

dynamic response of the absorber.

2.3.3 Derivation of Blade and Absorber Equations of Motion using Lagrange’s Equation

The blade and absorber equations of motion can also be derived via an energy

method, using Lagrange’s equation:

⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡+

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧⎥⎦

⎤⎢⎣

⎡ −+

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧⎥⎦

⎤⎢⎣

⎡ +∗

∗

∗∗

∗∗

aM

x00

x2a2a62

x100am31

r22

f

2

rfa

mL

r

a

ζ

ζ

ζ

ζ

ζ

γζνα

ν

ζναζ

ανζζ

2.16

iii

d

i

QqL

qr

qL

dtd

=∂∂

−∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

&& 2.17

30

where L is the Lagrangian, VTL −= , iq is the generalized coordinate, ζ and rx in

this case, and iQ is the generalized force. The damping in the absorber is captured with

the Rayleigh dissipation function, dr , where

In the Lagrangian, T is the kinetic energy, and V is the potential energy of the blade-

absorber system.

The kinetic energy of the system includes contributions from the blade and

absorber and is as follows:

where bv and av are defined in Eq. 2.5 and Eq. 2.8. The potential energy also includes

contributions from the blade and absorber and is as follows:

The differential equations of motion for the blade-absorber system can then be

determined by substituting the kinetic and potential energies into the Lagrangian and

applying Lagrange’s Equation (Eq. 2.17) for ζ=1q and r2 xq = , eliminating higher

order terms, and linearizing the equations of motion about the trim condition, which

results in the same equations of motion as in Eq. 2.16. While the generalized force would

be determined from the aerodynamic moment about the lag hinge, it is not considered in

this analysis, since the forcing terms on the right hand side of Eq. 2.16 will be set to zero

for the eigenvalue analysis.

2rad xc

21r &= 2.18

∫ ⋅+⋅=R

0 aaabb vvm21drvvm

21T 2.19

2ra

2 xk21k

21V += ζζ 220

31

2.4 Complex Eigenvalue Analysis

By setting the forcing terms on the right-hand side to zero and calculating the

eigenvalues of the system, the modal frequencies and damping of the coupled lag mode

and absorber mode can be determined.

The system of equations is in second order form and must be put in first order

form to calculate the eigenvalues:

where

The eigenvalues are of the form:

where nω are the modal frequencies, and nζ are the modal damping ratios. If the real

part of the eigenvalues is positive, the system is unstable. The modal damping of the

system is dependent on the absorber parameters, mα , fα , a , and aζ .

The blade lag/absorber system behavior is examined using the two-degree-

freedom model (Eq. 2.16) over a range of absorber parameters given in Table 2-4. For all

of the simulations in this section, the values for the rotor parameters used are

rev70 /.=ζν (corresponding to a soft-inplane hingeless rotor) and 0L =ζ (implying no

inherent damping in the blade lag mode).

AYY =& 2.21

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

=

r

r

x

xY

ζ

ζ&

&

2.22

2nnnnn 1is ζωωζ −±−= 2.23

32

2.5 Results

2.5.1 Modal Frequencies and Damping Ratios

Figures 2-4a - 2-6a show variation in damping ratios of the coupled absorber and

the blade lag modes, versus frequency ratio, fα , for inboard ( 30a .= ), mid-span

( 50a .= ) and outboard ( 70a .= ) locations of the absorber, respectively. For each of

these figures, the isolated absorber mode damping ratio is 30a .=ζ , and the series of

curves shown correspond to the five different values of absorber-to-blade mass ratio, mα .

From the figures it is seen that the maximum possible damping in the blade lag mode

occurs at a frequency ratio of 1f =α , and does not exceed 15% critical damping, which

is half of the isolated absorber mode damping. Further, from Figure 2-4a it is evident that

for the inboard absorber location, a higher absorber mass is required to achieve the

maximum damping of 15% in the lag mode. On the other hand, as the absorber moves

outboard (Figure 2-5a and Figure 2-6a), the maximum possible damping is transferred to

Table 2-4: Absorber parameter values used in simulation

Absorber Parameter Values Considered

a 0.3, 0.5, 0.7 aζ 0.3, 0.5, 0.7

fα 0.5 – 1.5

mα .01, .02, .03, .04, .05

33

the lag mode even for the smaller values of mass ratio, mα , if the frequency ratio is close

to 1.

Corresponding to the modal damping ratios in Figures 2-4a - 2-6a, Figures 2-4b -

2-6b also show the system modal frequencies versus fα . The lag mode frequency

remains relatively uniform, whereas the absorber mode frequency increases at a nearly

linear rate with increasing fα . For weaker Coriolis coupling (inboard location of the

absorber of 30a .= , Figure 2-4), the modal frequencies show little interaction. For

stronger Coriolis coupling (mid-span or outboard absorber locations, Figure 2-5 and

Figure 2-6) and larger mα values, the modal frequencies show much greater interaction.

Figures 2-7a - 2-9a show damping ratios of the coupled absorber and the blade lag

modes, as a function of frequency ratio, fα , when the isolated absorber mode damping

ratio is increased to 50a .=ζ . For the inboard location of the absorber ( 30a .= ,

Figure 2-7a), very little damping is transferred to the blade lag mode. However, as the

absorber moves outboard (Figure 2-8a and Figure 2-9a), and for higher values of mα ,

more damping is transferred to the blade lag mode. The maximum damping that can be

transferred to the lag mode (at a frequency ratio of 1f =α ) is 25% critical, which is

again half of the isolated absorber mode damping of 50% critical. The system modal

frequencies corresponding to the modal damping ratios are also presented in Figures 2-7b

- 2-9b.

Figures 2-10a - 2-12a present modal damping ratios, as a function of frequency

ratio, fα , when aζ is increased even further to 0.7. The maximum damping transferable

34

to the lag mode is 35% critical, again half of the isolated absorber mode damping. To

transfer this maximum damping to the lag mode an outboard absorber location ( 70a .= ,

see Figure 2-12a), larger mass ratio ( 050m .=α ), and frequency tuning ( 1f =α ) are

required. As the absorber moves inboard (Figure 2-10a and Figure 2-11a) the damping

transferred to the lag mode is smaller (and less than the ceiling value of 0.35), for the

values of mα considered. Results did show (not presented), that for the higher damped

absorbers ( 50a .=ζ and 70a .=ζ ) and inboard absorber locations (Figures2-7a, 2-10a,

and 2-11a) the lag mode damping ratios would reach the ceiling values possible (0.25 for

Figure 2-7a, and 0.35 for Figures 2-10a and 2-11a) if larger absorber masses (larger

values of mα ) were used.

From the results of Figures 2-4a - 2-12a, it appears that increased values of

absorber damping, aζ , are not necessarily advantageous. For high aζ values, the

damping transferred to the lag mode could actually be quite modest for inboard or mid-

span absorber locations and small absorber mass values. The physical reason for this

could be that an overly damped absorber would undergo less radial motion, and thus the

Coriolis forces and the mechanism for transfer of damping to the lag mode are

diminished. Of course, if the absorber mass is large enough and the absorber location is

outboard enough, a higher damped absorber would transfer more damping to the lag

mode than a lower damped absorber (compare Figure 2-12a to Figure 2-6a and Figure 2-

9a). For a lower damped absorber ( 30a .=ζ , Figures 2-4a - 2-6a), even lower mass

ratios and inboard absorber locations can transfer the maximum damping (15% critical in

this case) to the lag mode. It should be noted, however, that a highly damped absorber

35

even with a moderate mα , and mid-span location, could still deliver sufficient damping

to the rotor lag mode (albeit less than the ceiling value of a21 ζ ). In Figure 2-11a, for

example, for 50a .= and 050m .=α , the lag mode damping ratio is 0.15 (comparable to

the ceiling value of 0.15 for the 30a .=ζ case in Figures 2-4a - 2-6a), but it holds this

over a large frequency range (as opposed to just near 1f =α ).

2.5.2 Frequency Response

By introducing a harmonic excitation force on the right hand side of the lag

equation of motion in Eq. 2.16, Frequency Response Functions (FRFs) for the absorber

displacement, rx , and blade lag displacement, ζ , can be obtained. Of particular interest

is the displacement amplitude of the absorber at a frequency of 1/rev, the dominant

excitation frequency in forward flight. Like the modal damping ratios, the amplitude of

1/rev absorber displacement is dependent on the absorber parameters: fα , a , and aζ .

Figure 2-13 shows the blade lag degree-of-freedom frequency response function

(FRF), with and without the absorber. For the no-absorber case (solid blue line), the lag

mode is undamped and shows a resonance peak at 0.7/rev. When the absorber is

introduced, the FRF (solid green line) looks similar to the classical FRF for a system with

a damped vibration absorber. The FRF presented in Figure 2-13 is for the parameter

values 70a .= , 30a .=ζ , 030m .=α , and 1f =α (for the damped case) and is for

illustration purposes only, with no tuning requirements considered. Figure 2-14 shows

36

the FRF for both the absorber mass amplitude as well as the blade lag amplitude, and

Figure 2-15 presents the ratio of the two – essentially the amplitude of the absorber mass

(as a percentage of radius), per degree of blade lag motion. Of particular interest in

Figure 2-15 is the value at 1/rev, since the system would be excited at this frequency (and

its higher harmonics) in forward flight. Figures 2-16 - 2-24 show the amplitude of the

absorber mass per degree of blade lag motion at 1/rev for variations in absorber

parameters listed in Table 2-4, with Figures 2-16 - 2-18 representing the 1/rev motions

for 30a .=ζ (corresponding to the modal damping seen in Figures 2-4 - 2-6). Similarly,

Figures 2-19 - 2-21 represent the motions for 50a .=ζ (corresponding to the modal

damping seen in Figures 2-7 - 2-9), and Figures 2-22 - 2-24 represent motions for

70a .=ζ (corresponding to the modal damping in Figures 2-10 - 2-12). From Figures 2-

16 - 2-24 ,it is seen that the 1/rev motion of the absorber can be reduced by moving the

absorber mass inboard (reducing a ) or by increasing the amount of absorber damping,

aζ . Although reducing a would result in a corresponding reduction in lag mode

damping, the damping can be recovered by slightly increasing the mass ratio (from mα

of 1-3% to 3-5%).

In all cases, the amplitude of the absorber mass varies from approximately 0.7%

to 4.3% radius per degree of lag motion. This translates to an absorber mass dynamic

motion range of about ± 1.4 to ± 8.4 inches per degree of lag motion for a BO-105-sized

rotor (radius 16.2 ft), and about ± 2.3 to ± 13.8 inches per degree of lag motion for a UH-

60-sized rotor (radius 26.8 ft).

37

2.6 Summary

The simple two-degree-of-freedom model described in this chapter has shown that

the radial vibration absorber can introduce damping in the lag mode through Coriolis

coupling. Even for mass ratios as low as 1% to 5% of the blade mass, a significant

amount of damping can be introduced into the rotor lag mode for most combinations of

absorber parameters, with up to one-half of the absorber mode damping transferred to the

lag mode. In general, as a and mα increase, the amount of damping in the lag mode also

increases. The maximum amount of lag damping occurs at a frequency tuning ratio, fα ,

of 1, but a considerable amount of lag damping is possible even when fα is decreased to

0.5 or increased to 1.5.

While the amount of damping that can be transferred to the lag mode is an

important consideration of the radial absorber, another concern is the dynamic

displacement amplitude of the absorber that is required to achieve these damping levels.

The absorber amplitude at 1/rev periodic motion was examined using the two-degree-of-

freedom model and was found not to be excessively large for all combinations of

absorber parameters.

38

XH

YH

ZH

xR

x

zR, z

ζ

XH

YH

ZH

xR

x

zR, z

ζ

XH

YH

ZH

xR

x

zR, z

XH

YH

ZH

xR

x

zR, z

XH

YH

ZH

xR

x

zR, z

ζζ

Figure 2-1: Coordinate systems used in two-degree-of-freedom model

39

a

ζ

ΩElastic Restoring / Damping Moments

Inertial Force

AerodynamicForce

r

Inertial Force

Coriolis Forcea

ζ


Inertial Force

AerodynamicForce

r

Inertial Force

Coriolis Forcea

ζ

a

ζ

a

ζ


Inertial Force

AerodynamicForce

r

Inertial Force

Coriolis Force

ΩΩElastic Restoring / Damping Moments

Inertial Force

AerodynamicForce

r

Inertial Force

Coriolis Force

Elastic Restoring / Damping Moments

Inertial Force

AerodynamicForce

r

Inertial Force

Coriolis Force

Elastic Restoring / Damping Moments

Inertial Force

AerodynamicForce

r

Inertial Force

Coriolis Force

Figure 2-2: Forces and moments acting on the blade contributing to moments about the lag hinge

Ω

a Inertial Force / Centrifugal Force

Spring Force / Damping Force

Coriolis Force

Ω



ΩΩ





Coriolis Force

Figure 2-3: Forces acting on absorber

40

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

0.05

0.1

0.15

0.2

0.25

0.3

Frequency Ratio, αf

Mo

dal

Dam

pin

g R

atio

Increasing αm

0.01

0.03

0.05

0.01

0.03

0.05

Lag Mode

Absorber Mode

(a) Modal damping ratios

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2


Mod

al F

requ

ency

(/r

ev)

Absorber Mode

Absorber Mode

Lag Mode

Lag Mode

Increasing αm

(b) Modal frequencies

Figure 2-4: Modal damping ratios (a) and frequencies (b) vs frequency ratio, fα ( 30a .= and 30a .=ζ )

41

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

0.05

0.1

0.15

0.2

0.25

0.3

Mod

al D

ampi

ng R

atio


Absorber Mode

Lag Mode

Increasing αm

0.01

0.05

0.01

0.05


0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2


Mod

al F

requ

ency

(/r

ev)

Absorber Mode

Absorber Mode

Lag Mode

Lag Mode

Increasing αm



42

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

0.05

0.1

0.15

0.2

0.25

0.3

Mod

al D

ampi

ng R

atio


Absorber Mode

Lag Mode

Increasing αm

0.01

0.05

0.01

0.05


0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2


Mod

al F

requ

ency

(/r

ev)

Absorber Mode

Absorber Mode

Lag Mode

Lag Mode

Increasing αm



43

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Mod

al D

ampi

ng R

atio


Absorber Mode

Lag Mode

Increasing αm


0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2


Mod

al F

requ

ency

(/r

ev)

Absorber Mode

Absorber Mode

Lag Mode

Lag Mode



44

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Mod

al D

ampi

ng R

atio


Absorber Mode

Lag Mode

Increasing αm

0.01

0.03

0.05

0.01

0.03

0.05


0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2


Mod

al F

requ

ency

(/r

ev)

Absorber Mode

Absorber Mode

Lag Mode

Lag Mode

Increasing αm



45

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Mod

al D

ampi

ng R

atio


Absorber Mode

Lag Mode

Increasing αm

0.01

0.05

0.01

0.05


0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2


Mod

al F

requ

ency

(/r

ev)

Absorber Mode

Absorber Mode

Lag Mode

Lag Mode

Increasing αm



46

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Mod

al D

ampi

ng R

atio


Absorber Mode

Lag Mode Increasing αm


0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2


Mod

al F

requ

ency

(/r

ev)

Absorber Mode

Lag Mode



47

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Mod

al D

ampi

ng R

atio


Absorber Mode

Lag Mode Increasing αm

0.01

0.05

0.01

0.05


0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2


Mod

al F

requ

ency

(/r

ev)

Absorber Mode

Lag Mode



48

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Mod

al D

ampi

ng R

atio


Absorber Mode

Lag Mode

Increasing αm

0.01

0.03

0.05

0.01

0.03

0.05


0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2


Mod

al F

requ

ency

(/r

ev)

Absorber Mode

Absorber ModeLag Mode

Lag Mode

Increasing αm



49

0 0.5 1 1.50

1

2

3

4

5

6

7

8

9

10

Frequency ( /rev)

Undamped case

With absorber

Fre

qu

ency

Res

po

nse

Mag

nit

ud

e, ζ

0/F0

Figure 2-13: FRF of the blade lag amplitude for undamped (no absorber) and damped

(with absorber) cases ( 70a .= , 30a .=ζ , 030m .=α , 1f =α )

0 0.5 1 1.50

2

4

6

8

10

12

14

16

18

20

Frequency ( /rev)

Fre

qu

ency

Res

po

nse

Mag

nit

ud

e

Figure 2-14: Frequency response function – absorber and blade lag amplitude

( 70a .= , 30a .=ζ , 030m .=α , 1f =α )

0F0x

0F0ζ

50

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

1

2

3

4

5

6

Frequency ( /rev)

Response at 1/rev

Fre

qu

ency

Rep

on

se M

agn

itu

de

Figure 2-15: Frequency response function - absorber amplitude in %R per degree of lag

motion ( 70a .= , 30a .=ζ , 030m .=α , 1f =α )

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Frequency Ratio,αf

Figure 2-16: 1/rev absorber amplitude per degree of lag motion vs frequency ratio, fα

( 30a .=ζ and 30a .= )

0

0x

ζ

⎟⎟⎠

⎞⎜⎜⎝

⎛deg%R0

0x

ζ

51

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Frequency Ratio,αf


( 30a .=ζ and 50a .= )

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Frequency Ratio,αf


( 30a .=ζ and 70a .= )

⎟⎟⎠

⎞⎜⎜⎝

⎛deg%R0

0x

ζ

⎟⎟⎠

⎞⎜⎜⎝

⎛deg%R0

0x

ζ

52

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Frequency Ratio,αf

Figure 2-19: 1/rev absorber amplitude per degree of lag motion vs frequency ratio, fα ,

( 50a .=ζ and 30a .= )

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Frequency Ratio,αf


( 50a .=ζ and 50a .= )

⎟⎟⎠

⎞⎜⎜⎝

⎛deg%R0

0x

ζ

⎟⎟⎠

⎞⎜⎜⎝

⎛deg%R0

0x

ζ

53

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Frequency Ratio,αf


( 50a .=ζ and 70a .= )

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Frequency Ratio,αf


( 70a .=ζ and 30a .= )

⎟⎟⎠

⎞⎜⎜⎝

⎛deg%R0

0x

ζ

⎟⎟⎠

⎞⎜⎜⎝

⎛deg%R0

0x

ζ

54

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Frequency Ratio,αf


( 70a .=ζ and 50a .= )

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Frequency Ratio,αf


( 70a .=ζ and 70a .= )

⎟⎟⎠

⎞⎜⎜⎝

⎛deg%R0

0x

ζ

⎟⎟⎠

⎞⎜⎜⎝

⎛deg%R0

0x

ζ

Chapter 3

Modeling and Aeromechanical Stability Analysis of a Rotor System with a Radial Vibration Absorber

While the two-degree-of-freedom system analyzed in the previous chapter

examined the ability of a radial absorber to transfer damping via Coriolis coupling into

the lag mode of an isolated blade in the rotating frame of reference, this chapter describes

the model used to examine the effect of the radial absorber on rotor-fuselage

aeromechanical stability characteristics. A classical six-degree-of-freedom rotor-body

aeromechanical stability analysis, with two cyclic flap modes, two cyclic lag modes,

body pitch and body roll, similar to the model developed in [54], is augmented to account

for the radial absorber. As with the flap and lag equations of motion, the absorber

equations of motion are transformed to the nonrotating frame using the Multiblade

Coordinate Transformation to give two cyclic absorber equations and a total of eight

degrees of freedom in the non-rotating coordinate system. Only the cyclic modes are

taken into account for the ground resonance analysis since the collective and differential

modes are not coupled to the body states. The rotor blade is modeled as a rigid body

undergoing flap and lag motions about spring-restrained flap and lag hinges. For

simplicity, the flap and lag hinges are assumed to be collocated at a distance e from the

rotor hub. Additionally, the hub is assumed to undergo rigid body pitch, yα , and roll,

xα , motions. Once derived, the equations of motion are linearized about the equilibrium

position, and the perturbation equations of motion are analyzed for stability.

56

The decay rates and modal frequencies for the regressing lag and absorber modes, as well

as the fuselage pitch and roll modes are determined by evaluating the eigenvalues of the

system, and this is done over a range of rotational speeds.

3.1 Coordinate System and Ordering Scheme

3.1.1 Coordinate System

The coordinate systems used in this analysis are shown in Figure 3-1. The inertial

frame of reference is defined as the hub-fixed nonrotating coordinate system



upward. The hub-fixed rotating coordinate system ( RRR zyx ,, ) with unit vectors,

RRR kji ˆ,ˆ,ˆ , is attached to the hub and rotates with the blades at an angular velocity of

Rk̂Ω , relative to the hub-fixed nonrotating coordinate system. The blade coordinate

system ( zyx ,, ), with unit vectors, kji ˆ,ˆ,ˆ , is also attached to the hub, where the x axis is

coincident with the blade. The transformation between the blade coordinate system and

the hub-fixed rotating coordinate system is through the following Euler angles:

where β is the blade flap angle, and ζ is the blade lag angle.

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

−=

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

R

R

R

kji

0kji

ˆˆˆ

cossinsincossincossin

sinsincoscoscos

ˆˆˆ

βζβζβζζ

βζβζβ 3.1

57

3.1.2 Ordering Scheme

In order to reduce the complexity of the equations of motion, an ordering scheme

is applied. The order of magnitude of the quantities used in the equations of motion is

shown in Table 3-1.

The nonlinear equations of motion are simplified by eliminating terms of order 3ε or

higher.

3.2 Position, Velocity, and Acceleration of Blade and Absorber

The accelerations of the blade and absorber are derived in a similar fashion as in

section 2.2, but the motion of the hub must also be included for a ground resonance

analysis.

Table 3-1: Ordering scheme

Variable Symbol Order

Blade/Rotor Properties bI , e , h O(1)

Absorber Properties am , a , aζ O(1) Absorber/Blade Ratios fα , mα O(1) Flap DOF

β , ∗

β , ∗∗

β O(ε )

Lag DOF ζ ,

∗

ζ , ∗∗

ζ O(ε )

Absorber DOF rx , rx

∗

, rx∗∗

O(ε )

Hub DOF yx αα , O(ε )

58

3.2.1 Blade

The velocity and acceleration of the blade are calculated by first determining the


The velocity can be determined by taking the first time derivative of the position vector.

Since the position vector is defined in a rotating reference frame, its first and second time

derivatives are given by Eq. 2.3 and Eq. 2.4. The rotating reference frame is rotating at

an angular velocity, RRyRx kji ˆˆˆ Ω++= ααω && . While the rotor acceleration is considered

to be zero for this analysis, the motion of the hub is also included in the angular velocity,

with nonzero acceleration terms.

The acceleration of an arbitrary point on the blade in the rotating coordinate

system is then

irr bˆ= 3.2

( ) [ ]( )[ ][ ]

( ) ( ) ( )( ) [ ]

( )[ ][ ] ( )

( ) [ ]( ) [ ] [ ]( ) [ ]

R

yx

yxyx

yx

R

yx

yx

2

R

yx

yx

2

b

k

er2

r2errer

j

er2erh

er2er2erer

i

erh

er2erer2r

a

ˆ

sincos

)sin()cos(sincoscossin

ˆ

sincos

sincos

ˆ

cossin

cossin)()(

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

+−

−+Ω++−

−−+−

+

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

Ω−−+−+

−+−

−Ω−−Ω−+−−

+⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

−−+

−−−

−−−Ω−+Ω−

=

ψαψαζ

ψαψαψαψαζ

ψαψαβ

ζζψαψαβ

ψαψαβ

ββζζ

ψαψαβ

ψαψαβ

ββζ

&&&

&&&&&&

&&&&&&

&&&&&

&&&

&&&

&&&&

&&&

&&&

3.3

59

3.2.2 Absorber

The acceleration of the absorber can be similarly determined, starting with its

position vector:

where a is the radial offset of the absorber from the hub, and rx is the displacement of

the absorber. The acceleration of the absorber is then

3.3 Derivation of Equations of Motion

The differential equations of motion for the blade and absorber system were

derived using Newtonian mechanics. The forces acting on the blade and absorber can be

ixar raˆ)( += 3.4

( ) ( ) ( )( )[ ][ ]

( ) [ ][ ]

( ) ( ) ( )( ) [ ]

( )[ ][ ][ ]

( ) ( )[ ]( ) [ ]( ) [ ][ ]

R

yxr

rryxr

yxr

yxrr

R

yxrrrr

yxr

yxr

r2

rr

R

ryxr

rryxr

yxr

rr2

r

a

k

x2x2xxa2

exaxaexa

j

x2x2x2x

exahexa2

exa2exaexa

i

x2x2

x2xexa2

exahexaexa2xa

a

ˆ

cossinsincos

sincoscossin

sincos

sincossincos

ˆ

cossin

cossin

cossin

⎪⎪

⎭

⎪⎪

⎬

⎫

⎪⎪

⎩

⎪⎪

⎨

⎧

−

++++Ω+

++−+−

−++−+

+

⎪⎪

⎭

⎪⎪

⎬

⎫

⎪⎪

⎩

⎪⎪

⎨

⎧

+−−Ω+

−+−++−

+−+

−Ω−+−Ω−++−+−

+

⎪⎪

⎭

⎪⎪

⎬

⎫

⎪⎪

⎩

⎪⎪

⎨

⎧

−−

+Ω++−−+

−−−++−−+−Ω−++Ω+−

=

ψαψα

ββψαψα

ψαψαζ

ψαψαβ

ψαψαβζζ

ψαψαβ

ψαψαβ

ββζζ

ββψαψαβ

ζψαψαβ

ψαψαβββζ

&&

&&&&&&

&&&&

&&&&&&

&&&&&&&&

&&&&

&&&

&&&

&&&&&

&&&&&&

&&&&

&&&

3.5

60

determined from the accelerations of the blade and absorber. These forces and moments

are then used to determine the differential equations of motion.

3.3.1 Flap Equation of Motion

The forces and moments acting on a blade element in the radial and vertical

directions contribute to the flap equation of motion. These forces and moments are as

follows:

Table 3-2: Forces and moments contributing to blade flapping

Force/ Moment

Magnitude Moment Arm about Flap Hinge

Force due to blade in x-direction ⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−−++−+Ω−−Ω

)]cos()sin(][)([)()(

)(ψαψαβ

ββζ

yx

2

erherer2r

mdr&&&&

&&&

β)( er −

Force due to absorber in x-direction ⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−−−+++−++Ω−+−Ω+

ryxr

rr2

ra xexah

exaexa2xam

&&&&&&

&&&

)]cos()sin(][)([)()()(

ψαψαβββζ

β)( exa r −+

Force due to blade in z-direction ⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

+Ω

−−−−−

)]sin()cos([

)]cos()sin([)()(

ψαψα

ψαψαβ

yx

yx

r2

rermdr

&&

&&&&&&

)( er −

Force due to absorber in z-direction

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

−+Ω+

−−+−−+−

ryxr

yxr

r

a

xxa2xa

exam

&&&&

&&&&

&&

βψαψα

ψαψαβ

)]sin()cos([)()]cos()sin()[(

)(

)( exa r −+

Aerodynamic Force

βF )( er −

Spring Moment

ββk --

61

To obtain the blade flap equation of motion, the moments are summed

about the flap hinge with flap up taken as positive, which results in the following

equation:

∫

∫

∫

−=+

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−−−+++−++Ω−+−Ω+

−++

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

++Ω+

+−++−+−++

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−−++−+Ω−−Ω

−+

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

+Ω

+−+−−

R

e

ryxr

rr2

rra

ryxr

yxrrra

yx

2R

e

yx

yxR

e

drerFk

xexahexaexa2xa

exam

xxa2xaexa

exam

erherer2r

ermdr

r2rer

ermdr

)(

)]cos()sin(][)([)()()(

)(

)]sin()cos([)()]cos()sin()[()(

)(

)]cos()sin(][)([)()(

))((

)]sin()cos([)]cos()sin([)(

))((

ββ β

ψαψαβββζ

β

βψαψα

ψαψαβ

ψαψαβββζ

β

ψαψα

ψαψαβ

&&&&&&

&&&

&&&&

&&&&&&

&&&&

&&&

&&

&&&&&&

3.6

βIdrermR

e

2 =−∫ )( and βSdrermR

e=−∫ )(

β

ββ I

SeRS

)( −=∗

eRee−

= , eR

hh−

= , eR

aa−

= , and eR

xx rr −=

3.72

m

2a a3

Ieam

αβ

=− )( and a3

Ieam

ma α

β

=− )( (for a uniform blade)

22

Ik

Se1Ω

++= ∗

β

βββν

( )drerFI

1MR

e2Aero ∫ −

Ω= β

ββγ

Using the following definitions:

with ( ) and , where ( ) ( )ψd

d=

∗

( )∗•

Ω= ( ) ( )∗∗••

Ω= 2 and dividing throughout by

yields the non-dimensional form of the flap equation in the rotating coordinate frame:

2I Ωβ

62

Eq. 3.8 is linearized about the trim condition, and the resulting perturbation

equation is transformed to the nonrotating frame using the Multiblade Coordinate

Transformation (MCT). This transformation yields the following cyclic flap equations:

( ) ( )( )

( )( )

( )Aero

yxrm

yx2

m

yx

rmrmm

2m

yx

rmrmm

2m

rm2

m

mrrmm2

m22

m

Mxa32

a31

xa3xa3ea3a3Se1

2

xa3xa3hea3

a3heS1

xa2a312

xa3xa3ea3a3a31

β

β

β

β

γψαψααζ

ψαψαζα

ψαψαααα

α

ψαψαααβα

αβ

βαζβα

βαααανβα

=⎟⎠⎞

⎜⎝⎛ −⎟⎠⎞

⎜⎝⎛ ++

⎟⎠⎞

⎜⎝⎛ ++−

⎟⎠⎞

⎜⎝⎛ +⎥⎥⎦

⎤

⎢⎢⎣

⎡

++

++++

⎟⎠⎞

⎜⎝⎛ −⎥⎥⎦

⎤

⎢⎢⎣

⎡

++++

++++

++−

++++++

∗∗

∗∗∗∗

∗∗∗

∗∗∗∗∗

∗∗∗

∗∗

)cos()sin(

)sin()cos(

)sin()cos(

)cos()sin(

3.8

( ) ( )

( )

( ) ( )[ ][ ] Aero

cxm2

m

y0m2

m0

s1c102

m

c1m2

m2

c1s1c12

m

Mea3a3Se12

hea3a3heS1

a312

ea3a32a31

ββ

β

β

γδααα

αβααβ

ζζβα

βαανβββα

=++++

+++++−

⎟⎠⎞

⎜⎝⎛ ++−

+++⎟⎠⎞

⎜⎝⎛ −++

∗∗

∗∗∗

∗

∗∗∗

3.9

( ) ( )

( )

( ) ( )[ ][ ] Aero

sym2

m

x0m2

m0

c1s102

m

s1m2

m2

s1c1s12

m

Mea3a3Se12

hea3a3heS1

a312

ea3a32a31

ββ

β

β

γδααα

αβααβ

ζζβα

βαανβββα

=++++

++++++

⎟⎠⎞

⎜⎝⎛ −+−

+++⎟⎠⎞

⎜⎝⎛ −−+

∗∗

∗∗∗

∗

∗∗∗

3.10

63

In Eq. 3.9 and Eq. 3.10, 0β is the steady blade coning angle. The cyclic

perturbation aerodynamic flap moments are determined by Eq. 3.11

The perturbation aerodynamic flap moment, βδM , will be determined in section 3.3.6.1.

3.3.2 Lag Equation of Motion

The forces and moments acting on a blade element in the radial and in-plane

directions contribute to the lag equation of motion. These forces and moments are as

follows:

∑=

=bN

1ii

Aeroi

b

Aeroc M

N2M )cos(ψγδγδ ββ

∑=

=bN

1ii

Aeroi

b

Aeros M

N2M )sin(ψγδγδ ββ

3.11

64

To obtain the blade lag equation of motion, the moments are summed about the

lag hinge with lag back taken as positive, which results in the following equation:

Table 3-3: Forces and moments contributing to blade lead-lag motions

Force/Moment Magnitude Moment Arm about Lag Hinge

Force due to blade in x- direction ⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−−++−+Ω−−Ω

)]cos()sin(][)([)()(

)(ψαψαβ

ββζ

yx

2

erherer2r

mdr&&&&

&&&

ζ)( er −

Force due to absorber in x-direction

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

−−+++−−++

Ω−+−Ω+

)]cos()sin(][)([)(

)()(

ψαψαβββ

ζ

yxr

rr

r2

r

a

exahxexa

exa2xa

m&&&&

&&&&

&

ζ)( exa r −+

Force due to blade in y-direction

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

+−+

++−

+Ω−+Ω−−−

)]sin()cos(][)([

)]sin()cos([)(

)()()(

)(

ψαψαβ

ψαψαβ

ββζζ

yx

yx

2

erh

er2

er2erer

mdr&&&&

&&&

&&&

)( er −

Force due to absorber in y-direction

⎪⎪

⎭

⎪⎪

⎬

⎫

⎪⎪

⎩

⎪⎪

⎨

⎧

Ω−

++−+++Ω−+

+Ω−+−−+

rr

yxr

r

2rr

a

x2xexah

exa2

exaexa

m

&&&

&&&&

&

&&

ζ

ψαψαβββ

ζζ

)]sin()cos(][)([)(

)()(

)( exa r −+

Aerodynamic Force

ζF )( er −

Spring Moment ζζk --

Damping Moment

ζζ &c --

65

Using the definitions from the previous section with the following additional

definitions:

and dividing throughout by 2I Ωβ yields the non-dimensional form of the lag equation in

the rotating coordinate frame:

∫

∫

∫

−=++

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

+−++

++−+

+Ω−++Ω−+−−+

−++

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

Ω−

++−+++Ω−++Ω−+−−+

−++

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−−++−+Ω−−Ω

−+

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

+−+

++−

+Ω−+Ω−−−

−

R

e

yxr

yxr

r2

rr

ra

rr

yxr

r2

rr

ra

yx

2R

e

yx

yx

2

R

e

drerFkc

exahexa2

exa2exaexa

exam

x2xexah

exa2exaexaexam

erherer2r

ermdr

erher2

er2erer

ermdr

)(

)]sin()cos(][)([)]sin()cos([)(

)()()(

)(

)]sin()cos(][)([)()()(

)(

)]cos()sin(][)([)()(

))((

)]sin()cos(][)([)]sin()cos([)(

)()()(

))((

ζζζ ζζ

ψαψαβ

ψαψαβ

ββζζ

ζ

ζ

ψαψαβββζζ

ψαψαβββζ

ζ

ψαψαβ

ψαψαβ

ββζζ

&

&&&&

&&&

&&&

&&&

&&&&

&&&

&&&&

&&&

&&&&

&&&

&&&

3.12

22

Ik

SeΩ

+= ∗

β

ζβζν

∫ −Ω

=R

e2Aero drerF

I1M )(ζζ

β

γ 3.13

66

Eq. 3.14 is linearized about the trim condition, and the resulting perturbation

equation is transformed to the nonrotating frame using the MCT. This transformation

yields the following cyclic lag equations:

In Eq. 3.15 and 3.16, 0β is the steady coning blade coning angle. The cyclic

perturbation aerodynamic lag moments are determined by

( ) ( )( )

( )( )( )

( ) Aerorrmm

yx2

m

yx

rmm

2m

2ra2

m

rmrm22

m

Mxx3a32

a31

xh3ha3S

a31

Ic

xS2a312

xa3xea3a31

ζ

β

β

ζ

ζ

γαα

ψαψαβα

ψαψααα

βα

ζζββα

ζαανζα

=+−

⎟⎠⎞

⎜⎝⎛ +++

⎟⎠⎞

⎜⎝⎛ +⎥⎥⎦

⎤

⎢⎢⎣

⎡

++

+++

Ω++++

+−+++

∗

∗∗∗

∗∗∗∗

∗

∗∗∗∗

∗

∗∗

)sin()cos(

)sin()cos(

)(

3.14

( ) ( )

( )

( ) ( )[ ] Aerocs1rc1rmxm0

2m

s1c12s1c12

m0

c1m2

c1s1c12

m

Mxxa6ha3Sa31

Ic

a312

ea32a31

ζβ

β

ζ

ζ

γδαααβα

ζζββαβ

ζανζζζα

=⎟⎠⎞

⎜⎝⎛ +−++++

⎟⎠⎞

⎜⎝⎛ +

Ω+⎟

⎠⎞

⎜⎝⎛ +++

++⎟⎠⎞

⎜⎝⎛ −++

∗∗∗∗

∗∗

∗∗∗

3.15

( ) ( )

( )

( ) ( )[ ] Aerosc1rs1rmym0

2m

c1s12c1s12

m0

s1m2

s1c1s12

m

Mxxa6ha3Sa31

Ic

a312

ea32a31

ζβ

β

ζ

ζ

γδαααβα

ζζββαβ

ζανζζζα

=⎟⎠⎞

⎜⎝⎛ −−++++

⎟⎠⎞

⎜⎝⎛ −

Ω+⎟

⎠⎞

⎜⎝⎛ −++

++⎟⎠⎞

⎜⎝⎛ −−+

∗∗∗∗

∗∗

∗∗∗

3.16

67

The perturbation aerodynamic lag moment, ζδM , will be determined in section 3.3.6.2.

3.3.3 Absorber Equation of Motion

The forces acting on the absorber in the radial direction contribute to the absorber

equation of motion. Since the absorber spring and damper forces act along the blade, the

components of the absorber acceleration in the hub-fixed rotating coordinate system

( RRR zyx ,, ) are resolved into the blade coordinate system ( zyx ,, ). These forces are as

follows:

∑=

=bN

1ii

Aeroi

b

Aeroc M

N2M )cos(ψγδγδ ζζ

∑=

=bN

1ii

Aeroi

b

Aeros M

N2M )sin(ψγδγδ ζζ

3.17

Table 3-4: Forces contributing to absorber radial motion

Force Magnitude

Force in radial direction

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

−−

++Ω+−−Ω−+−Ω+

)]cos()sin()[()]sin()cos([)(

)()(

ψαψαβ

ψαψαβζ

yx

yxr

rr2

r

a

ehxa2

xexa2xam

&&&&

&&

&&&

Spring Force ra xk

Damping Force ra xc &

68

The absorber equation of motion is obtained by summing forces on the absorber

in the radial direction, radialaaradial amF∑ =

Similar to the flap and lag equations, the absorber equation of motion can be non-

dimensionalized by using the following definitions:

where fα is the ratio of the absorber natural frequency to the lag natural frequency as

defined in Chapter 2. Dividing throughout by )( eRm 2a −Ω , the following absorber

equation of motion results:

Eq. 3.20 is linearized about the trim condition, and the resulting perturbation equation is

transformed to the nonrotating frame using the MCT. This transformation yields the

following cyclic absorber equations:

0xkxceh

xa2xexa2xa

m rara

yx

yxr

rr2

r

a =++⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

−−−

+Ω+++Ω−++Ω+−

&

&&&&

&&

&&&

)]cos()sin()[()]sin()cos([)(

)()(

ψαψαβ

ψαψαβζ

3.18

1m

k2

a

a2

2a22

f −Ω

=Ω

=ω

να ζ

Ω=

a

afa m

c2 ζναζ

eRee−

= , eR

hh−

= , eR

aa−

= , and eR

xx rr −=

3.19

axx2eh

a2ea2x

r22

frfayx

yxr

=++⎟⎠⎞

⎜⎝⎛ −−−

⎟⎠⎞

⎜⎝⎛ ++−+

∗∗∗∗∗

∗∗∗∗∗

ζζ ναναζψαψαβ

ψαψαβζ

)cos()sin()(

)sin()cos()( 3.20

69

3.3.4 Body Roll and Pitch Equations of Motion

The body roll perturbation equation of motion is as follows:

and the body pitch perturbation equation of motion is as follows:

The perturbation forces and moments, Yδ , Hδ , xMδ , and yMδ , are obtained by

calculating the blade root shear forces and moments and then summing these forces and

moments over the number of blades. As seen in Eq. 3.23 and Eq. 3.24, there are both

inertial and aerodynamic contributions to the perturbation forces and moments.

3.3.5 Inertial Contributions to Perturbation Forces and Moments

To obtain the inertial contributions to the perturbation forces and moments, the

individual blade root shear forces and moments must first be calculated. Both the blade

0xxx2eh

a2ea2xx2x

c1r22

fs1rc1rfay0

x0s1c1c1rs1rc1r

=+⎟⎠⎞

⎜⎝⎛ ++−+

+⎟⎠⎞

⎜⎝⎛ +−+⎟

⎠⎞

⎜⎝⎛ −+

∗∗∗

∗∗∗∗∗

ζζ ναναζαβ

αβζζ

)(

)( 3.21

0xxx2eh

a2ea2xx2x

s1r22

fc1rs1rfax0

y0c1s1s1rc1rs1r

=+⎟⎠⎞

⎜⎝⎛ −+−−

+⎟⎠⎞

⎜⎝⎛ −−+⎟

⎠⎞

⎜⎝⎛ −−

∗∗∗

∗∗∗∗∗

ζζ ναναζαβ

αβζζ

)(

)( 3.22

AeroAeroxxxxxxxx YhMYhMKCI δδδδααα −+−=++ &&& 3.23

AeroAeroyyyyyyyy HhMHhMKCI δδδδααα −+−=++ &&& 3.24

70

and absorber contribute to the blade root shear forces and moments. All shear forces and

moments must be considered, except the blade root lag moment, which is used to

determine the rotor torque and not required in this analysis. Refer to Figure 3-2 for the

blade root shear forces and moments directions.

The blade root radial shear, rSδ , is obtained by summing the forces on the blade

in the radial direction:

The blade root vertical shear, zSδ , is obtained by summing the forces on the blade

in the vertical direction:

The blade root drag shear, xSδ , is obtained by summing the forces on the blade in

the chordwise direction:

The blade root flap moment, βδM , is similarly obtained:

⎪⎪

⎭

⎪⎪

⎬

⎫

⎪⎪

⎩

⎪⎪

⎨

⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛

−−++−+−Ω−−Ω+

+

⎟⎟⎠

⎞⎜⎜⎝

⎛

−−++−+

Ω−−−+Ω

=∫

)]cos()sin(][)([)()()(

)]cos()sin(][)([)(

)(]cos)([)(

ψαψαβββζ

ψαψαβββ

ζβ

δδ

yx

r2

ra

yx

2R

er

erheaxea2xa

m

erher

er2eremdr

S

&&&&

&&&&&

&&&&&&

&

3.25

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛

−+Ω

−−−−−+

⎟⎟⎠

⎞⎜⎜⎝

⎛

+Ω

−−−−−

=∫

ryx

yxa

yx

yxR

e

z

xa2aea

m

r2rer

mdr

S

&&&&

&&&&&&

&&

&&&&&&

βψαψα

ψαψαβ

ψαψα

ψαψαβ

δδ



)(

3.26

⎪⎪

⎭

⎪⎪

⎬

⎫

⎪⎪

⎩

⎪⎪

⎨

⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛

+−+−Ω−−Ω−+−−

+

⎟⎟⎠

⎞⎜⎜⎝

⎛

+−+−Ω−−Ω−+−−

=∫

)]sin()cos(][)([)()()(

)]sin()cos(][)([)()()(

)(

ψαψαβββζζ

ψαψαβββζζ

δδ

yx

2

a

yx

2R

ex

eahea2eaea

m

erher2erer

mdr

S

&&&&

&&&

&&&&

&&&

3.27

71

Finally, the blade root pitching moment, φδM , is obtained:

Eq. 3.25 - Eq. 3.29 are nondimensionalized using the definitions in sections 3.3.1

- 3.3.3, resulting in the following expressions for the inertial contributions to the

perturbation forces and moments:

⎪⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛

−−+−−−+Ω−+Ω+−

−+

⎟⎟⎠

⎞⎜⎜⎝

⎛

−+Ω

−−−−−−+

⎟⎟⎠

⎞⎜⎜⎝

⎛

−−+−−−Ω−+Ω−

−+

⎟⎟⎠

⎞⎜⎜⎝

⎛

+Ω

−−−−−−

=∫

∫

)]cos()sin(][)([)()()(

)(


)(

)]cos()sin(][)([)()(

)()(


)()(

ψαψαβββζ

β

βψαψα

ψαψαβ

ψαψαβββζ

β

ψαψα

ψαψαβ

δδ β

yx

r2

ra

ryx

yxa

yx

2R

e

yx

yxR

e

erheaxea2xa

eam

xa2aea

eam

erherer2r

ermdr

r2rer

ermdr

M

&&&&

&&&&&

&&&&

&&&&&&

&&&&

&&&

&&

&&&&&&

3.28

⎪⎪

⎭

⎪⎪

⎬

⎫

⎪⎪

⎩

⎪⎪

⎨

⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛

+−+−Ω−−Ω−+−−

−+

⎟⎟⎠

⎞⎜⎜⎝

⎛

+−+−Ω−−Ω−+−−

−

=∫

)]sin()cos(][)([)()()(

)(

)]sin()cos(][)([)()()(

)()(

ψαψαβββζζ

β

ψαψαβββζζ

β

δδ φ

yx

2

a

yx

2R

e

eahea2eaea

eam

erher2erer

ermdr

M

&&&&

&&&

&&&&

&&&

3.29

72

( ) ( )

( )( )

( ) ⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎠⎞

⎜⎝⎛ −⎥⎦

⎤⎢⎣

⎡

+++

+

−++−

+++++

Ω=

∗∗∗∗∗

∗∗

∗∗∗

)cos()sin( ψαψααβα

ααζα

ββααα

δ β

yx

m0

mb

rmrmm

0mmmb

2r

a212Mh

x2x2a212

a21e2a2Me1

SS 3.30

( )

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

⎟⎠⎞

⎜⎝⎛ −⎥⎦

⎤⎢⎣

⎡

+++

−

⎟⎠⎞

⎜⎝⎛ +⎥⎦

⎤⎢⎣

⎡

+++

−

−+−

Ω=

∗∗∗∗∗

∗∗∗

∗∗∗∗

)cos()sin(

)sin()cos(

ψαψααα

ψαψααα

βαβα

δ β

yx

mm

b

yx

mm

b

0mm

2z

e2a2Me1

e2a2Me1

2

x2a21

SS 3.31

( ) ( )( )

( )

( ) ⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

++−

⎟⎠⎞

⎜⎝⎛ +⎥⎦

⎤⎢⎣

⎡

+++

−

++−+

Ω=

∗

∗∗∗∗∗

∗∗∗

ββαα

ψαψααβα

αζαζα

δ β

0mm

yx

m0

mb

rmmm

2x

e2a212

a212Mh

x2a21a21

SS )sin()cos( 3.32

( ) ( )( )

( ) ⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

⎟⎠⎞

⎜⎝⎛ ++++−

⎟⎠⎞

⎜⎝⎛ −⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

++

+++−

++−

+++−+−

Ω=

∗∗∗

∗∗∗∗

∗

∗

∗

∗∗∗

)sin()cos(

)cos()sin(

ψαψααα

ψαψαβαβ

αα

ζβαβα

βααβα

δ

β

β

β

β

ββ

yxm2

m

yx

0m0

m2

m

02

mr0m

m2

m2

m

2

ea3Sea312

ha3Sh

ea3Sea31

a312xa3

ea3Sea31a31

IM 3.33

( ) ( )( ) ⎪

⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎠⎞

⎜⎝⎛ ++−

++++−Ω= ∗∗∗∗

∗

∗∗∗

)sin()cos( ψαψααβ

βαζβαζβαδ

β

βφyxm0

r0m02

m02

m2

a3Sh

xa6a31a31IM 3.34

73

with the following additional definition:

The rotor side force, Yδ , and rotor drag, Hδ , can be determined using the

following equations:

Substituting Eq. 3.30 and Eq. 3.32 into Eq. 3.36 and summing over bN blades

yields the following for the rotor side force and drag:

The rotor roll moment, xMδ , and pitch moment, yMδ , can be determined using

the following equations:

βSMeR

M bb

)( −=∗ 3.35

( )∑=

−=bN

1ii

ixi

ir SSY ψδψδδ cossin

( )∑=

+=bN

1ii

ixi

ir SSH ψδψδδ sincos

3.36

( ) ( )

( ) ( )[ ] ⎪⎪

⎭

⎪⎪

⎬

⎫

⎪⎪

⎩

⎪⎪

⎨

⎧

++++

⎟⎠⎞

⎜⎝⎛ +−+

+++

Ω=

∗∗∗

∗∗

∗∗∗∗

xmb0m

s1rs1rms10m

s10mc1m

b2

2Mha312

xx2a3

a31a31

2N

SY

ααβα

αββα

ββαζα

δ β 3.37

( ) ( )

( ) ( )[ ] ⎪⎪

⎭

⎪⎪

⎬

⎫

⎪⎪

⎩

⎪⎪

⎨

⎧

+++−

⎟⎠⎞

⎜⎝⎛ +−−

+++−

Ω=

∗∗∗

∗∗

∗∗∗∗

ymb0m

c1rc1rmc10m

c10ms1m

b2

2Mha312

xx2a3

a31a31

2NSH

ααβα

αββα

ββαζα

δ β 3.38

( )[ ]∑=

++=bN

1ii

ii

iz

ix MSeMM ψδψδδδ φβ cossin

( )[ ]∑=

++−=bN

1ii

ii

iz

iy MSeMM ψδψδδδ φβ sincos

3.39

74

Substituting Eq. 3.31, Eq. 3.33, and Eq. 3.34 into Eq. 3.39 and summing over

blades yields the following for the rotor inertial roll and pitch moments: bN

( )( )

( )[ ]( )[ ]

( )[ ]( )

( )( ) ⎪

⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

++

++−

+−

++++

+++−

+++−

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++

+++−

Ω=

∗∗∗

∗∗

∗∗

∗∗∗

∗∗

∗∗

∗

∗

s1rmm0

c1rmm0s1r0m

c12

m0

c1m2

m

s1m2

m

ym2

m

x

m0

m2

m

b2x

xa3e2

xa3e22xe2

a31

a3Se2a312

a3Sea31

a3Se2a312

a3Sh2

a3Se2a31

2NIM

ααβ

ααββα

ζαβ

βαα

βαα

ααα

ααβ

αα

δβ

β

β

β

β

β 3.40

( )( )

( )[ ]( )[ ]

( )[ ]( )

( )( ) ⎪

⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

+−

+++

+−

++++

++++

++++

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++

+++−

Ω=

∗∗∗

∗∗

∗∗

∗∗∗

∗∗

∗∗

∗

∗

c1rmm0

s1rmm0c1r0m

s12

m0

s1m2

m

c1m2

m

xm2

m

y

m0

m2

m

b2y

xa3e2

xa3e22xe2

a31

a3Se2a312

a3Sea31

a3Se2a312

a3Sh2

a3Se2a31

2N

IM

ααβ

ααββα

ζαβ

βαα

βαα

ααα

ααβ

αα

δβ

β

β

β

β

β 3.41

Eq. 3.37 and Eq. 3.40 can then be substituted into Eq. 3.23, while Eq. 3.38 and

Eq. 3.41 can be substituted into Eq. 3.24, which are then moved to the left hand side of

the equations. The inertial contributions from the rotor hub forces and moments in the

perturbation body roll equation of motion are as follows:

75

The inertial contributions from the rotor hub forces and moments in the perturbation body

pitch equation of motion are as follows:

( )( )( )

( ) ( )[ ]( )[ ]( ) ( )

( )( )

( )( )[ ]

( ) ( )[ ][ ]( )[ ] ⎪

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎨

⎧

++−

+++++

+−

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+

++−

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

++

++

+++

+

−+

++++

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++

++

Ω=−

∗∗

∗∗

∗∗∗

∗

∗

∗∗

∗

∗

∗∗

∗∗∗∗

∗∗∗

∗∗

∗

s1rm0bm0

s1m0

ym2

m

c1rmbm0

c1m

2m

x

m0

m

mb22

m

s1rbm0

c1m2

m0

s1m0

2m

b2x

xa6SMhe

ah3a3Se4a312

xa3SMe2

a3Se2

a312

a3Sh4

a3Se2

1SMh2a31xSMhe

a3Sha31

a3She

a31

2NIYhM

αβαβ

βαβααα

ααβ

βα

α

α

αβ

α

αααβ

ζααβ

βαβ

α

δδ

β

β

β

β

β

β

β

β

β

β

β

3.42

( )( )( )( ) ( )[ ]

( )[ ]( ) ( )

( )( )

( ) ( )[ ][ ]

( ) ( )[ ][ ]( )[ ] ⎪

⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎨

⎧

+++

−+++−

+−

+++−

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+

+++

+++

+

−+

++++

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++

++−

Ω=−

∗∗

∗∗

∗∗∗

∗∗

∗∗

∗

∗

∗∗

∗∗∗∗

∗∗∗

∗∗

∗

c1rm0bm0

c1m0

xm2

m

s1rmbm0

s1m2

m

y

m0

m

mb22

m

c1rbm0

s1m2

m0

c1m0

2m

b2y

xa6SMhe

ah3a3Se4a312

xa3SMe2

a3Se2a312

a3Sh4

a3Se2

1SMh2a31xSMhe

a3Sha31

a3She

a31

2N

IHhM

αβαβ

βαβααα

ααβ

βαα

α

αβ

α

αααβ

ζααβ

βαβ

α

δδ

β

β

β

β

β

β

β

β

β

β

β 3.43

76

The aerodynamic contributions must next be calculated to complete all equations

of motion.

3.3.6 Aerodynamic Contributions to Perturbation Forces and Moments

There are perturbation aerodynamic moments in all equations of motion, with the

exception of the absorber equation of motion. The steady and perturbation values of the

perpendicular and tangential velocity components are required to calculate the

perturbation aerodynamic moments. For axisymmetric flow (ground or hover

conditions), the velocity components are as follows:

These steady and perturbation velocity components are used to calculate the steady and

perturbation lift and drag:

xR

uT =Ω

3.44

λ=ΩRuP 3.45

( ) ⎟⎠⎞

⎜⎝⎛ ++−−=

Ω

∗∗∗

ψαψαβζδ sincos yx0T xhxR

u 3.46

( ) ⎟⎠⎞

⎜⎝⎛ −++=

Ω

∗∗∗

ψαψαββδ cossin yx0P xhxR

u 3.47

[ ]TP2

Tz uuuca21F −= θρ 3.48

⎥⎦

⎤⎢⎣

⎡ −+= 2PTP

d2Tx uuu

aC

uca21F θρ 3.49

( ) ( ) ( )[ ]2TTPPTTz uuuuu2uca

21F δθδθδρδ +−+−= 3.50

( ) ( )⎥⎦

⎤⎢⎣

⎡−−+⎟

⎠⎞

⎜⎝⎛ += TPPTPP

dTTx uuu2uuu

aCu2uca

21F δθθδθδρδ 3.51

77

3.3.6.1 Perturbation Aerodynamic Flap Moment

The perturbation aerodynamic flap moment used in Eq. 3.11 is calculated as

follows:

For this analysis, the perturbation blade pitch, δθ , is considered to be zero.

Introducing Eq. 3.44 - Eq. 3.47 into Eq. 3.52 and evaluating the integral, the

perturbation aerodynamic flap moment is:

The terms on the right hand side of Eq. 3.9 and Eq. 3.10 can be determined using

Eq. 3.11 and Eq. 3.53 as follows:

( ) ( ) ( )[ ]

∫

∫

∫∫

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎟⎠⎞

⎜⎝⎛Ω

+⎟⎠⎞

⎜⎝⎛

Ω−

Ω

+⎟⎠⎞

⎜⎝⎛

Ω−

ΩΩ=

+−+−Ω

=

Ω≈−

Ω=

1

0 2TTP

PTT

R

0

2TTPPTT2

R

0 z2

R

e z2Aero

xdx

Ru

Ru

Ru

Ru

Ru2

Ru

2

rdruuuuu2uca21

I1

rdrFI

1drerFI

1M

δθδ

θδ

γ

δθδθδρ

δδγδ

β

βββ

)(

3.52

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

⎟⎠⎞

⎜⎝⎛ −⎟⎟⎠

⎞⎜⎜⎝

⎛++

⎟⎠⎞

⎜⎝⎛ +⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −+

+⎟⎠⎞

⎜⎝⎛ −

−=

∗∗

∗∗

∗∗

ψαψαβ

ψαψαλθ

βλθ

βζλθ

γγδβ

cossin

sincos43

4

yx0

yx0Aero

6h

81

64h

81

6

M 3.53

78

3.3.6.2 Perturbation Aerodynamic Lag Moment

The perturbation aerodynamic lag moment used in Eq. 3.17 is calculated as

follows:

Like in the equation for the flap moment, the perturbation blade pitch, δθ , is considered

to be zero in the lag moment equation.

Introducing Eq. 3.44 - Eq. 3.47 into Eq. 3.56 and evaluating the integral, the

perturbation aerodynamic lag moment is:

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛+−⎟

⎠⎞

⎜⎝⎛ ++

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ +⎟⎠⎞

⎜⎝⎛ −

−=∗∗

∗∗

y0

s1c1

x0s1c1Aero

6h

81

81

64h

6M

c

αβ

ββ

αλθβλθζζλθ

γγδβ

434 3.54

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟

⎠⎞

⎜⎝⎛ −+

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −⎟⎠⎞

⎜⎝⎛ −

−=∗∗

∗∗

x0

c1s1

y0c1s1Aero

6h

81

81

64h

6M

s

αβββ

αλθβλθζζλθ

γγδβ

434 3.55

( ) ( )

∫

∫

∫∫

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎟⎠⎞

⎜⎝⎛

ΩΩ+⎟

⎠⎞

⎜⎝⎛

Ω−

ΩΩ+

⎟⎠⎞

⎜⎝⎛

Ω+

ΩΩ=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+−

+⎟⎠⎞

⎜⎝⎛ +

Ω=

Ω≈−

Ω=

1

0TPPTP

PdTT

R

0

TPPTP

Pd

TT2

R

0 x2

R

e x2Aero

xdx

Ru

Ru

Ru2

Ru

Ru

Ru

aC

Ru2

Ru

2

rdruuu2uu

ua

Cu2uca

21

I1

rdrFI

1drerFI

1M

δθθδ

θδγ

δθθδ

θδρ

δδγδ

β

ββζ )(

3.56

79

The terms on the right hand side of Eq. 3.15 and Eq. 3.16 can be determined using

Eq. 3.17 and Eq. 3.57:

3.3.6.3 Perturbation Fuselage Aerodynamic Roll and Pitch Moments

As with the inertial contributions to the fuselage roll and pitch moments, to

calculate the fuselage aerodynamic roll and pitch moments (very last terms on the right

hand side of Eq. 3.23 and Eq. 3.24), the perturbation aerodynamic blade root shear forces

and moments must first be calculated. The blade root perturbation flap and lag moments

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

⎟⎠⎞

⎜⎝⎛ −⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −+

⎟⎠⎞

⎜⎝⎛ +⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ ++⎟

⎠⎞

⎜⎝⎛ +−

⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ +−

=

∗∗

∗∗

∗∗

ψαψαλθβλθ

ψαψαλθβλθ

βλθζλθ

γγδζ

cossin

sincos43

4

yx0

yxd

0d

d

Aero

26h

38

6aC1

4aC1h

386aC1

M 3.57

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ +⎟⎠⎞

⎜⎝⎛ −−

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ ++⎟

⎠⎞

⎜⎝⎛ ++

⎟⎠⎞

⎜⎝⎛ +⎟⎠⎞

⎜⎝⎛ +

−=

∗∗

∗

∗

y0s1c1

xd

0d

s1c1d

Aero

26h

3838

6aC1

4aC1h

6aC1

Mc

αλθβλθββλθ

αλθβλθ

ζζλθ

γγδζ 43

4

3.58

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −−⎟

⎠⎞

⎜⎝⎛ −⎟⎠⎞

⎜⎝⎛ −−

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ ++⎟

⎠⎞

⎜⎝⎛ ++

⎟⎠⎞

⎜⎝⎛ −⎟⎠⎞

⎜⎝⎛ +

−=

∗∗

∗

∗

x0c1s1

yd

0d

c1s1d

Aero

26h

3838

6aC1

4aC1h

6aC1

Ms

αλθβλθββλθ

αλθβλθ

ζζλθ

γγδζ 43

4

3.59

80

were calculated in sections 3.3.6.1 and 3.3.6.2. The three perturbation blade root shear

forces and pitching moment are calculated below.

The perturbation blade root drag shear, AeroxSδ , is calculated as follows:

Substituting Eq. 3.44 - Eq. 3.47 into Eq. 3.60 and evaluating the integral yields the

following equation for the perturbation blade root drag shear:

The perturbation blade root radial shear, AerorSδ , is calculated as follows:

( ) ( )

∫

∫ ∫

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎟⎠⎞

⎜⎝⎛

ΩΩ+⎟

⎠⎞

⎜⎝⎛

Ω−

ΩΩ+

⎟⎠⎞

⎜⎝⎛

Ω+

ΩΩ⎟⎟⎠

⎞⎜⎜⎝

⎛ Ω=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+−

+⎟⎠⎞

⎜⎝⎛ +

==

1

0TPPTP

PdTT2

R

0

R

0

TPPTP

Pd

TTx

Aerox

dx

Ru

Ru

Ru2

Ru

Ru

Ru

aC

Ru2

Ru

RI

2

druuu2uu

ua

Cu2u

ca21drFS

δθθδ

θδγ

δθθδ

θδρδδ

β

3.60

⎪⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

⎟⎠⎞

⎜⎝⎛ −⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −+

⎟⎠⎞

⎜⎝⎛ +

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎟⎠⎞

⎜⎝⎛ ++

⎟⎠⎞

⎜⎝⎛ +

−

⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ +−

⎟⎟⎠

⎞⎜⎜⎝

⎛ Ω=

∗∗

∗∗

∗∗

ψαψαλθβλθ

ψαψαλθβ

λθ

βλθζλθ

γδ β

cossin

sincos

3

3

yx0

yx

d0

d

d

2Aerox

22

h3

2aC2

aC

h

32aC2

RI

2S 3.61

81


following equation for the perturbation blade root radial shear:

The perturbation blade root vertical shear, AerozSδ , is calculated as follows:


following equation for the perturbation blade root vertical shear:

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡

ΩΩ−⎟

⎠⎞

⎜⎝⎛Ω

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎟⎠⎞

⎜⎝⎛Ω

+⎟⎠⎞

⎜⎝⎛

Ω−

Ω+

⎟⎠⎞

⎜⎝⎛

Ω−

ΩΩ

⎟⎟⎠

⎞⎜⎜⎝

⎛ Ω−=

−−=

∫

∫

∫ ∫

1

0TP

2T

1

0 2TTP

PTT

02

R

0

R

0 zz0Aeror

dxR

uR

uR

uca21

dx

Ru

Ru

Ru

Ru

Ru2

Ru

RI

2

drFdrFS

θρβ

δθδ

θδ

βγ

βδβδ

β 3.62

( )

⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

⎟⎠⎞

⎜⎝⎛ +⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎠⎞

⎜⎝⎛ −+

−+

⎟⎠⎞

⎜⎝⎛ −⎟⎠⎞

⎜⎝⎛ ++

⎟⎠⎞

⎜⎝⎛ −−+⎟

⎠⎞

⎜⎝⎛ −

⎟⎟⎠

⎞⎜⎜⎝

⎛ Ω−=

∗∗

∗∗

∗∗

ψαψαλθβ

λθβ

ψαψαββ

βλθββζλθβ

γδ β

sincos

cossin

33

yx

00

yx00

00

2Aeror

232

h

h21

31

231

22

RI

2S 3.63

( ) ( ) ( )[ ]

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎟⎠⎞

⎜⎝⎛Ω

+⎟⎠⎞

⎜⎝⎛

Ω−

Ω+

⎟⎠⎞

⎜⎝⎛

Ω−

ΩΩ⎟⎟⎠

⎞⎜⎜⎝

⎛ Ω=

+−+−==

∫

∫∫

1

0 2TTP

PTT2

R

0

2TTPPTT

R

0 zAeroz

dx

Ru

Ru

Ru

Ru

Ru2

Ru

RI

2

druuuuu2uca21drFS

δθδ

θδ

γ

δθδθδρδδ

β 3.64

82

The perturbation blade root pitching moment, AeroMφδ , is calculated as follows:


following equation for the perturbation blade root pitching moment:

( )⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

⎟⎠⎞

⎜⎝⎛ +⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+−−

⎟⎠⎞

⎜⎝⎛ −⎟⎠⎞

⎜⎝⎛ +−

+⎟⎠⎞

⎜⎝⎛ −−

⎟⎟⎠

⎞⎜⎜⎝

⎛ Ω=

∗∗

∗∗

∗∗

ψαψαλθβλθ

ψαψαβ

βζλθ

γδ β

sincos

cossin

3

yx0

yx0

2Aeroz

232h

h21

31

31

22

RI

2S 3.65

⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

ΩΩ+

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛Ω

−ΩΩ

+⎟⎠⎞

⎜⎝⎛Ω+

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎟⎠⎞

⎜⎝⎛

ΩΩ+⎟

⎠⎞

⎜⎝⎛

Ω−

ΩΩ+

⎟⎠⎞

⎜⎝⎛

Ω+

ΩΩ

Ω=

ΩΩΩ++=

∫∫

∫

∫∫∫

1

01

0TTm

2PTPd

2T

1

0TPPTP

PdTT

0

2

1

0TTm2R

0 x

R

0 x0Aero

dxR

uR

ua

Cc2

xdxR

uR

uR

ua

CR

u

xdx

Ru

Ru

Ru2

Ru

Ru

Ru

aC

Ru2

Ru

I2

dxR

uR

ua

CcIdrrFdrFrM

δ

θβ

δθθδ

θδ

β

γ

δγβδβδ

β

βφ

3.66

83

The rotor aerodynamic side force, AeroYδ , and drag force, AeroHδ , can be

determined using the following equations:

As seen in the above equations, the steady terms for AerorS and Aero

xS are also required.

They are calculated as follows:

⎪⎪⎪⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪⎪⎪

⎨

⎧

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ +⎟⎟⎠

⎞⎜⎜⎝

⎛++−

⎥⎦

⎤⎢⎣

⎡−++

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎟⎠⎞

⎜⎝⎛ −⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −+

⎟⎠⎞

⎜⎝⎛ +

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎟⎠⎞

⎜⎝⎛ ++

⎟⎠⎞

⎜⎝⎛ +

−

⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −−

Ω=

∗∗∗

∗∗

∗∗

∗∗

ψαψαβ

ζ

λλθβ

ψαψαλθβλθ

ψαψαλθ

β

λθ

βλθζλθ

β

γδ βφ

sincos

8

cossin

sincos

4

3

84

yx0m

2d

yx0

yx

d0

d

d

0

2Aero

32h

31

aCc

46aC1

26h

38

6aC1

4aC1h

36aC1

IM 3.67

( ) ( )( )

∑

∑

=

=

⎟⎟⎠

⎞⎜⎜⎝

⎛

−−

−=

−−−=

b

b

N

1i iiiAero

xiiiAero

r

iiAero

xiiAero

r

N

1iii

iAeroxii

iAeror

Aero

SS

SS

SSY

ζψζψ

ψδψδ

ζψζψδδ

sincos

cossin

cossin

,,

,,

,,

3.68

( ) ( )( )

∑

∑

=

=

⎟⎟⎠

⎞⎜⎜⎝

⎛

−+

+=

−+−=

b

b

N

1i iiiAero

xiiiAero

r

iiAero

xiiAero

r

N

1iii

iAeroxii

iAeror

Aero

SS

SS

SSH

ζψζψ

ψδψδ

ζψζψδδ

cossin

sincos

sincos

,,

,,

,,

3.69

⎟⎠⎞

⎜⎝⎛ −+

Ω=

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛Ω

−ΩΩ

+⎟⎠⎞

⎜⎝⎛Ω

Ω== ∫∫

2d2

1

0

2PTPd

2T

2R

0 xAerox

2aC

31

RI

2

dxR

uR

uR

ua

CR

uR

I2

drFS

λθλγ

γ

β

β

3.70

84

Substituting Eq. 3.61, Eq. 3.63, Eq. 3.70, and Eq. 3.71 into Eq. 3.68 and Eq. 3.69 and

summing over bN blades yields the following for the rotor aerodynamic side and drag

forces:

⎟⎠⎞

⎜⎝⎛ −

Ω−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

ΩΩ−⎟

⎠⎞

⎜⎝⎛Ω

Ω−=−= ∫∫

23RI

2

dxR

uR

uR

uR

I2

drFS

0

2

1

0TP

2T

R

0 0

2

z0Aeror

λθβγ

θβγβ

β

β

3.71

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎨

⎧

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎟⎠⎞

⎜⎝⎛ +++

⎟⎟⎠

⎞⎜⎜⎝

⎛++

+

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −+

⎟⎠⎞

⎜⎝⎛ −

+

⎟⎠⎞

⎜⎝⎛ +⎟⎠⎞

⎜⎝⎛ −−

⎟⎠⎞

⎜⎝⎛ −−⎟

⎠⎞

⎜⎝⎛ −+

⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −⎟⎠⎞

⎜⎝⎛ ++

⎟⎠⎞

⎜⎝⎛ −+−⎟

⎠⎞

⎜⎝⎛ −⎟⎠⎞

⎜⎝⎛ −

Ω=

∗

∗

∗

∗

∗

∗

x

d0

20d

y2

0

0

s1c1

s1c1s10

s10s1c1d

c12d

c1s10

b2

Aero

2aC

32

31

2aCh

3232

32

3h

3

233

232aC

32

2aC

31

232

2N

RI

2Y

αθλβ

βθλ

αλθλθβ

λθβ

ββλθ

βλθβββ

ζλθβζζθλ

ζλθλζζλθβ

γδ β 3.72

85

The rotor aerodynamic roll moment, Aerox

Mδ , and pitch moment, Aeroy

Mδ , can be

determined using the following equations:

As seen in the above equations, the steady terms for AerozS , AeroMβ , and AeroMφ are also

required. They are calculated as follows:

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎨

⎧

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎟⎠⎞

⎜⎝⎛ +++

⎟⎟⎠

⎞⎜⎜⎝

⎛++

−

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −+

⎟⎠⎞

⎜⎝⎛ −⎟⎠⎞

⎜⎝⎛ −+

⎟⎠⎞

⎜⎝⎛ −−⎟

⎠⎞

⎜⎝⎛ ++

⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −⎟⎠⎞

⎜⎝⎛ +−

⎟⎠⎞

⎜⎝⎛ −++⎟

⎠⎞

⎜⎝⎛ +⎟⎠⎞

⎜⎝⎛ −

Ω=

∗

∗

∗

∗

∗

∗

y

d0

20d

x00

c1s1

c1s1c10

c10c1s1d

s12d

s1c10

b2

Aero

2aC

32

31

2aC

h

32323

23h

3

233

232aC

32

2aC

31

232

2N

RI

2H

αθλβ

βθλ

αλθλθβλθβ

ββλθ

βλθβββ

ζλθβζζθλ

ζλθλζζλθβ

γδ β 3.73

( ) ( ) ( )[ ]( )( )∑

∑

=

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++−

++=

−+−+=

b

b

x

N

1i iiiAero

iiiAero

ziAero

iiAero

iiAero

ziAero

N

1iii

iAeroii

iAeroz

iAeroAero

MeSM

MSeM

MeSMM

ζψζψ

ψδψδδ

ζψζψδδ

φβ

φβ

φβ

sincos

cossin

cossin

,,,

,,,

,,,

3.74

( ) ( ) ( )[ ]( )( )∑

∑

=

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−+−

++−=

−+−+−=

b

b

y

N

1i iiiAero

iiiAero

ziAero

iiAero

iiAero

ziAero

N

1iii

iAeroii

iAeroz

iAeroAero

MeSM

MSeM

MeSMM

ζψζψ

ψδψδδ

ζψζψδδ

φβ

φβ

φβ

cossin

sincos

sincos

,,,

,,,

,,,

3.75

86

Substituting Eq. 3.53, Eq. 3.65, Eq. 3.66, Eq. 3.76, Eq. 3.77, and Eq. 3.78 into Eq. 3.74

and Eq. 3.75 summing over bN blades yields the following for the rotor aerodynamic roll

and pitch moments:

⎟⎠⎞

⎜⎝⎛ −

Ω=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

ΩΩ−⎟

⎠⎞

⎜⎝⎛Ω

Ω== ∫∫

23RI

2

dxR

uR

uR

uR

I2

drFS

2

1

0TP

2T

R

0

2

zAeroz

λθγ

θγ

β

β

3.76

⎟⎠⎞

⎜⎝⎛ −

Ω=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

ΩΩ−⎟

⎠⎞

⎜⎝⎛Ω

Ω== ∫∫

34RI

2

xdxR

uR

uR

uI2

drrFM

2

1

0TP

2T

R

0

2z

Aero

λθγ

θγ

β

ββ

3.77

⎥⎦

⎤⎢⎣

⎡+⎟⎟

⎠

⎞⎜⎜⎝

⎛−+Ω=

⎟⎠⎞

⎜⎝⎛Ω

Ω+

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛Ω

−ΩΩ

+⎟⎠⎞

⎜⎝⎛Ω

Ω=

⎟⎠⎞

⎜⎝⎛Ω

Ω+=

∫

∫

∫∫

aCc

31

46aC

81I

dxR

ua

CcI

xdxR

uR

uR

ua

CR

uI2

dxR

ua

CcIdrrFM

m2

d2

1

0

2Tm2

1

0

2PTPd

2T2

1

0

2Tm2R

0 x0Aero

λθλγ

γ

γ

γβ

β

β

β

βφ

3.78

87

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎨

⎧

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −−

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −−

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎟⎠⎞

⎜⎝⎛ +++

⎟⎠⎞

⎜⎝⎛ ++++++

−

⎟⎟⎠

⎞⎜⎜⎝

⎛−++

⎟⎠⎞

⎜⎝⎛ +⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −⎟⎠⎞

⎜⎝⎛ +−

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛++−

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ ++−

⎟⎠⎞

⎜⎝⎛ ++⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −−

Ω=

∗

∗

∗

∗∗

∗

∗

y0

y2

0

x

00dm

0

d0

m

c1

2d

s1c10c1s1

s1

2d

0m

c1d

0m

c1s1

b2Aero

43e

283

2622e

43h

6aC

41

aC

3c

64e

aC

31

61h

aC

2ch

6e

81

46aC

81

386e

81

4aC

81

aC

6c

6aC

41

aC

3c

6e

843e

64

2N

IMx

αλθλθβ

αλθβλθλθ

αβθλββ

θλβ

βλθλ

ββλθβββ

ζλβ

ζθλβ

ζθθζλθλθ

γδ β

3.79

88

Eq. 3.72, Eq. 3.73, Eq. 3.79, and Eq. 3.80 can then be substituted into Eq. 3.23 and

Eq. 3.24 and moved to the left hand side of the equations. The aerodynamic

contributions from the rotor hub forces and moments in the perturbation body roll and

pitch equations of motion are as follows:

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎨

⎧

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −+

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −+

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎟⎠⎞

⎜⎝⎛ +++

⎟⎠⎞

⎜⎝⎛ ++++++

−

⎟⎟⎠

⎞⎜⎜⎝

⎛−++

⎟⎠⎞

⎜⎝⎛ −⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ +⎟⎠⎞

⎜⎝⎛ ++

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+++

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ ++−

⎟⎠⎞

⎜⎝⎛ ++⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −

Ω=

∗

∗

∗

∗∗

∗

∗

x0

x2

0

y

00dm

0

d0

m

s1

2d

c1s10s1c1

c1

2d

0m

s1d

0m

s1c1

b2Aeroy

43e

283

2622e

43h

6aC

41

aC

3c

64e

aC

31

61h

aC

2ch

6e

81

46aC

81

386e

81

4aC

81

aC

6c

6aC

41

aC

3c

6e

843e

64

2N

IM

αλθλθβ

αλθβλθλθ

αβθλββ

θλβ

βλθλ

ββλθβββ

ζλβ

ζθλβ

ζθθζλθλθ

γδ β

3.80

89

( )

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎨

⎧

⎥⎦

⎤⎢⎣

⎡−++++−

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −−

⎥⎦⎤

⎢⎣⎡ ++−

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

+⎟⎟⎠

⎞⎜⎜⎝

⎛++

⎟⎟⎠

⎞⎜⎜⎝

⎛+

+

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −+

⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −+

⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −

+

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛+++

⎟⎠⎞

⎜⎝⎛ ++⎟

⎠⎞

⎜⎝⎛ ++

⎟⎠⎞

⎜⎝⎛ ++++⎟

⎠⎞

⎜⎝⎛ +

+

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −−

⎥⎦⎤

⎢⎣⎡ +++

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+⎟⎠⎞

⎜⎝⎛ ++

⎟⎠⎞

⎜⎝⎛ +

+

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −

=Ω

−

∗

∗

∗

∗

∗

∗

c1

2

0d

s10

c10

s1m

2d

0

2d

y

200

2

00

x

0m20

2

d2d20

d0

c10

s10

c1md

0

d

s10

b2

AeroAero

4h

61

6aC

81

6e

81

3843

3h

6h

6e

8

aC

6c

4aC

81

2aC

61h

43

2h

23

43h

43e

283

22he

43

2h

32h

aCch

41

2aC

21h

6aC

41

2aC

32

4e

31h

6e

81

3826h

h61

6e

81

aC

3c

6aC

41

4aC

31h

43h

43e

64

NIYhM2

x

βλβθλ

βλθβλθ

ζθβθθ

ζλβ

λ

α

λθβλθβ

λθβλθβ

λθλθ

α

ββ

θλθλβ

θλβ

βλθβλθ

ββ

ζθλβ

θλ

ζλθβλθλθ

γδδ

β

3.81

90

( )

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

⎨

⎧

⎥⎦

⎤⎢⎣

⎡−++++−

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −+

⎥⎦⎤

⎢⎣⎡ ++−

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

+⎟⎟⎠

⎞⎜⎜⎝

⎛++

⎟⎟⎠

⎞⎜⎜⎝

⎛+

−

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −+

⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −+

⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −

−

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛+++

⎟⎠⎞

⎜⎝⎛ ++⎟

⎠⎞

⎜⎝⎛ ++

⎟⎠⎞

⎜⎝⎛ ++++⎟

⎠⎞

⎜⎝⎛ +

+

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −−

⎥⎦⎤

⎢⎣⎡ ++−

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+⎟⎠⎞

⎜⎝⎛ ++

⎟⎠⎞

⎜⎝⎛ +

+

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −−

=Ω

−

∗

∗

∗

∗

∗

∗

s1

2

0d

c10

s10

c1m

2d

0

2d

x

200

2

00

y

0m20

2

d2d20

d0

s10

c10

s1md

0

d

c10

b2

AeroAero

4h

61

6aC

81

6e

81

3843

3h

6h

6e

8

aC

6c

4aC

81

2aC

61h

43

2h

23

43h

43e

283

22he

43

2h

32h

aCch

41

2aC

21h

6aC

41

2aC

32

4e

31h

6e

81

3826h

h61

6e

81

aC

3c

6aC

41

4aC

31h

43h

43e

64

NI

HhM2y

βλβθλ

βλθβλθ

ζθβθθ

ζλβ

λ

α

λθβλθβ

λθβλθβ

λθλθ

α

ββ

θλθλβ

θλβ

βλθβλθ

ββ

ζθλβ

θλ

ζλθβλθλθ

γ

δ

β

3.82

91

Finally, the perturbation equations of motion in the nonrotating frame of reference

can be expressed as:

where [ ]M , [ ]C , and [ ]K are the 8x8 mass, damping, and stiffness matrices, and

{ } [ ]yxs1rc1rs1c1s1c1T xxq ααζζββ= . The damping matrix has contributions from

both inertial and aerodynamic terms, and the stiffness matrix has contributions from

inertial, aerodynamic, and elastic terms. The terms of each of the matrices are listed in

Appendix A.

3.4 Complex Eigenvalue Analysis

The decay rates and modal frequencies for the regressing lag and absorber modes,

as well as the fuselage pitch and roll modes are determined by evaluating the eigenvalues

of the system, and this is done over a range of rotational speeds. As with the two-degree-

of-freedom system, the system of equations in Eq. 3.83 are in second order form and

must be put in first order form to calculate the eigenvalues, which are used to determine

the aeromechanical stability characteristics of the system. The eigenvalues are complex;

the modal decay rates are determined from the real part of the eigenvalues. If the real

part of any eigenvalue is positive, that part of the system is unstable.

[ ] [ ] [ ] { }0qKqCqM =⎭⎬⎫

⎩⎨⎧+

⎭⎬⎫

⎩⎨⎧+

⎭⎬⎫

⎩⎨⎧ ∗∗∗

3.83

92

3.5 Results

The baseline rotor/support properties used in the simulations are from the model

tested by the US Army Aeroflightdynamics Directorate at Ames in 1980 [55]. See

Table C-1 for rotor data. The range of absorber properties examined is given in Table 3-

5. Note that the absorber damping ratios used do not include 70a .=ζ as was the case

for the two-degree-of-freedom system. This is due to the fact that the damping

transferred to the lag mode at this high value of absorber damping is small unless a larger

absorber mass is used; therefore, this value will not be considered in this analysis.

Since the results for the undamped baseline system were generated for the present

analysis from derived equations of motion (without the absorber), they are compared with

results taken from [55] in Figures 3-3a (modal frequencies) and 3-3b (modal decay rates).

In both figures, the data from [55] are plotted in dashed lines (theoretical) and asterisks

(experimental), while baseline results from the present analysis calculated for comparison

with damped results are plotted in solid lines. As can be seen from the figures, there is

generally excellent agreement with Bousman’s results. The one exception is that of the

decay rate for the roll mode (plotted in green), which is slightly higher than the

Table 3-5: Absorber parameter values used in ground resonance simulations

Absorber Parameter Values Considered

a 0.3, 0.5, 0.7 aζ 0.3, 0.5

mα .01, .03, .05

93

experimental data, but it is still in good agreement with Bousman’s theoretical result for

that mode.

For a aζ value of 0.3, and an inboard absorber location of 30a .= , Figures 3-4a -

3-6a show the modal frequencies and Figures 3-4b - 3-6b show the corresponding modal

decay rates of the important modes as a function of rotor speed, for absorber mass ratios,

mα , of 0.01, 0.03, and 0.05, respectively. For all modal decay rate figures, the decay rate

of the absorber mode is significantly larger than the other three modes and is not shown.

Included on all figures (plotted in black) are the results for the undamped baseline system

(with no absorber present) for comparison. The baseline system shows an instability near

a rotational speed of 750 RPM where the regressing lag mode coalesces with the body

roll mode. Since the absorber frequency, expressed as 2

a

aa m

kΩ−=ω , and rotor lag

frequency, expressed as 220 I

Se Ω+=

ζ

ζζζ ωω , are functions of the rotational speed, Ω ,

the frequency ratio, fα , is also a function of Ω . Thus,

Consequently, the absorber frequency has to be tuned to the lag frequency ( 1f =α ) at a

selected value of Ω , and this tuning is done at 750 RPM (the rotational speed where the

regressing lag mode couples with the body roll mode for the baseline system). At this

rotor speed, maximum lag damping augmentation is observed in Figures 3-4b - 3-6b.

220

2

a

a

af

IS

e

mk

Ω+

Ω−==

ζ

ζζ

ζ ωωω

α 3.84

94

However, increases in lag damping, relative to the baseline, are observed over a range of

rotational speeds. Rotational speeds of 540 RPM to 830 RPM correspond to an fα range

of 1.5 – 0.5, which was the frequency ratio range examined in Chapter 2. Additionally,

smaller increases in lag damping relative to the baseline occur beyond this range, as the

lag damping does not go to zero beyond fα of 1.5 or 0.5.

Figures 3-7 - 3-9 and Figures 3-10 - 3-12 show similar results for mid-span and

outboard absorber locations, 50a .= and 70a .= , respectively. Comparing the modal

decay rate results in Figures 3-4 - 3-12 it can be observed that the greatest increases in lag

damping (and improvements in aeromechanical stability characteristics) are generally

realized for larger values of mα (compare Figures 3-4b - 3-6b), and larger values of a

(compare Figure 3-4b, to Figure 3-7b or Figure 3-10b).

For 30a .= , the variations of the regressing absorber mode frequency as a

function of RPM are seen in Figure 3-4a - 3-6a. The regressing absorber frequency

crosses the regressing lag frequency near 750 RPM (the rotor speed at which the absorber

was tuned), but shows little interaction with the other modes elsewhere. For larger a or

mα (Figures 3-8a - 3-12a), however, the regressing absorber and lag modes strongly

couple over a large rotational speed range and change the overall aeromechanical stability

characteristics.

Figures 3-13 - 3-21 show modal frequencies and damping results for a higher

damped absorber ( 50a .=ζ ). Comparing the modal decay rates for different absorber

locations and mass ratios, it is once again observed that the larger augmentation in

regressing lag mode damping is achieved when the absorber is moved outboard and/or

95

the mass ratio is increased. Comparing the modal frequencies for different absorber

locations and mass ratios, it is again seen that for the inboard absorber location the

regressing absorber mode frequency only crosses the regressing lag mode frequency at

the tuning RPM, but for mid-span or outboard locations, or as the mass ratio is increased,

the absorber and lag regressing modes are highly coupled. There is only one combination

of absorber parameters where the amount of damping that is transferred to the lag mode

may be insufficient in completely alleviating the instability that was present in the

baseline case (refer to Figure 3-13b). This could be predicted by the amount of damping

available in the lag mode from the two-degree-of-freedom analysis in Figure 2-7.

3.6 Summary

From the results presented in this chapter, it is clearly shown that the radial

vibration absorber has a significant influence on helicopter aeromechanical stability. For

the example rotor used in the simulations, for all but one combination of absorber

parameters considered, the instability in the regressing lag mode was completely

eliminated, and in most cases, the stability margins are significant. For the remaining

case, the instability was reduced, but not completely eliminated. These results indicate

that not only is a radial absorber able to transfer damping to the lag mode via Coriolis

coupling, the damping transferred is also sufficient to prevent ground resonance.

96

XH

YH

ZH

xR

zR

ζ

β x

e

z β

ψ

XH

YH

ZH

xR

zR

ζ

β x

e

z β

XH

YH

ZH

xR

zR

ζζ

ββ x

e

z ββ

ψψ

Figure 3-1: Coordinate systems used in ground resonance analysis

ψ

Ω

z

y

x

Fz

My

Mz

Fy

FxMx

Mβ

Mζ

Mφ

Sz

SrSx

ψ

Ω

z

y

x

Fz

My

Mz

Fy

FxMx

Mβ

Mζ

Mφ

Sz

SrSx

ψψ

ΩΩ

z

y

x

Fz

My

Mz

Fy

FxMx

Mβ

Mζ

Mφ

Sz

SrSx

z

y

x

Fz

My

Mz

Fy

FxMx

Mβ

Mζ

Mφ

Sz

SrSx

Figure 3-2: Blade root shear forces and moments

97

200 300 400 500 600 700 800 9000

1

2

3

4

5

6

7

Rotor Speed (RPM)

Fre

quen

cy (

Hz)

AFDD theoretical

AFDD experimental

Current analysis

(a) Modal frequencies

200 300 400 500 600 700 800 900−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Rotor Speed (RPM)

Dec

ay R

ate

(/se

c)

AFDD experimental

AFDD theoretical

Current analysis

(b) Modal decay rates

Figure 3-3: Comparison of baseline results with results from [55]

98

500 550 600 650 700 750 800 850 9000

5

10

15

Rotor Speed (RPM)

Fre

quen

cy (

Hz)

Regressing absorber mode

Regressing lag mode

Roll mode

Pitch mode


500 550 600 650 700 750 800 850 900−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Rotor Speed (RPM)

Dec

ay R

ate

( /s

ec)

Regressing lag mode

Roll mode

Pitch mode

Baseline case

Unstable


Figure 3-4: Modal frequencies and decay rates vs RPM ( 0103030a ma .,.,. === αζ )

99

500 550 600 650 700 750 800 850 9000

5

10

15

Rotor Speed (RPM)

Fre

quen

cy (

Hz)


Regressing lag mode

Roll mode

Pitch mode

BaselineWith absorber


500 550 600 650 700 750 800 850 900−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Rotor Speed (RPM)

Dec

ay R

ate

( /s

ec)

Coupled lag/roll modes

Pitch mode

Unstable



100

500 550 600 650 700 750 800 850 9000

5

10

15

Rotor Speed (RPM)

Fre

quen

cy (

Hz)


Regressing lag mode

Roll mode

Pitch mode


500 550 600 650 700 750 800 850 900−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Rotor Speed (RPM)

Dec

ay R

ate

( /s

ec)


Pitch mode

Unstable



101

500 550 600 650 700 750 800 850 9000

5

10

15

Rotor Speed (RPM)

Fre

quen

cy (

Hz)


Regressing lag mode

Roll mode

Pitch mode


500 550 600 650 700 750 800 850 900−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Rotor Speed (RPM)

Dec

ay R

ate

( /s

ec)


Pitch mode

Unstable



102

500 550 600 650 700 750 800 850 9000

5

10

15

Rotor Speed (RPM)

Fre

quen

cy (

Hz)

Coupled lag/absorber modes

Reg lag mode(baseline case)

Roll mode

Pitch mode



500 550 600 650 700 750 800 850 900−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Rotor Speed (RPM)

Dec

ay R

ate

( /s

ec)

Regressing lag mode

Roll mode

Unstable



103

500 550 600 650 700 750 800 850 9000

5

10

15

Rotor Speed (RPM)

Fre

quen

cy (

Hz)




500 550 600 650 700 750 800 850 900−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Rotor Speed (RPM)

Dec

ay R

ate

( /s

ec)

Regressing lag mode

Roll mode

Pitch mode

Unstable



104

500 550 600 650 700 750 800 850 9000

5

10

15

Rotor Speed (RPM)

Fre

quen

cy (

Hz)


Roll mode

Pitch mode


500 550 600 650 700 750 800 850 900−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Rotor Speed (RPM)

Dec

ay R

ate

( /s

ec)

Regressing lag mode

Roll mode

Pitch mode

Unstable



105

500 550 600 650 700 750 800 850 9000

5

10

15

Rotor Speed (RPM)

Fre

quen

cy (

Hz)


Roll mode

Pitch mode

Regressing lag mode(baseline case)


500 550 600 650 700 750 800 850 900−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Rotor Speed (RPM)

Dec

ay R

ate

( /s

ec)

Regressing lag mode

Roll mode

Pitch mode

Unstable



106

500 550 600 650 700 750 800 850 9000

5

10

15

Rotor Speed (RPM)

Fre

quen

cy (

Hz)



Roll mode

Pitch mode


500 550 600 650 700 750 800 850 900−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Rotor Speed (RPM)

Dec

ay R

ate

( /s

ec)

Regressing lag mode

Roll mode

Pitch mode

Unstable



107

500 550 600 650 700 750 800 850 9000

5

10

15

Rotor Speed (RPM)

Fre

quen

cy (

Hz)


Regressinglag mode

Roll mode

Pitch mode


500 550 600 650 700 750 800 850 900−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Rotor Speed (RPM)

Dec

ay R

ate

( /s

ec)

Regressing lag mode

Roll mode

Pitch mode

Unstable



108

500 550 600 650 700 750 800 850 9000

5

10

15

Rotor Speed (RPM)

Fre

quen

cy (

Hz)


Regressing lag mode

Roll mode

Pitch mode


500 550 600 650 700 750 800 850 900−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Rotor Speed (RPM)

Dec

ay R

ate

( /s

ec)

Regressing lag mode

Roll mode

Pitch mode

Unstable



109

500 550 600 650 700 750 800 850 9000

5

10

15

Rotor Speed (RPM)

Fre

quen

cy (

Hz)


Regressing lag mode

Roll mode

Pitch mode



500 550 600 650 700 750 800 850 900−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Rotor Speed (RPM)

Dec

ay R

ate

( /s

ec)


Pitch mode

Unstable



110

500 550 600 650 700 750 800 850 9000

5

10

15

Rotor Speed (RPM)

Fre

quen

cy (

Hz)


Regressing lag mode

Roll mode

Pitch mode


500 550 600 650 700 750 800 850 900−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Rotor Speed (RPM)

Dec

ay R

ate

( /s

ec)

Regressing lag mode

Roll mode

Pitch mode

Unstable



111

500 550 600 650 700 750 800 850 9000

5

10

15

Rotor Speed (RPM)

Fre

quen

cy (

Hz)


Regressing lag mode

Roll mode

Pitch mode


500 550 600 650 700 750 800 850 900−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Rotor Speed (RPM)

Dec

ay R

ate

( /s

ec)

Regressing lag mode

Roll mode

Pitch mode

Unstable



112

500 550 600 650 700 750 800 850 9000

5

10

15

Rotor Speed (RPM)

Fre

quen

cy (

Hz)



Pitch mode


500 550 600 650 700 750 800 850 900−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Rotor Speed (RPM)

Dec

ay R

ate

( /s

ec)

Regressing lag mode

Roll mode

Pitch mode

Unstable



113

500 550 600 650 700 750 800 850 9000

5

10

15

Rotor Speed (RPM)

Fre

quen

cy (

Hz)


Regressing lag mode

Roll mode

Pitch mode


500 550 600 650 700 750 800 850 900−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Rotor Speed (RPM)

Dec

ay R

ate

( /s

ec)

Regressing lag mode

Roll mode

Pitch mode

Unstable



114

500 550 600 650 700 750 800 850 9000

5

10

15

Rotor Speed (RPM)

Fre

quen

cy (

Hz)



Roll mode

Pitch mode


500 550 600 650 700 750 800 850 900−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Rotor Speed (RPM)

Dec

ay R

ate

( /s

ec)

Regressing lag mode

Roll mode

Pitch mode

Unstable



115

500 550 600 650 700 750 800 850 9000

5

10

15

Rotor Speed (RPM)

Fre

quen

cy (

Hz)


Roll mode

Pitch mode


500 550 600 650 700 750 800 850 900−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Rotor Speed (RPM)

Dec

ay R

ate

( /s

ec)

Regressing lag mode

Roll mode

Pitch mode

Unstable



Chapter 4

Comparison with Chordwise Damped Vibration Absorber

One alternative to root-end auxiliary lag dampers, suggested by researchers at

Penn State [39-45], was to introduce lag mode damping through an embedded chordwise

damped vibration absorber (also called chordwise inertial damper) (see Figures 4-1 and

4-2) in the outboard region of the blade. For the correct choice of system design

parameters, the results showed that a significant amount of lag damping could be

introduced, and aeromechanical stability could be improved. However, the chordwise

absorber has stringent restrictions on stroke-length due to space limitations, and there are

concerns that the motion of the absorber mass in the chordwise direction, which results in

the movement of the blade center of gravity, can have a detrimental influence on blade

aeroelastic stability (pitch-flap flutter). Additionally, the magnitude of the absorber mass

required for satisfactory damping augmentation was quite large (on the order of 10% of

the blade mass). This chapter compares the amount of lag damping achieved by the

radial absorber versus chordwise absorber on the same rotor. The study also compares

the absorber response amplitudes under periodic loading, representative of forward flight,

required to achieve those levels of damping for both absorbers.

117

4.1 Analysis

While Kang presented lag damping results in [41], these results can not be

directly compared with the results from Chapter 2, since the systems are analyzed using

different damper parameters, as well as different rotor systems. For example, Kang uses

a complex stiffness approach to model his spring-damper system; where in this analysis,

the damper is modeled as a system with a spring and linear viscous damper. In this

chapter, both absorbers are analyzed using similar nondimensional formulations of the

differential equations of motion.

The governing linearized, nondimensional differential equations of motion for the

two-degree-of-freedom radial absorber are as follows, where the equations of motion are

derived and the absorber parameters are described in Chapter 2:

The equations of motion for the chordwise absorber are similarly derived from the results

in [41], resulting in the following:

( )

( )( ) ⎭

⎬⎫

⎩⎨⎧

=⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡+

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧⎥⎦

⎤⎢⎣

⎡−

−−+

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧⎥⎦

⎤⎢⎣

⎡ −+

∗

∗

∗∗

∗∗

aM

x00

x2ea2ea60

x100ea31

r

22f

2

rfa

m

r

2m

ζ

ζ

ζ

ζ

ζνα

νζναζ

α

ζα

4.1

( ) ( )( )

⎭⎬⎫

⎩⎨⎧

=⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−−

+⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧⎥⎦

⎤⎢⎣

⎡ −+

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧⎥⎦

⎤⎢⎣

⎡

−−−−−+

∗

∗

∗∗

∗∗

cc

22f

m2

cfac

cm

cc

cm2

cm

aM

xee3

x2a2a60

x1erer3er31

ζ

ζ

ζ

ζ

ζναανζ

ναζα

ζαα

4.2

118

where cx is the chordwise motion of the absorber mass, cr is the radial location of the

absorber, and ca is the absorber position relative to the feathering axis of the blade, all

nondimensionalized by the rotor radius. The remaining absorber parameters in Eq. 4.2

correspond to the same radial absorber parameters in Eq. 4.1, and are described in

Chapter 2. It is assumed that there is no inherent damping in the lag mode of the rotor

blade for both systems. For both systems of equations, by setting the forcing terms on the

right-hand side to zero and obtaining the eigenvalues of the system, the modal damping

of the coupled lag mode and absorber mode can be calculated. Further, by introducing a

harmonic excitation force for the lag equation of motion, frequency response functions

for the absorber displacement, rx and cx , and blade lag displacement, ζ , can be

obtained. The frequency response functions can then be used to calculate the amplitude

of the absorber displacement (or the stroke-length of the absorber mass) per degree

amplitude of blade lag motion, when the blade is undergoing periodic lead-lag motion in

forward flight conditions. Of particular interest is the dynamic displacement amplitude

of the absorber at a frequency of 1/rev, the dominant excitation frequency in forward

flight.

4.2 Blade Lag Damping and Absorber Response

Figures 4-3 - 4-11 show the amount of damping that can be transferred to the lag

mode for both types of absorber, using the same absorber parameters listed in Table 2-4.

As seen in the figures, for the parameters examined, the blade with the radial absorber is

able to reach much higher levels of damping than the blade with the chordwise absorber.

119

Considering a lightly damped absorber ( 30a .=ζ ), at an inboard ( 30a .= ) absorber

location (Figure 4-3), the maximum amount of damping transferred to the lag mode by

the chordwise absorber is 1.1% critical, compared with 15% critical for the radial

absorber, and this maximum damping occurs at a mass ratio of 0.05 for both absorbers.

As the absorber moves outboard, the amount of lag mode damping achieved for both

absorber concepts increases. However, the radial absorber outperforms the chordwise

absorber at a mid-span location ( 50a .= ): 3.3% critical damping in the lag mode for the

chordwise absorber vs. 15% critical damping in the lag mode for the radial absorber

(Figure 4-4). At the mid-span location, the maximum damping transferred to the lag

mode occurs at the maximum mass ratio considered for the chordwise absorber, whereas

for the radial absorber, the maximum lag mode damping is achieved even at a low mass

ratio of 0.02. Similarly, for an outboard absorber location ( 70a .= ), the damping ceiling

of 15% critical damping in the lag mode is achieved at a mass ratio of 0.01 for the radial

absorber, while the maximum lag mode damping for the chordwise absorber is only 7.5%

critical damping, and that is achieved again only for the largest mass ratio considered in

the analysis (Figure 4-5).

The same trend holds true for larger values of absorber damping ( 50a .=ζ and

70a .=ζ ) (see Figures 4-6 - 4-11). At these levels of absorber damping, the maximum

lag mode damping for the chordwise absorber is less than that obtained for the lightly

damped chordwise absorber for all absorber locations. However, for the radial absorber

for all levels of absorber damping considered, the maximum lag mode damping is one-

half of the isolated absorber mode damping, although a larger mass ratio is required for

120

inboard or mid-span absorber locations to reach this damping ceiling, as described in

Chapter 2. For example, a mass ratio of 0.05 is required to reach the damping ceiling of

25% critical damping in the lag mode for a mid-span absorber and an isolated absorber

damping value of 0.5 (Figure 4-7), while a mass ratio of 0.03 is required to reach the

same damping ceiling at an outboard absorber location ( 70a .= ) with the same absorber

damping (Figure 4-8).

In Figures 4-15a - 4-23a, the absorber response is plotted as a percentage of the

blade radius per degree of lag motion. For all absorber parameters evaluated, the

response of the radial absorber per degree of lag motion is generally twice that of the

chordwise absorber. However, the chordwise absorber stroke is limited by the chord

length, and the response of the chordwise absorber can also be nondimensionalized by the

chord length. Figures 4-15b - 4-23b show both systems’ response amplitudes per degree

of lag motion, where the radial absorber’s response is nondimensionalized by the radius,

and the chordwise absorber’s response is nondimensionalized by a notional chord length

of 0.08R. If the chord length is a smaller fraction of the radius, the response amplitude of

the chordwise absorber becomes an even greater percentage of the chord length. In

general, as the amount of damping transferred to the lag mode increases, the response

amplitude of both absorbers increases. The largest response amplitude for the chordwise

absorber occurs when the amount of lag mode damping is also the largest, and is ±26% of

the chord length (Figure 4-17b). For the same absorber parameters ( 70a .= , 30a .=ζ ,

and 050m .=α ), the radial absorber is able to transfer twice as much damping to the lag

121

mode (15% vs 7.5% critical), and the response amplitude is only slightly more than ±4%

of the blade radius (Figure 4-17b).

Most recently, the performance of the chordwise absorber has been studied with

the absorber located at the tip of the rotor blade, assuming the absorber will be

incorporated into the tip mass system of the blade [43-44]. Although this would not be

practical for a radial vibration absorber, the results are shown here for comparison with

the chordwise absorber. Figures 4-12 - 4-14 and 4-24 - 4-26 show a comparison of the

performance of the two systems when the absorber mass is located at the rotor tip. As

seen in Figure 4-12, at this location, the chordwise absorber is able to achieve a similar

level of damping to that of the radial absorber, with similar mass ratios, using an absorber

damping ratio of 30% critical. However, to achieve this level of damping, the chordwise

absorber response amplitude is approximately ±37% of the chord, as seen in Figure 4-

24b, where the response of the radial absorber with the same parameters is approximately

±5% of the radius.

4.3 Advantages and Disadvantages of Both Systems

The absorber response described above only considers the dynamic component of

the total response. However, the total response of the absorber contains a static

component and a dynamic component. A major factor in the future design of the radial

absorber is the large centrifugal force field in which the absorber will be required to

operate. The static displacement of the absorber due to the centrifugal force is dependent

on the rotor speed, the radial offset of the absorber from the hub, the absorber mass, and

122

the absorber spring stiffness. Using the spring stiffness required to achieve the desired

tuning to the lag natural frequency produces a large static displacement, resulting in the

absorber essentially “pegged” at the end of the rotor blade. Therefore, a frequency-

dependent spring stiffness is required for the absorber, with a high static stiffness to

withstand the centrifugal force, yet a low enough dynamic stiffness to still achieve the

desired tuning to the lag frequency of the blade.

The issue of the absorber operating in a high centrifugal force environment is also

a concern for the chordwise absorber, if the absorber has an initial offset from the feather

axis of the blade. For this reason, a fluid elastic absorber is under development for use as

a chordwise absorber. Research has shown that the fluid elastic damper can achieve

damping levels in the lag mode of 3-5% critical damping [41-44].

Another major issue that affects the chordwise absorber, but not the radial

absorber, has to do with the fact that the motion of the absorber mass causes a shift in the

blade center of gravity. This can cause the pitch divergence boundary to be adversely

affected by the addition of the absorber, and the pitch mode becomes more unstable as

the absorber tuning frequency decreases or the absorber mass increases [41]. The motion

of the radial absorber does not cause the blade chordwise center of gravity to shift and

would not results in any pitch instabilities.

4.4 Summary

The radial vibration absorber has been compared with the chordwise damped

vibration absorber. It was shown that for the absorber parameters considered for the

123

radial absorber, the chordwise absorber generally transfers much less damping to the lag

mode than the radial absorber. To achieve the levels of lag damping of the radial

absorber, the chordwise absorber must either use a much larger mass or be positioned

further outboard on the rotor blade. Additionally, the chordwise absorber has a

significant restriction on stroke length, and this will be a major consideration in the

design of the chordwise absorber. However, one disadvantage of the radial absorber as

compared with the chordwise absorber is the static displacement of the radial absorber

due to the extremely large centrifugal force. This will be a major consideration in the

design of the radial vibration absorber.

124

Figure 4-1: Embedded chordwise damped vibration absorber [42]

Elastic Axis

xc

acma

y

ζ

ka, ca

Elastic Axis

xc

ma

ζ

ka, ca ac

Elastic Axis

xc

acma

y

ζ

ka, ca

Elastic Axis

xc

ma

ζ

ka, ca ac

Figure 4-2: Embedded chordwise absorber schematic (redrawn from [44])

125

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

0.05

0.1

0.15

0.2

0.25

0.3


Mo

dal

Dam

pin

g R

atio

s (%

Cri

tica

l) Absorber Mode

Lag Mode

RadialChordwise

Figure 4-3: Modal damping ratios vs frequency ratio, fα ( 30a .= and 30a .=ζ )

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

0.05

0.1

0.15

0.2

0.25

0.3


Mo

dal

Dam

pin

g R

atio

s (%

Cri

tica

l)

Absorber Mode

Lag Mode

RadialChordwise


126

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

0.05

0.1

0.15

0.2

0.25

0.3


Mo

dal

Dam

pin

g R

atio

s (%

Cri

tica

l)

Absorber Mode

Lag Mode

RadialChordwise


0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5


Mo

dal

Dam

pin

g R

atio

s (%

Cri

tica

l)

Absorber Mode

Lag Mode

RadialChordwise


127

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5


Mo

dal

Dam

pin

g R

atio

s (%

Cri

tica

l)Absorber Mode

Lag Mode

RadialChordwise


0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5


Mo

dal

Dam

pin

g R

atio

s (%

Cri

tica

l)

Absorber Mode

Lag Mode

RadialChordwise


128

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7


Mo

dal

Dam

pin

g R

atio

s (%

Cri

tica

l)

Absorber Mode

Lag Mode

RadialChordwise


0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7


Mo

dal

Dam

pin

g R

atio

s (%

Cri

tica

l) Absorber Mode

Lag Mode

RadialChordwise


129

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7


Mo

dal

Dam

pin

g R

atio

s (%

Cri

tica

l)

Absorber Mode

Lag Mode

RadialChordwise


0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

0.05

0.1

0.15

0.2

0.25

0.3


Mo

dal

Dam

pin

g R

atio

s (%

Cri

tica

l)

Absorber Mode

Lag Mode

RadialChordwise


130

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5


Mo

dal

Dam

pin

g R

atio

s (%

Cri

tica

l)

Absorber Mode

Lag Mode

RadialChordwise


0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7


Mo

dal

Dam

pin

g R

atio

s (%

Cri

tica

l)

Absorber Mode

Lag Mode

RadialChordwise


131

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50.5

1

1.5

2


Ab

sorb

er R

esp

on

se (

%R

per

deg

ree

of

lag

mo

tio

n)

Radial

Chordwise

(a) %R per degree for both systems

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.51

1.2

1.4

1.6

1.8

2


Rad

ial A

bso

rber

Res

po

nse

(%

R p

er d

egre

e o

f la

g m

oti

on

)

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.57

8

9

10

11

12

Ch

ord

wis

e D

amp

er R

esp

on

se (

%c

per

deg

ree

of

lag

mo

tio

n)

(b) %R per degree for radial absorber, %c per degree for chordwise absorber

Figure 4-15: 1/rev absorber amplitude per degree of lag motion ( 30a .= and 30a .=ζ )

132

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50.5

1

1.5

2

2.5

3

3.5


Ab

sorb

er R

esp

on

se (

%R

per

deg

ree

of

lag

mo

tio

n)

Radial

Chordwise


0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.51.5

2

2.5

3

3.5


Rad

ial A

bso

rber

Res

po

nse

(%

R p

er d

egre

e o

f la

g m

oti

on

)

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.512

14

16

18

20

Ch

ord

wis

e D

amp

er R

esp

on

se (

%c

per

deg

ree

of

lag

mo

tio

n)



133

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.51

1.5

2

2.5

3

3.5

4

4.5


Ab

sorb

er R

esp

on

se (

%R

per

deg

ree

of

lag

mo

tio

n)

Radial

Chordwise


0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.52

3

4

5


Rad

ial A

bso

rber

Res

po

nse

(%

R p

er d

egre

e o

f la

g m

oti

on

)

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.515

20

25

30

Ch

ord

wis

e D

amp

er R

esp

on

se (

%c

per

deg

ree

of

lag

mo

tio

n)



134

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3


Ab

sorb

er R

esp

on

se (

%R

per

deg

ree

of

lag

mo

tio

n)

Radial

Chordwise


0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50.9

1

1.1

1.2

1.3


Rad

ial A

bso

rber

Res

po

nse

(%

R p

er d

egre

e o

f la

g m

oti

on

)

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.56

6.5

7

7.5

8

Ch

ord

wis

e D

amp

er R

esp

on

se (

%c

per

deg

ree

of

lag

mo

tio

n)



135

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50.8

1

1.2

1.4

1.6

1.8

2

2.2


Ab

sorb

er R

esp

on

se (

%R

per

deg

ree

of

lag

mo

tio

n)

Radial

Chordwise


0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.51.6

1.8

2

2.2


Rad

ial A

bso

rber

Res

po

nse

(%

R p

er d

egre

e o

f la

g m

oti

on

)

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.510

11

12

13

Ch

ord

wis

e D

amp

er R

esp

on

se (

%c

per

deg

ree

of

lag

mo

tio

n)



136

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.51

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3


Ab

sorb

er R

esp

on

se (

%R

per

deg

ree

of

lag

mo

tio

n)

Radial

Chordwise


0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.52.3

2.4

2.5

2.6

2.7

2.8

2.9

3

3.1


Rad

ial A

bso

rber

Res

po

nse

(%

R p

er d

egre

e o

f la

g m

oti

on

)

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.514

14.5

15

15.5

16

16.5

17

17.5

18

Ch

ord

wis

e D

amp

er R

esp

on

se (

%c

per

deg

ree

of

lag

mo

tio

n)



137

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1


Ab

sorb

er R

esp

on

se (

%R

per

deg

ree

of

lag

mo

tio

n)

Radial

Chordwise


0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50.6

0.8

1

1.2


Rad

ial A

bso

rber

Res

po

nse

(%

R p

er d

egre

e o

f la

g m

oti

on

)

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.54

5

6

7

Ch

ord

wis

e D

amp

er R

esp

on

se (

%c

per

deg

ree

of

lag

mo

tio

n)



138

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2


Ab

sorb

er R

esp

on

se (

%R

per

deg

ree

of

lag

mo

tio

n)

Radial

Chordwise


0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.51.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9


Rad

ial A

bso

rber

Res

po

nse

(%

R p

er d

egre

e o

f la

g m

oti

on

)

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.57

7.5

8

8.5

9

9.5

10

10.5

11

Ch

ord

wis

e D

amp

er R

esp

on

se (

%c

per

deg

ree

of

lag

mo

tio

n)



139

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6


Ab

sorb

er R

esp

on

se (

%R

per

deg

ree

of

lag

mo

tio

n)

Radial

Chordwise


0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.51.5

2

2.5


Rad

ial A

bso

rber

Res

po

nse

(%

R p

er d

egre

e o

f la

g m

oti

on

)

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.510

15

20

Ch

ord

wis

e D

amp

er R

esp

on

se (

%c

per

deg

ree

of

lag

mo

tio

n)



140

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.51.5

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5


Ab

sorb

er R

esp

on

se (

%R

per

deg

ree

of

lag

mo

tio

n)

Radial

Chordwise


0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.53

4

5

6

7


Rad

ial A

bso

rber

Res

po

nse

(%

R p

er d

egre

e o

f la

g m

oti

on

)

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.520

25

30

35

40

Ch

ord

wis

e D

amp

er R

esp

on

se (

%c

per

deg

ree

of

lag

mo

tio

n)



141

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.51.5

2

2.5

3

3.5

4

4.5


Ab

sorb

er R

esp

on

se (

%R

per

deg

ree

of

lag

mo

tio

n)

Radial

Chordwise


0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.53

3.5

4

4.5


Rad

ial A

bso

rber

Res

po

nse

(%

R p

er d

egre

e o

f la

g m

oti

on

)

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.520

22

24

26

Ch

ord

wis

e D

amp

er R

esp

on

se (

%c

per

deg

ree

of

lag

mo

tio

n)



142

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.51

1.5

2

2.5

3

3.5


Ab

sorb

er R

esp

on

se (

%R

per

deg

ree

of

lag

mo

tio

n) Radial

Chordwise


0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.52.2

2.4

2.6

2.8

3

3.2

3.4

3.6

3.8


Rad

ial A

bso

rber

Res

po

nse

(%

R p

er d

egre

e o

f la

g m

oti

on

)

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.514

15

16

17

18

19

20

21

22

Ch

ord

wis

e D

amp

er R

esp

on

se (

%c

per

deg

ree

of

lag

mo

tio

n)



Chapter 5

Elastic Blade Analysis

While the radial absorber was shown in Chapters 2 and 3 to achieve good results

for lag damping and aeromechanical stability, the effects of a discrete mass embedded in

the rotor blade, moving radially, on the blade spanwise and root loads, as well as rotor

hub loads and vibration levels, must still be examined. In this chapter, the rotor blade is

analyzed as an elastic beam undergoing coupled flap and lag bending under aerodynamic

loading and the blade and hub loads due to the addition of the radial absorber are

examined (relative to a baseline rotor with no absorber).

The rotor blade is modeled structurally as a slender elastic beam undergoing flap

and lag bending. The baseline elastic blade model used in this analysis is based on the

formulation in the University of Maryland Advanced Rotor Code (UMARC) [59]. The

elastic torsion and axial degrees of freedom of the rotor blade are neglected. As in

previous chapters, the absorber is modeled as a spring-mass-damper system moving

radially within the blade. The governing differential equations of motion are derived

using Hamilton’s Principle and then spatially discretized using the finite element method.

The aerodynamic forces and moments are determined using a quasi-steady aerodynamic

model, with the rotor inflow calculated using a linear inflow model, also based on the

formulation in [59]. Finally, the blade and hub loads are determined based on a coupled

propulsive trim-rotor response analysis.

144

5.1 Coordinate Systems

The coordinate systems used in this analysis are a combination of those used in

UMARC for the elastic blade analysis, with an additional coordinate system specifically

for modeling the embedded absorber. The coordinate system with absorber is shown in

Figure 5-1.

5.2 Ordering Scheme

As discussed in Chapter 2, an ordering scheme is applied to reduce the complexity

of the equations of motion. In the elastic blade/absorber analysis, terms of order 2ε are

retained, while terms of order 3ε and higher are dropped. The order of the terms used in

the development of the elastic blade equations of motion is the same as in [59]. The

order of terms specific to the radial absorber is listed in Table 5-1.

5.3 Elastic Blade Model

The rotor blade is assumed to undergo elastic flap and lag bending, with the

elastic torsion and axial degrees of freedom of the rotor blade neglected. The continuous

blade is spatially discretized using the finite element method with eight degrees of

Table 5-1: Absorber terms in ordering scheme

Absorber Terms Order a , ak , am , ac O(1)

rx O(ε )

145

freedom for each element. The blade degrees of freedom are v , v′ , w , and w′ at the

nodes of each beam element.

The blade discretized nonlinear governing differential equations of motion are

derived starting with Hamilton’s principle. If the system is subject to nonconservative

forces, such as the aerodynamic forces acting on a rotor blade, virtual work terms must be

included. The generalized Hamilton’s principle, which includes the virtual work term, is

as follows:

where Uδ is the variation of strain energy, Tδ is the variation of kinetic energy, and

Wδ is the virtual work done by external forces. A completed, detailed derivation of the

strain energy and kinetic energy of the rotor blade can be found in [59]. The absorber

adds additional terms to the strain and kinetic energies and will be described in detail in

section 5.4.

Using Eq. 5.1, the variations in strain energy, kinetic energy, and virtual work of

the thi element can be written in matrix form:

where iM , iC , and iK are the elemental mass, stiffness, and damping matrices, which

include both linear structural and aerodynamic terms. iF is the load vector, which

includes constant and nonlinear structural and aerodynamic terms, ( )iAi FFF += , and

iq is the elemental vector of the degrees of freedom for the thi element, where

( ) 0dtWTU2

1

t

t=−−=Π ∫ δδδδ 5.1

( ) ( )iiiiiiiTiiii FqKqCqMqWTU −++=−− &&&δδδδ 5.2

146

[ ]22221111Ti wwvvwwvvq ′′′′= . The elemental structural mass matrix can be expressed

as

where

The elemental structural damping matrix can be expressed as

where

The elemental structural stiffness matrix can be expressed as

where

⎥⎦

⎤⎢⎣

⎡=

ww4x4

4x4vvi M0

0MM 5.3

dxHmHMMlel

Twwvv ∫== 5.4

⎥⎦

⎤⎢⎣

⎡=

4x4wv

vwvvi 0C

CCC 5.5

dxHHme2dxHHme2Clel

Tglel

Tgvv ∫∫ ′+′−= θcos

dxHHme2dxHmH2CClel

Tglel

Tpwvvw ∫∫ ′−−=−= θβ sin

5.6

⎥⎦

⎤⎢⎣

⎡=

wwwv

vwvvi KK

KKK 5.7

147

The structural load vector can be expressed as

where

The shape functions, H , are the same for both flap and lag:

( ) dxHHEIEI

dxHHFdxHHmK

lel

T2y

2z

lel

TAlel

T2vv

∫∫∫

′′′′++

′′+Ω−=

θθ sincos

( ) dxHHEIEIKKlel

Tyzwvvw ∫ ′′′′−== θθ cossin

( ) dxHHEIEIdxHHFKlel

T2y

2zlel

TAww ∫∫ ′′′′++′′= θθ cossin

and ∫ Ω=1

x

2A dmF ξξ

5.8

⎭⎬⎫

⎩⎨⎧

=w

v

FF

F 5.9

( ) dxHdvmv2dxHdwwvvm2

dxHxmedxHmeF

lel

T1

xlel

Tx

0

lel

Tglel

Tgv

∫ ∫∫ ∫

∫∫′⎥⎦

⎤⎢⎣⎡ ′−⎥⎦

⎤⎢⎣⎡ ′′+′′+

′−=

ξξ

θθ

&&&

coscos

dxHdvmw2

dxHxmedxxHmeF

lel

T1

x

lel

Tglel

Tpgw

∫ ∫

∫∫′⎥⎦

⎤⎢⎣⎡ ′−

′−−=

ξ

θβ

&

sin

5.10

⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛−

+⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛

+⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛

=

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

=

2

el

3

el

2

el

3

el

2

el

3

el

2

el

3

el

4

3

2

1

T

lx

lx

lx3

lx2

xlx2

lx

1lx3

lx2

HHHH

H 5.11

148

The blade pitch angle, θ , in the above equations is the rigid pitch angle due to control

system settings and blade pre-twist and is determined as follows:

where 75θ is the collective pitch at the 75% span location, c1θ and s1θ are the lateral and

longitudinal cyclic pitch controls settings, and twθ is the blade pre-twist angle referenced

to the 75% span location.

The virtual work used in Eq. 5.1 is due to the external aerodynamic loads on the

rotor blade. The general expression for the virtual work done by the aerodynamic loads

for the flap-lag model is

where AvL is the distributed airload on the blade in the chordwise direction and A

wL is the

distributed airload on the blade in the vertical direction. The virtual work of the thi

element is determined using a quasi-steady aerodynamic model with a linear inflow

model, again based on the formulation in [59]. Using this model, the section airloads are

calculated as functions of airfoil properties and blade and wind velocities. The airloads

are motion dependent; the loads that are linear functions of blade velocities and

displacements are formulated as aerodynamic damping and stiffness matrices and added

to the structural matrices. The constant and nonlinear aerodynamic loads are added to the

constant and nonlinear structural forcing vector. Compressibility effects, Mach number

effects, reverse flow and retreating blade stall are not modeled in this analysis. A

summary of the aerodynamic loads development is provided in Appendix B.

( )750xtws1c175 .sincos −+++= θψθψθθθ 5.12

( )∫ +=R

0

Aw

Avb dxwLvLW δδδ 5.13

149

Once the elemental matrices and vectors are determined, the global equations of

motion for the rotor blade can be assembled, ensuring compatibility between degrees of

freedom at adjoining element nodes. The compatibility conditions are continuity of

displacement and slope for flap and lag bending between elements. This results in the

blade governing finite element differential equations of motion:

5.4 Absorber Model

The absorber is modeled as a single degree of freedom spring-mass-damper

system moving radially within the rotor blade. The absorber contributes additional strain

and kinetic energy terms to the system, as well as a virtual work term due to the fact that

the absorber is a damped system. These additional terms are also substituted into Eq. 5.1,

which results in a modification in the blade equations of motion.

The strain energy of the absorber comes from the spring and can be written as

with the variation in strain energy as

The kinetic energy of the absorber is dependent on its velocity and can be written as

FKqqCqM =++ &&& 5.14

2raa xk

21U = 5.15

rraa xxkU δδ = 5.16

aaaa vvm21T ⋅= 5.17

150

with the variation in kinetic energy as

The absorber velocity is determined by first starting with its position vector:

where u is a kinematic axial deflection of the rotor blade (and the absorber position) due

to foreshortening, and uδ is its variation:

The velocity can then be determined by taking the first time derivative of the position

vector, as described in Eq. 2.3 , where

The resulting absorber velocity and variation in velocity in component form are as

follows:

Eq. 5.18 becomes (after integration by parts with respect to time for use in Eq. 5.1)

aaaa vvmT δδ ⋅= 5.18

( ) kwjviuxar raˆˆˆ ++++= 5.19

( )dxwv21u

x

0

22∫ ′+′−=

( )∫ ′′+′′−=x

odxwwvvu δδδ

5.20

( ) kiki pppˆˆˆcosˆsin Ω+Ω≈+Ω= βββω 5.21

( )

vwvvwv

wuxvvwaxvv

vxvvxv

paz

paz

pray

pray

rax

rax

δβδδ

β

δβδδδδ

βδδδ

Ω+=

Ω+=

Ω−Ω+Ω+=

Ω−+Ω+=Ω−=Ω−=

&

&

&

&

&

&

5.22

151

The ordering scheme is applied to the absorber terms, and the terms of order 3ε and

higher are discarded.

The damper associated with the absorber is modeled as a viscous damper, which

contributes to the virtual work of the system. The virtual work due to the damper is

The variations in strain and kinetic energies and the virtual work term are

substituted into Eq. 5.1 and again since the virtual displacements, rxδ , vδ , and wδ are

arbitrary, this results in the absorber contributions to the blade-absorber equations of

motion. The resulting blade-absorber equations of motion can be written as

In Eq. 5.25, bbM , bbC , and bbK represent the global mass, damping, and stiffness

matrices of the baseline rotor blade (without the absorber), modified slightly due to the

addition of the absorber. Since the absorber is modeled as a discrete system, most of the

absorber terms are simply added to the degrees of freedom in the blade global equations

of motion at the thk global node at which the absorber is located (see Figure 5-2).

Specifically,

( )( )( ) ( )

( )( )( ) ⎥

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

Ω++Ω−Ω−−+

Ω+Ω+Ω+Ω−−+

′′+′′Ω+−

Ω−+Ω+Ω+−

= ∫

wwaxv2w

vvvw2x2v

dxwwvvax

xwaxv2x

mT

2p

2rp

2p

2p

22pr

a

o

2r

rpr2

r

aa

δβββ

δββ

δδ

δβ

δ

&&&

&&&&

&&&

5.23

rraa xxcW δδ &−= 5.24

⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡+

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡−

+⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡

aa

bb

aaaTba

babb

aaaTba

babb

aaa

bb

FF

qq

KKKK

qq

CCCC

qq

M00M

&

&

&&

&& 5.25

152

However, the v′δ and w′δ terms in Eq. 5.23 are integrated using the shape functions in

Eq. 5.11 over each finite element for all elements located inboard of the absorber and

added to the blade elemental stiffness matrices before they are assembled into the global

stiffness matrix, as shown in Eq. 5.27:

This term contributes to the centrifugal stiffening of the rotor blade.

The coupling ( ba ) and absorber ( aa ) terms essentially add an additional row and

column to the blade global matrices due to the absorber:

The Coriolis coupling terms as well as other coupling terms in Eq. 5.25 appear only at the

node where the absorber is located; i.e., except for the terms described in Eq. 5.29, all

terms in baC and baK are zero.

In the load vector in Eq. 5.25, bbF includes the constant and nonlinear blade

structural terms and aerodynamic forcing terms. The absorber contributes additional

structural forcing terms:

( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) 2

akkbbkkbb

pakkbbkkbb

pakkbbkkbb

akkbbkkbb

akkbbkkbb

mvvKvvK

m2vwCvwCm2wvCwvCmwwMwwM

mvvMvvM

Ω−=

Ω+=

Ω−=+=+=

,,

,,,,

,,,,

β

β&&&&

&&&&

&&&&&&&&

&&&&&&&&

5.26

( )∫ ′′+Ω==lel

Tr

2awwvv dxHHxamKK 5.27

2aaaa

aaa

aaa

mkK

cCmM

Ω−=

==

5.28

( )( ) p

2akba

akba

mwK

m2vC

βΩ=

Ω=& 5.29

153

where aaF is the static centrifugal force on the absorber mass.

5.5 Blade and Absorber Responses

Once the global blade-absorber equations of motion are assembled, the kinematic

boundary conditions are enforced. For a hingeless rotor, the flap and lag displacements

and rotations are assumed to be zero at the root. These constraints are applied to the

finite element model by removing the first four rows and columns from the global

matrices.

It is possible for the set of discretized equations of motion to involve a significant

number of degrees of freedom, depending on the number of elements chosen to represent

the rotor blade. To reduce computational time, the blade-absorber equations can be

transformed into modal space using the eigenvectors of Eq. 5.31

where

from Eq. 5.25, with any number of modes chosen to represent the rotor blade. Using the

eigenvectors associated with the selected modes, Φ , the displacement, velocity, and

acceleration can be written in terms of the modes shapes and the generalized coordinates,

( ) ( )amF

amwFwF2

aaa

p2

akbbkbb

Ω=

Ω−= β 5.30

0qKqM =+&& 5.31

⎥⎦

⎤⎢⎣

⎡=

aa

bb

M00M

M and ⎥⎦

⎤⎢⎣

⎡=

aaTba

babb

KKKK

K 5.32

154

pq Φ= , where p is the vector of generalized coordinates. Introducing into Eq. 5.25, the

modal equations of motion can then be obtained as

where

ΦΦ= MM T~ , ΦΦ= CC T~ , ΦΦ= KK T~ , and FF TΦ=~ .

5.5.1 Absorber Static Displacement

In the two-degree-of-freedom and ground resonance analyses, the absorber spring

stiffness has been considered to be a constant, and its value determined from the tuning

requirements of the system. The absorber response contains a static component, due to

the centrifugal force on the absorber, and a dynamic component. The static displacement

of the absorber due to the centrifugal force is dependent on the rotor speed, the radial

offset of the absorber from the hub, the absorber mass, and the absorber spring stiffness.

Using the spring stiffness required to achieve the desired tuning to the lag natural

frequency produces a large static displacement, resulting in the absorber essentially

“pegged” at the end of the rotor blade. Therefore, a frequency-dependent spring stiffness

is required for the absorber, with a high static stiffness to withstand the centrifugal force,

yet a low enough dynamic stiffness to still achieve the desired tuning to the lag frequency

of the blade. In the present analysis the static and dynamic responses are calculated

separately. The static displacement is calculated using the static stiffness, while the

dynamic response is calculated using the lower dynamic stiffness corresponding to the

FpKpCpM ~~~~ =++ &&& 5.33

155

lag natural frequency. The absorber static spring stiffness is chosen such that the static

displacement of the absorber due to the centrifugal force is 2.5% of the blade radius. The

dynamic spring stiffness is chosen such that the absorber is tuned to the fundamental lag

natural frequency.

5.5.2 Coupled Rotor-Absorber Response/Trim Calculation

The rotor/absorber response and vehicle trim (vehicle orientation and controls) are

coupled and must be obtained in an iterative process. The process begins with an initial

estimate of the vehicle orientation and controls. Using this initial estimate, the steady-

state rotor (and absorber) responses are calculated. The steady-state flap, lag, and

absorber modal responses of Eq. 5.25 are calculated using the Harmonic Balance

Method, retaining at least five harmonics in the solution to accurately determine the 4/rev

vibratory hub loads. The static response is solved separately using Eq. 5.34 with the

larger static stiffness of the absorber in aaK :

The dynamic response is then solved using Eq. 5.35, replacing the stiffness in aaK with

the lower, dynamically tuned stiffness. aaF , which contains the centrifugal force term, is

set to zero for the dynamic response.

( ) 0dEOM21 2

0=∫ ψ

ππ

5.34

156

In Eqs. 5.34 and 5.35, the EOM term refers to Eq. 5.25. The physical response, q , is

determined from the modal response using the transformation pq Φ= . Because the

equations of motion are nonlinear, the response is solved using Newton’s method for

solving nonlinear equations [59].

For a given blade/absorber response, the blade root forces and moments can be

determined by integrating the loads along the span, and the rotor hub loads are calculated

from the blade root loads. The fuselage is modeled as a rigid body and steady loads from

the rotor system, horizontal tail, tail rotor, and fuselage are considered for trim in steady,

level flight. Eq. 5.36 is used to calculate the vehicle equilibrium state (see Figure 5-3):

( )

( )

( )

( )

( )

( ) 0d5EOM1

0d5EOM1

0d2EOM1

0d2EOM1

0dEOM1

0dEOM1

2

0

2

0

2

0

2

0

2

0

2

0

=Ψ

=Ψ

=Ψ

=Ψ

=Ψ

=Ψ

∫

∫

∫

∫

∫

∫

ψπ

ψπ

ψπ

ψπ

ψπ

ψπ

π

π

π

π

π

π

sin

cos

sin

cos

sin

cos

M

5.35

( ) ( ) ( )( ) ( )

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

−+−+++−−++−+−++++

−−−++++

−+

=

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

scgftrtrscgfzfH

hthtscgsfscgsyfH

trtrscgsscgsfxfH

htsH

sH

ssH

trsH

sH

f

sH

sH

xf

Vz

Vy

Vx

Vz

Vy

Vx

xYxTyDMMxLxhWDxhWMMzhTyhWyhYMM

LWFFFTFFY

FFD

MMMFFF

z

y

x

yxz

zy

z

φααααα

φφφφφαφα

φφαα

coscossincoscossin

cossinsincossinsincoscos

sincossincos

5.36

157

If the vehicle is not in equilibrium, the vehicle orientation and controls are updated and

the entire procedure is repeated until it is determined that the vehicle is in equilibrium.

5.6 Blade Loads

There are several methods which can be used to determine the blade loads. The

radial distribution of forces and moments along the blade can be calculated using the

reaction force method or the force summation method. The blade root loads can be

determined using one of three methods: the reaction force method, force summation

method, or finite element constraint equation method [58, 59].

The blade root shear forces and moments are calculated using the finite element

constraint equation method, where the unconstrained global matrices and displacement

vectors are used to determine the loads at the blade root. Once the blade and absorber

responses are determined, they can be used to calculate the reaction forces at the

constraint:

Only the rows of the mass, stiffness, and damping matrices corresponding to the

constrained degrees of freedom are necessary to calculate the reaction forces and

moments at the constraint. For the case of a hingeless rotor modeled as a cantilever beam

with flap and lag deflections, this corresponds to the first four rows in the matrices.

Alternatively, the blade root loads can be determined using the reaction force

method and the force summation method. These two methods can also be used to

calculate the radial distribution of the shear forces and moments. Because the absorber is

FqKqCqMR −++= &&& 5.37

158

embedded within the rotor blade and can be located at any point along the blade span,

calculation of the forces and moments along the blade is necessary to determine what

impact the absorber has on these loads.

The reaction force method used in this analysis is based on the formulation in [58,

59]. In the method, the blade loads are calculated by solving the finite element governing

equations at the elemental level to determine the reactions at the elemental endpoints.

The reactions forces on an element are calculated by:

where the terms in the above equations are the unmodified elemental matrices and

vectors. The elemental matrices must be used and not the global matrices, as the internal

reactions between elements sum to zero in the global system. To account for the absorber

using this method, the absorber terms (Eq. 5.26 - Eq. 5.30) are added to the elemental

matrices of the element at which the absorber is located, instead of adding them to the

global finite element matrices. Because the absorber is located at a node, between two

elements, the absorber terms can be added to either of the two elements, providing the

terms are correctly placed at the appropriate nodal positions within the element.

In order to calculate the forces and moments in a particular direction using the

finite element constraint equation or reaction force method, a displacement in that same

direction must be included in the finite element model. In this analysis, elastic axial

deformation is neglected, which has little influence on the rotor flap and lag response

because the axial deformations are small. However, the axial forces are very large,

primarily due to the centrifugal force, and these forces cannot be captured using the

iiiiiiii FqKqCqMQ −++= &&& 5.38

159

constraint equation or reaction force method since there is no axial displacement included

in the blade model. The radial distribution of the blade forces and moments, as well as

the blade root loads, can also be calculated using the force summation method. In this

method, analytical expressions for the inertial and external forces are integrated along the

blade span.

The blade in-plane and vertical forces at any given point along the blade can be

calculated by (adapted from [58])

where AL contains respective components of the aerodynamic forcing. The inertial

loads, IL , are calculated from the respective components of the blade acceleration.

The absorber contributes additional inertial terms, calculated from the absorber

acceleration. The absorber acceleration is determined from the absorber velocity, using

Eq. 2.4, with the respective components of the absorber acceleration expressed as

After application of the ordering scheme, the inertial loads due to the absorber are as

follows:

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

+⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

+++

=⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

∫Iaw

Iav

Iau

R

xAw

Iw

Av

Iv

Au

Iu

z

y

x

LLL

dxLLLLLL

FFF

0

5.39

( )

( ) wxav2wa

vvw2x2va

wxav2xa

2p

2rp

2paz

2p

22pray

p2

r2

rax

βββ

ββ

β

Ω−+Ω+Ω+=

Ω−Ω−Ω−Ω+=

Ω++Ω−Ω−=

&&&

&&&&

&&&

5.40

( )[ ][ ]

( )[ ]rp2

kpkaIaw

kpk2

rkaIav

kp22

rkraIau

xav2wmL

w2vx2vmL

wxav2xmL

+Ω+Ω+=

Ω−Ω−Ω+=

Ω+Ω+−Ω−=

ββ

β

β

&&&

&&&&

&&&

5.41

160

In Eq. 5.41, kv , kw , kv& , kw& , kv&& , and kw&& are the flap and lag displacements, velocities,

and accelerations of the rotor blade at the absorber location. The forces in these

equations are not integrated along the length of the blade; rather they are added to the

blade integrated forces at the absorber location along the blade. The blade root loads can

also be calculated using Eq. 5.39, by substituting 0x0 = .

While the chordwise and vertical forces transmitted to the blade due to the

absorber can be modeled as the inertial forces acting at the location of the absorber mass,

the absorber radial force is transmitted to the rotor blade through its attachment points

and would be the force due to the spring stiffness and absorber displacement, xkF aSF =

for the model used in this analysis. However, since this is still an initial concept, the

location and type of attachment points are unknown (see Figure 5-4 for a schematic of the

absorber attached to the rotor blade). In order to account for these forces, the radial force

is taken to be the sum of the inertial forces acting at the point on the blade at which the

absorber mass is located.

where kv& is the blade lag velocity at the absorber location, and kw is the blade flap

displacement at the absorber location. These two forces are equivalent, which can be

seen from the absorber equation of motion:

( )r2

ararakp2

aka2

aIF xmxcxmwmvm2amF Ω−+−Ω+Ω+Ω= &&&& β 5.42

( ) kp2

aka2

ar2

aarara wmvm2amxmkxcxm βΩ+Ω+Ω=Ω−++ &&&& 5.43

161

In reality, this force would be distributed at the attachment points of the absorber, rather

than acting at a single point on the rotor blade. These forces are shown in a schematic in

Figure 5-4.

5.7 Hub Loads

Once the blade root loads are known, the hub forces and moments can be

calculated by summing the root loads from every blade, assuming all blades are identical:

where bN is the number of blades. Using a Fourier series expansion, the hub loads can

be expressed as a sum of the steady hub loads (used in the vehicle trim calculation) and

vibratory hub loads [59].

( )

( )

( )

( )

( )

( )∑

∑

∑

∑

∑

∑

=

=

=

=

=

=

+=

−+=

−−=

+=

−+=

−−=

b

b

b

b

b

b

N

1mp

mx

mz

Hz

N

1mpm

mzm

mym

mx

Hy

N

1mpm

mzm

mym

mx

Hx

N

1mp

mx

mz

Hz

N

1mpm

mzm

mym

mx

Hy

N

1mpm

mzm

mym

mx

Hx

MMM

MMMM

MMMM

FFF

FFFF

FFFF

β

βψψψ

βψψψ

β

βψψψ

βψψψ

sincossin

cossincos

sincossin

cossincos

5.44

162

5.8 Results

Two different helicopter/rotor systems are examined, and results are generated

using both models. The first is a full-scale, four-bladed BO-105-type hingeless rotor

helicopter model whose rotor and fuselage properties are given in Appendix C. The

second is the HART I (Higher Harmonic Control Aeroacoustic Rotor Test) 40% model of

the BO-105 rotor [61, 62], which is also a hingeless rotor system. Since the HART I

rotor is a wind tunnel model, notional fuselage properties are chosen in order to conduct a

propulsive trim analysis. The majority of the notional fuselage properties are in terms of

length or area, which are based on a fraction of the rotor radius or disk area. Therefore,

the same fractions of rotor radius and disk area that are used for the full-scale BO-105

model are also used in the 40% model (see Table C-4). An approximation for the gross

weight of the HART I rotor-fuselage model is taken from the hover results in [62]. The

HART I rotor and notional fuselage data are also given in Appendix C.

5.8.1 Full-Scale BO-105 Rotor-Fuselage Model Results

Simulation results are first obtained for a baseline rotor (without an absorber), and

then the radial absorber is added to the rotor to compare the changes in blade response

and loads relative to the baseline rotor. The mass of the rotor blade with the absorber is

adjusted such that both systems have the same total mass, and the comparisons can be

made on an equal basis. Six beam elements are used to model the rotor blade. As in

Chapters 2 and 3, the behavior of the system was examined over a range of absorber

parameters (given in Table 2-4), with the absorber frequency tuned to the first lag natural

163

frequency for all simulations ( 1f =α at the fundamental lag natural frequency). All

results are generated at an advance ratio, 350.=μ .

The baseline rotor is trimmed using the procedure described above, which results

in the following vehicle orientation and control settings:

These values are then used for all simulations with the absorber such that the

comparisons of the changes in blade and hub loads due to the addition of the absorber can

be made.

5.8.1.1 Eigenvalue Analysis

The eigenvalues of the system are first calculated for all absorber parameters in

order to determine the amount of damping that is transferred to the lag mode of the rotor

blade. These values of damping are also compared with the two-degree-of-freedom rigid

blade results and are listed in Table 5-3. As can be seen, the amount of damping

transferred to the lag mode agrees well for the most part with the two-degree-of-freedom

results when the absorber is located outboard. However, when the absorber is positioned

Table 5-2: Trim results (baseline BO-105-type rotor)

Vehicle Orientation/Control Setting Symbol Value (deg) Collective pitch 75θ 9.7 Lateral cyclic c1θ 2.0 Longitudinal cycle s1θ -7.9 Forward shaft tilt sα 4.7 Lateral shaft tilt sφ -2.7 Tail rotor collective pitch trθ 3.4

164

at a mid-span or inboard location, the amount of damping that is transferred to the lag

mode is much lower than that predicted for the rigid blade model. This is primarily due

to the difference in the mode shapes and corresponding difference in the lag velocity

distribution along the blade span between the rigid blade and the BO-105-type elastic

blade which has constant stiffness properties. For a hingeless rotor blade with a softer

hinge flexure (and for an articulated rotor), the difference in the blade lag velocity

distribution is more similar to that of the rigid blade, resulting in higher damping ratios

achieved in the lag mode. The mode shapes for the fundamental lag mode of a hingeless

blade with constant stiffness, a hingeless blade with a softer hinge flexure (HART I

rotor), and a rigid blade are shown in Figure 5-5.

165

5.8.1.2 Blade and Absorber Responses

The blade tip responses (flap and lag motion), nondimensionalized by the rotor

radius, are shown in Figures 5-6 - 5-14. For most of the inboard and mid-span absorber

Table 5-3: Comparison of lag mode damping – elastic and rigid blades

Absorber Parameters Lag Mode Damping (% critical)

a aζ mα Elastic Blade Rigid Blade 0.3 0.3 0.01 0.5 1.9 0.3 0.3 0.03 1.5 6.7 0.3 0.3 0.05 2.6 14.1 0.5 0.3 0.01 3.4 6.0 0.5 0.3 0.03 14.2 14.3 0.5 0.3 0.05 14.1 14.0 0.7 0.3 0.01 12.1 14.2 0.7 0.3 0.03 13.6 13.9 0.7 0.3 0.05 13.3 13.4 0.3 0.5 0.01 0.3 1.1 0.3 0.5 0.03 0.9 3.3 0.3 0.5 0.05 1.5 5.9 0.5 0.5 0.01 1.8 3.1 0.5 0.5 0.03 6.2 11.0 0.5 0.5 0.05 11.8 22.6 0.7 0.5 0.01 6.0 6.5 0.7 0.5 0.03 21.9 22.9 0.7 0.5 0.05 22.5 22.5 0.3 0.7 0.01 0.2 0.8 0.3 0.7 0.03 0.6 2.3 0.3 0.7 0.05 1.1 3.9 0.5 0.7 0.01 1.3 2.1 0.5 0.7 0.03 4.2 6.8 0.5 0.7 0.05 7.3 12.3 0.7 0.7 0.01 4.0 4.3 0.7 0.7 0.03 14.0 15.1 0.7 0.7 0.05 29.7 31.2

166

parameters examined, the dynamic peak-to-peak lag response shows a slight increase of

less than 5% over the baseline dynamic lag response. However, for an absorber located

at 70% of the blade radius ( 70a .= ), the absorber has a larger effect on the dynamic

blade tip lag response (see Figures 5-8, 5-11, and 5-14). For this absorber location, the

largest increase in dynamic response over the baseline occurs for an absorber damping

ratio of 30% critical and a mass ratio of 0.05. This increase in the dynamic peak-to-peak

response directly corresponds to the amount of damping that is transferred to the lag

mode by the absorber.

The addition of the absorber has little effect on the flap response. The peak-to-

peak flap response shows a maximum of a 0.7% increase over the baseline rotor. For the

cases of an outboard absorber location and a mass ratio of 0.05, the peak-to-peak flap

response decreases slightly from the baseline rotor flap response.

The absorber response, nondimensionalized by the rotor radius, is shown in

Figures 5-15 - 5-23. As discussed in section 5.5.1, each absorber is assumed to have a

static spring stiffness such that the static offset of the absorber is 2.5% of the rotor radius.

For this chosen static offset, the ratio between the static and dynamic stiffnesses,

dynamica

statica

kk

, varies from 9 to 20, depending on the absorber location along the rotor blade.

If a constant spring stiffness is used for the absorber, the static displacement would be

unacceptably large. For example, when the absorber is located inboard on the rotor blade

( 30a .= ), the static deflection of the absorber due to the centrifugal force would be

approximately 5 feet, putting the absorber almost 10 feet down the span of the blade.

When the absorber is located at mid-span, its theoretical static displacement would be 8.2

167

feet, and at the outboard location, the static deflection would be more than 11.5 feet. For

these two locations, with a constant, dynamically tuned spring stiffness, the absorber

mass would rapidly extend to the blade tip and remain pinned there by the centrifugal

force.

Since the static response of the absorber is arbitrary at this point due to the

method of selection of the static spring stiffness, it is more useful to look just at the

dynamic response amplitude of the absorber. Depending on the absorber parameters, the

dynamic response of the absorber varies from ±0.3% of the rotor radius to ±3.1% of the

rotor radius. The maximum absorber dynamic response occurs for absorber parameters

of 70a .= , 30a .=ζ , and 050m .=α . In general, the absorber dynamic response

amplitude increases as the absorber moves outboard (see for example Figures 5-15 - 5-

17). Additionally, the mass ratio has relatively little effect for inboard and mid-span

absorbers (refer to Figures 5-15 - 5-16 and Figures 5-18 - 5-19 ), but a larger mass ratio

increases the response amplitude for an outboard absorber, especially when absorber

mode damping is not too high (see Figures 5-17 and 5-20).

5.8.1.3 Blade Loads

Neither the spanwise loads nor the blade root loads exhibit significant variations

from the baseline rotor with the addition of the absorber for most of the absorber

parameters. The blade root shear forces and moments are shown in Figures 5-24 - 5-68.

For the majority of the absorber parameters examined, the blade root drag shear

shows a slight increase with the addition of the absorber when compared to the baseline

168

rotor, as seen in Figures 5-24 - 5-32. In general, the peak-to-peak blade root drag shear

has a less than 5% increase over the blade root drag shear of the baseline rotor. However,

like the lag response, when the absorber is located in the outboard portion of the rotor

blade ( 70a .= ), it has a larger effect on the blade root drag shear, with the largest

increase occurring for an absorber with a damping ratio of 30% critical and a mass ratio

of 0.05 (see Figure 5-26). For these absorber parameters, the peak-to-peak blade root

drag shear increases by approximately 32% over the baseline case. The blade root lag

moment shows similar trends to the drag shear, with the majority of the absorber

parameters resulting in a less than 5% increase in the peak blade root lag moment over

the baseline rotor and larger increases coming for an outboard absorber (refer to

Figures 5-33 - 5-41).

Like the flap response, the blade root vertical shear and flap moment do not

experience significant variations from the baseline rotor. For all absorber parameters

examined, the peak-to-peak values of blade root vertical shear (Figures 5-42 - 5-50) and

blade root flap moment (Figures 5-51 - 5-59) are all within 10% of the baseline case, and

most are within 4% of the baseline rotor.

Although the mass of the rotor blade is adjusted for the addition of the absorber,

such that the total mass of the system (baseline rotor or rotor plus absorber) is the same

for all cases, the blade root radial shear force for the rotor with the absorber is not exactly

the same as the baseline rotor due to the difference in the mass distribution between the

two rotors. However, for most cases examined, the variation in the blade root radial

shear force is small (less than 10%), as shown in Figures 5-60 - 5-68, with the largest

169

increases in peak-to-peak shear force occurring for the outboard absorber with low

damping.

The spanwise loads, shown in Figure 5-69 - 5-95, show a slight “jump” at the

absorber location due to the addition of the discrete mass of the absorber located along

the blade span. While a variation of the spanwise loads around the azimuth is expected

(and seen in the blade root loads), the addition of the absorber does not significantly

change the loads, particularly near the peak loads.

5.8.1.4 Hub Loads

The steady hub loads with and without the absorber are shown in Figure 5-96 - 5-

104. Like the blade loads, the hub loads do not change significantly with the addition of

the absorber for the majority of the absorber parameters examined. Since the hub loads

for the rotor with the absorber are calculated for the same set of control settings for the

trimmed rotor without the absorber, the actual steady hub loads are expected to differ

slightly from the loads shown in the figures and would require a slightly different set of

control settings to achieve trimmed flight.

The 4/rev vibratory hub loads with and without the absorber are shown in

Figures 5-105 - 5-113. At the airspeed examined ( 350.=μ ), the addition of the

absorber does not increase the vibratory loads significantly for the majority of absorber

parameters. In general, small changes in vibratory hub forces and moments are observed

(on the order of 1-5%). However, the vibratory hub loads exhibit a sensitivity to

vibration depending on the location of the absorber, as well as the mass ratio of the

170

absorber, which may be exploited for vibration reduction as described in Chapter 7,

section 7.2.2 . For example, for a mid-span absorber, the 4/rev forces, p4xF and p4

yF ,

exhibit little change from the baseline rotor (see Figure 5-106), while for an outboard

absorber, p4xF and p4

yF decrease by up to 10% over the baseline rotor (see for example

Figure 5-107). The 4/rev moments, p4xM , p4

yM , and p4zM , for the same mid-span

absorber increase slightly over the baseline rotor. However, for the outboard absorber,

the 4/rev moments decrease significantly as compared to the baseline rotor, with a

decrease of up to 10% for p4xM and p4

yM and 17% for p4zM .

5.8.2 HART Rotor Results

Similar results are obtained for the 40% scale HART I rotor. Using the notional

fuselage properties, the baseline rotor is trimmed at an advance ratio of 0.35, and the

control settings obtained are used for the rotor with the absorber added (see Table 5-4).

Table 5-4: Trim results (baseline HART I rotor)

Vehicle Orientation/Control Setting Symbol Value (deg) Collective pitch 75θ 6.0 Lateral cyclic c1θ 2.3 Longitudinal cycle s1θ -6.45 Forward shaft tilt sα 5.0 Lateral shaft tilt sφ -5.0 Tail rotor thrust trT 220 (lbf)

171

Since this is a wind tunnel model, and there is no tail rotor, the notional vehicle is

trimmed using a value for the tail rotor thrust instead of tail rotor collective pitch. For

this rotor, twenty beam elements are used to model the rotor blade. Results are shown for

an absorber damping ratio of 0.3, absorber location of 0.5, and three mass ratios: 0.01,

0.03, and 0.05.

5.8.2.1 Eigenvalue Analysis

The eigenvalues of the baseline HART I rotor are first calculated, and the first six

natural frequencies are compared with results from [60] (see Table 5-5) as a way to

validate the model used in this analysis. As is seen in Table 5-5, the frequencies agree

well with the results generated from the Rotorcraft Comprehensive Analysis System

(RCAS).

The eigenvalues are then calculated for the rotor with the absorber to determine

the amount of damping transferred to the lag mode of the rotor. These values are again

compared with the two-degree-of-freedom rigid blade results and listed in Table 5-6. As

described in section 5.8.1.1, since the HART I rotor has an inboard flexure, and, as a

Table 5-5: Comparison of baseline rotor modal frequencies with results from RCAS

Mode Frequency (Hz) RCAS frequency (Hz)

Lag 1 10.1 10.2 Flap 1 18.6 18.5 Flap 2 46.4 46.4 Lag 2 75.3 75.5 Flap 3 81.3 81.4 Lag 3 190.5 190.9

172

result, a spanwise velocity profile that is closer to that of a rigid blade than the constant

stiffness BO-105-type rotor examined previously, it is expected that the damping

transferred to the lag mode is more comparable to the rigid blade results than the BO-

105-type rotor that has a uniform stiffness.

5.8.2.2 Response and Loads Results

The blade tip responses, absorber responses, blade root and spanwise loads, and

hub loads are shown in Figures 5-114-5-125, and the trends are similar to those seen in

the results for the BO-105-type rotor. The peak-to-peak dynamic lag response (Figure 5-

114) increases as the absorber mass (and amount of damping transferred to the lag mode)

increases. The absorber has little effect on the peak-to-peak flap response (Figure 5-114),

with all results remaining within 4% of the baseline rotor. The largest dynamic response

of the absorber (Figure 5-115) occurs for an absorber mass ratio of 0.05 and is ± 0.9% of

the rotor radius.

The blade root forces and moments also exhibit no significant increases (or

decreases) over the baseline case. The largest increases occur in the peak-to-peak blade

root drag shear force (Figure 5-116) and lag moment (Figure 5-117) for an absorber mass

Table 5-6: Comparison of lag mode damping – HART I and rigid blades

Absorber Parameters Lag Mode Damping (% critical)

a aζ mα Elastic Blade Rigid Blade 0.5 0.3 0.01 12.4 6.0 0.5 0.3 0.03 14.1 14.3 0.5 0.3 0.05 14.1 14.0

173

ratio of 0.05 and show an increase of approximately 18% over the baseline case, with

smaller increases occurring for smaller mass ratios. The peak-to-peak blade root vertical

shear (Figure 5-118) and flap moment (Figure 5-119) remain within approximately 5% of

the baseline rotor for all absorber mass ratios. The peak-to-peak blade root radial shear

(Figure 5-120) remains within 4% of the baseline rotor value. As with the BO-105-type

rotor, there is a slight jump seen in the spanwise loads (Figures 5-121- 5-123) at the

absorber location, but no significant increase in the spanwise loads. In general, there is

little change in the steady hub loads (Figure 5-124) with the addition of the absorber,

while the 4/rev vibratory loads (Figure 5-125) decrease slightly when the absorber is

added to the rotor blades.

5.9 Summary

A rotor blade was modeled as an elastic beam undergoing coupled flap- and lag-

bending under aerodynamic loading, and the changes in the blade responses and loads, as

well as the hub loads due to the addition of the radial absorber (relative to a baseline rotor

with no absorber) were examined. It was shown that no significant increases in the blade

and hub loads were introduced due to the addition of the absorber for the rotor and most

of the absorber parameters examined. The dynamic blade lag response, blade root drag

shear, and blade root lag moment increases as the amount of damping transferred to the

lag mode increases. However, this does not have a detrimental effect on the hub loads,

and in fact, it may be possible to reduce the vibratory hub loads with the absorber. The

174

dynamic flap response, blade root vertical shear, and blade root flap moment remain

relatively unaffected by the addition of the absorber.

175

βp

βp

ψ

ψ

XH

YH

ZH, Z

X

Y, y

x

z

xa

ya

za

wv

xra

Undeformed elastic axis

Deformed elastic axis

βp

βp

ψ

ψ

XH

YH

ZH, Z

X

Y, y

x

z

xa

ya

za

wv

xra

βp

βp

ψ

ψ

XH

YH

ZH, Z

X

Y, y

x

z

βp

βp

ψ

ψ

XH

YH

ZH, Z

X

Y, y

x

z

xa

ya

za

wv

xra

Undeformed elastic axis

Deformed elastic axis

Figure 5-1: Elastic blade coordinate system with absorber

Absorber at kth

FEM node

v1

v2

w1

w2w'1

w'2

v'2

v'1

vN

wNv'N

w'N

vk

wk

xr

v'k

w'k

Absorber at kth

FEM node

v1

v2

w1

w2w'1

w'2

v'2

v'1

vN

wNv'N

w'N

vk

wk

xr

v'k

w'k

Figure 5-2: Spatial discretization of the rotor blade using the Finite Element Method

(with a radial absorber located at the kth finite element node)

176

C.G.

YfFy

Df

Fx

Ttr

xht

Mzf

TOP VIEW

xtr

Mz

LEFT SIDE VIEW

h

Fz

FxMy

Myf

Wxcg

DfLht

ztr

αs

h

Fy

Fz

Mx

Ttr

Mxf

W

ycg

φs

zht

REAR VIEW

C.G.

YfFy

Df

Fx

Ttr

xht

Mzf

TOP VIEW

xtr

Mz

C.G.

YfFy

Df

Fx

Ttr

xht

Mzf

TOP VIEW

xtr

Mz

LEFT SIDE VIEW

h

Fz

FxMy

Myf

Wxcg

DfLht

ztr

αs

LEFT SIDE VIEW

h

Fz

FxMy

Myf

Wxcg

DfLht

ztr

αs

h

Fy

Fz

Mx

Ttr

Mxf

W

ycg

φs

zht

REAR VIEW

h

Fy

Fz

Mx

Ttr

Mxf

W

ycg

φs

zht

REAR VIEW

Figure 5-3: Forces and moments exerted on a helicopter in level forward flight (figure

redrawn from [57])

177

½ka

½ka

½ka½ka

½ka

x

Inertial force

Spring forces transmitted to rotor blade

Spring forces on absorber mass

x

Inertial force

Spring forces transmitted to rotor blade

Spring forces on absorber mass Figure 5-4: Schematic of absorber mass and springs attached to rotor blade and

equivalent radial forces on blade due to absorber – spring and inertial forces

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.2

0.4

0.6

0.8

1

1.2

Spanwise Position (r/R)

Am

plit

ud

e

Hingeless blade with inboard flexureHingeless blade with constant stiffnessRigid blade

Figure 5-5: Fundamental lag mode shape – elastic and rigid blades

178

0 45 90 135 180 225 270 315 360−4

−2

0

2

4

6

8

10

Blade Azimuth, ψ (deg)

Bla

de

Tip

Fla

p a

nd

Lag

Res

po

nse

(%

R)

Flap

Lag

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05

Figure 5-6: Blade tip flap and lag responses ( 30a .= and 30a .=ζ )

0 45 90 135 180 225 270 315 360−4

−2

0

2

4

6

8

10


Bla

de

Tip

Fla

p a

nd

Lag

Res

po

nse

(%

R)

Flap

Lag

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


179

0 45 90 135 180 225 270 315 360−4

−2

0

2

4

6

8

10


Bla

de

Tip

Fla

p a

nd

Lag

Res

po

nse

(%

R)

Flap

Lag

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


0 45 90 135 180 225 270 315 360−4

−2

0

2

4

6

8

10


Bla

de

Tip

Fla

p a

nd

Lag

Res

po

nse

(%

R)

Flap

Lag

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


180

0 45 90 135 180 225 270 315 360−4

−2

0

2

4

6

8

10


Bla

de

Tip

Fla

p a

nd

Lag

Res

po

nse

(%

R)

Flap

Lag

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


0 45 90 135 180 225 270 315 360−4

−2

0

2

4

6

8

10


Bla

de

Tip

Fla

p a

nd

Lag

Res

po

nse

(%

R)

Flap

Lag

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


181

0 45 90 135 180 225 270 315 360−4

−2

0

2

4

6

8

10


Bla

de

Tip

Fla

p a

nd

Lag

Res

po

nse

(%

R)

Flap

Lag

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


0 45 90 135 180 225 270 315 360−4

−2

0

2

4

6

8

10


Bla

de

Tip

Fla

p a

nd

Lag

Res

po

nse

(%

R)

Flap

Lag

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


182

0 45 90 135 180 225 270 315 360−4

−2

0

2

4

6

8

10


Bla

de

Tip

Fla

p a

nd

Lag

Res

po

nse

(%

R)

Flap

Lag

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


0 45 90 135 180 225 270 315 360−1

0

1

2

3

4

5

6


Ab

sorb

er R

esp

on

se (

%R

)

αm

= 0.01α

m = 0.03

αm

= 0.05

Figure 5-15: Absorber response ( 30a .= and 30a .=ζ )

183

0 45 90 135 180 225 270 315 360−1

0

1

2

3

4

5

6


Ab

sorb

er R

esp

on

se (

%R

)

αm

= 0.01α

m = 0.03

αm

= 0.05


0 45 90 135 180 225 270 315 360−1

0

1

2

3

4

5

6


Ab

sorb

er R

esp

on

se (

%R

)

αm

= 0.01α

m = 0.03

αm

= 0.05


184

0 45 90 135 180 225 270 315 360−1

0

1

2

3

4

5

6


Ab

sorb

er R

esp

on

se (

%R

)

αm

= 0.01α

m = 0.03

αm

= 0.05


0 45 90 135 180 225 270 315 360−1

0

1

2

3

4

5

6


Ab

sorb

er R

esp

on

se (

%R

)

αm

= 0.01α

m = 0.03

αm

= 0.05


185

0 45 90 135 180 225 270 315 360−1

0

1

2

3

4

5

6


Ab

sorb

er R

esp

on

se (

%R

)

αm

= 0.01α

m = 0.03

αm

= 0.05


0 45 90 135 180 225 270 315 360−1

0

1

2

3

4

5

6


Ab

sorb

er R

esp

on

se (

%R

)

αm

= 0.01α

m = 0.03

αm

= 0.05


186

0 45 90 135 180 225 270 315 360−1

0

1

2

3

4

5

6


Ab

sorb

er R

esp

on

se (

%R

)

αm

= 0.01α

m = 0.03

αm

= 0.05


0 45 90 135 180 225 270 315 360−1

0

1

2

3

4

5

6


Ab

sorb

er R

esp

on

se (

%R

)

αm

= 0.01α

m = 0.03

αm

= 0.05


187

0 45 90 135 180 225 270 315 360−1200

−1000

−800

−600

−400

−200

0

200

400

Bla

de

Ro

ot

Dra

g S

hea

r (l

bf)


Baselineα

m = 0.01

αm

= 0.03α

m = 0.05

Figure 5-24: Blade root drag shear ( 30a .= and 30a .=ζ )

0 45 90 135 180 225 270 315 360−1200

−1000

−800

−600

−400

−200

0

200

400

Bla

de

Ro

ot

Dra

g S

hea

r (l

bf)


Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


188

0 45 90 135 180 225 270 315 360−1200

−1000

−800

−600

−400

−200

0

200

400

Bla

de

Ro

ot

Dra

g S

hea

r (l

bf)


Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


0 45 90 135 180 225 270 315 360−1200

−1000

−800

−600

−400

−200

0

200

400

Bla

de

Ro

ot

Dra

g S

hea

r (l

bf)


Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


189

0 45 90 135 180 225 270 315 360−1200

−1000

−800

−600

−400

−200

0

200

400

Bla

de

Ro

ot

Dra

g S

hea

r (l

bf)


Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


0 45 90 135 180 225 270 315 360−1200

−1000

−800

−600

−400

−200

0

200

400

Bla

de

Ro

ot

Dra

g S

hea

r (l

bf)


Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


190

0 45 90 135 180 225 270 315 360−1200

−1000

−800

−600

−400

−200

0

200

400

Bla

de

Ro

ot

Dra

g S

hea

r (l

bf)


Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


0 45 90 135 180 225 270 315 360−1200

−1000

−800

−600

−400

−200

0

200

400

Bla

de

Ro

ot

Dra

g S

hea

r (l

bf)


Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


191

0 45 90 135 180 225 270 315 360−1200

−1000

−800

−600

−400

−200

0

200

400

Bla

de

Ro

ot

Dra

g S

hea

r (l

bf)


Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


0 45 90 135 180 225 270 315 360−5000

−4000

−3000

−2000

−1000

0

1000

2000

3000

Bla

de

Ro

ot

Lag

Ben

din

g M

om

ent

(ft−

lbf)


Baselineα

m = 0.01

αm

= 0.03α

m = 0.05

Figure 5-33: Blade root lag bending moment ( 30a .= and 30a .=ζ )

192

0 45 90 135 180 225 270 315 360−5000

−4000

−3000

−2000

−1000

0

1000

2000

3000

Bla

de

Ro

ot

Lag

Ben

din

g M

om

ent

(ft−

lbf)


Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


0 45 90 135 180 225 270 315 360−5000

−4000

−3000

−2000

−1000

0

1000

2000

3000

Bla

de

Ro

ot

Lag

Ben

din

g M

om

ent

(ft−

lbf)


Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


193

0 45 90 135 180 225 270 315 360−5000

−4000

−3000

−2000

−1000

0

1000

2000

3000

Bla

de

Ro

ot

Lag

Ben

din

g M

om

ent

(ft−

lbf)


Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


0 45 90 135 180 225 270 315 360−5000

−4000

−3000

−2000

−1000

0

1000

2000

3000

Bla

de

Ro

ot

Lag

Ben

din

g M

om

ent

(ft−

lbf)


Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


194

0 45 90 135 180 225 270 315 360−5000

−4000

−3000

−2000

−1000

0

1000

2000

3000

Bla

de

Ro

ot

Lag

Ben

din

g M

om

ent

(ft−

lbf)


Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


0 45 90 135 180 225 270 315 360−5000

−4000

−3000

−2000

−1000

0

1000

2000

3000

Bla

de

Ro

ot

Lag

Ben

din

g M

om

ent

(ft−

lbf)


Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


195

0 45 90 135 180 225 270 315 360−5000

−4000

−3000

−2000

−1000

0

1000

2000

3000

Bla

de

Ro

ot

Lag

Ben

din

g M

om

ent

(ft−

lbf)


Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


0 45 90 135 180 225 270 315 360−5000

−4000

−3000

−2000

−1000

0

1000

2000

Bla

de

Ro

ot

Lag

Ben

din

g M

om

ent

(ft−

lbf)


Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


196

0 45 90 135 180 225 270 315 3600

500

1000

1500

2000

2500

3000

Bla

de

Ro

ot

Ver

tica

l Sh

ear

(lb

f)


Baselineα

m = 0.01

αm

= 0.03α

m = 0.05

Figure 5-42: Blade root vertical shear ( 30a .= and 30a .=ζ )

0 45 90 135 180 225 270 315 3600

500

1000

1500

2000

2500

3000

Bla

de

Ro

ot

Ver

tica

l Sh

ear

(lb

f)


Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


197

0 45 90 135 180 225 270 315 3600

500

1000

1500

2000

2500

3000

Bla

de

Ro

ot

Ver

tica

l Sh

ear

(lb

f)


Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


0 45 90 135 180 225 270 315 3600

500

1000

1500

2000

2500

3000

Bla

de

Ro

ot

Ver

tica

l Sh

ear

(lb

f)


Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


198

0 45 90 135 180 225 270 315 3600

500

1000

1500

2000

2500

3000

Bla

de

Ro

ot

Ver

tica

l Sh

ear

(lb

f)


Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


0 45 90 135 180 225 270 315 3600

500

1000

1500

2000

2500

3000

Bla

de

Ro

ot

Ver

tica

l Sh

ear

(lb

f)


Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


199

0 45 90 135 180 225 270 315 3600

500

1000

1500

2000

2500

3000

Bla

de

Ro

ot

Ver

tica

l Sh

ear

(lb

f)


Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


0 45 90 135 180 225 270 315 3600

500

1000

1500

2000

2500

3000

Bla

de

Ro

ot

Ver

tica

l Sh

ear

(lb

f)


Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


200

0 45 90 135 180 225 270 315 3600

500

1000

1500

2000

2500

3000

Bla

de

Ro

ot

Ver

tica

l Sh

ear

(lb

f)


Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


0 45 90 135 180 225 270 315 3600

1000

2000

3000

4000

5000

6000

Bla

de

Ro

ot

Fla

p B

end

ing

Mo

men

t (f

t−lb

f)


Baselineα

m = 0.01

αm

= 0.03α

m = 0.05

Figure 5-51: Blade root flap bending moment ( 30a .= and 30a .=ζ )

201

0 45 90 135 180 225 270 315 3600

1000

2000

3000

4000

5000

6000

Bla

de

Ro

ot

Fla

p B

end

ing

Mo

men

t (f

t−lb

f)


Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


0 45 90 135 180 225 270 315 3600

1000

2000

3000

4000

5000

6000

Bla

de

Ro

ot

Fla

p B

end

ing

Mo

men

t (f

t−lb

f)


Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


202

0 45 90 135 180 225 270 315 3600

1000

2000

3000

4000

5000

6000

Bla

de

Ro

ot

Fla

p B

end

ing

Mo

men

t (f

t−lb

f)


Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


0 45 90 135 180 225 270 315 3600

1000

2000

3000

4000

5000

6000

Bla

de

Ro

ot

Fla

p B

end

ing

Mo

men

t (f

t−lb

f)


Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


203

0 45 90 135 180 225 270 315 3600

1000

2000

3000

4000

5000

6000

Bla

de

Ro

ot

Fla

p B

end

ing

Mo

men

t (f

t−lb

f)


Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


0 45 90 135 180 225 270 315 3600

1000

2000

3000

4000

5000

6000

Bla

de

Ro

ot

Fla

p B

end

ing

Mo

men

t (f

t−lb

f)


Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


204

0 45 90 135 180 225 270 315 3600

1000

2000

3000

4000

5000

6000

Bla

de

Ro

ot

Fla

p B

end

ing

Mo

men

t (f

t−lb

f)


Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


0 45 90 135 180 225 270 315 3600

1000

2000

3000

4000

5000

6000

Bla

de

Ro

ot

Fla

p B

end

ing

Mo

men

t (f

t−lb

f)


Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


205

0 45 90 135 180 225 270 315 3602.7

2.75

2.8

2.85

2.9

2.95x 10

4

Bla

de

Ro

ot

Rad

ial S

hea

r (l

bf)


Baselineα

m = 0.01

αm

= 0.03α

m = 0.05

Figure 5-60: Blade root radial shear ( 30a .= and 30a .=ζ )

0 45 90 135 180 225 270 315 3602.7

2.75

2.8

2.85

2.9

2.95x 10

4

Bla

de

Ro

ot

Rad

ial S

hea

r (l

bf)


Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


206

0 45 90 135 180 225 270 315 3602.7

2.75

2.8

2.85

2.9

2.95x 10

4

Bla

de

Ro

ot

Rad

ial S

hea

r (l

bf)


Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


0 45 90 135 180 225 270 315 3602.7

2.75

2.8

2.85

2.9

2.95x 10

4

Bla

de

Ro

ot

Rad

ial S

hea

r(lb

f)


Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


207

0 45 90 135 180 225 270 315 3602.7

2.75

2.8

2.85

2.9

2.95x 10

4

Bla

de

Ro

ot

Rad

ial S

hea

r (l

bf)


Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


0 45 90 135 180 225 270 315 3602.7

2.75

2.8

2.85

2.9

2.95x 10

4

Bla

de

Ro

ot

Rad

ial S

hea

r (l

bf)


Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


208

0 45 90 135 180 225 270 315 3602.7

2.75

2.8

2.85

2.9

2.95x 10

4

Bla

de

Ro

ot

Rad

ial S

hea

r (l

bf)


Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


0 45 90 135 180 225 270 315 3602.7

2.75

2.8

2.85

2.9

2.95x 10

4

Bla

de

Ro

ot

Rad

ial S

hea

r (l

bf)


Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


209

0 45 90 135 180 225 270 315 3602.7

2.75

2.8

2.85

2.9

2.95x 10

4

Bla

de

Ro

ot

Rad

ial S

hea

r (l

bf)


Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


0.2 0.4 0.6 0.8 1−1500

−1000

−500

0

500


Sp

anw

ise

Dra

g S

hea

r (l

bf)

ψ = 45 deg

0.2 0.4 0.6 0.8 1−1500

−1000

−500

0

500


Sp

anw

ise

Dra

g S

hea

r (l

bf)

ψ = 135 deg

0.2 0.4 0.6 0.8 1−1500

−1000

−500

0

500


Sp

anw

ise

Dra

g S

hea

r (l

bf)

ψ = 225 deg

0.2 0.4 0.6 0.8 1−1500

−1000

−500

0

500


Sp

anw

ise

Dra

g S

hea

r (l

bf)

ψ = 315 deg

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05

Figure 5-69: Spanwise drag shear ( 30a .= and 30a .=ζ )

210

0.2 0.4 0.6 0.8 1−1500

−1000

−500

0

500


Sp

anw

ise

Dra

g S

hea

r (l

bf)

ψ = 45 deg

0.2 0.4 0.6 0.8 1−1500

−1000

−500

0

500


Sp

anw

ise

Dra

g S

hea

r (l

bf)

ψ = 135 deg

0.2 0.4 0.6 0.8 1−1500

−1000

−500

0

500


Sp

anw

ise

Dra

g S

hea

r (l

bf)

ψ = 225 deg

0.2 0.4 0.6 0.8 1−1500

−1000

−500

0

500


Sp

anw

ise

Dra

g S

hea

r (l

bf)

ψ = 315 deg

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


0.2 0.4 0.6 0.8 1−1500

−1000

−500

0

500


Sp

anw

ise

Dra

g S

hea

r (l

bf)

ψ = 45 deg

0.2 0.4 0.6 0.8 1−1500

−1000

−500

0

500


Sp

anw

ise

Dra

g S

hea

r (l

bf)

ψ = 135 deg

0.2 0.4 0.6 0.8 1−1500

−1000

−500

0

500


Sp

anw

ise

Dra

g S

hea

r (l

bf)

ψ = 225 deg

0.2 0.4 0.6 0.8 1−1500

−1000

−500

0

500


Sp

anw

ise

Dra

g S

hea

r (l

bf)

ψ = 315 deg

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


211

0.2 0.4 0.6 0.8 1−1500

−1000

−500

0

500


Sp

anw

ise

Dra

g S

hea

r (l

bf)

ψ = 45 deg

0.2 0.4 0.6 0.8 1−1500

−1000

−500

0

500


Sp

anw

ise

Dra

g S

hea

r (l

bf)

ψ = 135 deg

0.2 0.4 0.6 0.8 1−1500

−1000

−500

0

500


Sp

anw

ise

Dra

g S

hea

r (l

bf)

ψ = 225 deg

0.2 0.4 0.6 0.8 1−1500

−1000

−500

0

500


Sp

anw

ise

Dra

g S

hea

r (l

bf)

ψ = 315 deg

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


0.2 0.4 0.6 0.8 1−1500

−1000

−500

0

500


Sp

anw

ise

Dra

g S

hea

r (l

bf)

ψ = 45 deg

0.2 0.4 0.6 0.8 1−1500

−1000

−500

0

500


Sp

anw

ise

Dra

g S

hea

r (l

bf)

ψ = 135 deg

0.2 0.4 0.6 0.8 1−1500

−1000

−500

0

500


Sp

anw

ise

Dra

g S

hea

r (l

bf)

ψ = 225 deg

0.2 0.4 0.6 0.8 1−1500

−1000

−500

0

500


Sp

anw

ise

Dra

g S

hea

r (l

bf)

ψ = 315 deg

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


212

0.2 0.4 0.6 0.8 1−1500

−1000

−500

0

500


Sp

anw

ise

Dra

g S

hea

r (l

bf)

ψ = 45 deg

0.2 0.4 0.6 0.8 1−1500

−1000

−500

0

500


Sp

anw

ise

Dra

g S

hea

r (l

bf)

ψ = 135 deg

0.2 0.4 0.6 0.8 1−1500

−1000

−500

0

500


Sp

anw

ise

Dra

g S

hea

r (l

bf)

ψ = 225 deg

0.2 0.4 0.6 0.8 1−1500

−1000

−500

0

500


Sp

anw

ise

Dra

g S

hea

r (l

bf)

ψ = 315 deg

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


0.2 0.4 0.6 0.8 1−1500

−1000

−500

0

500


Sp

anw

ise

Dra

g S

hea

r (l

bf)

ψ = 45 deg

0.2 0.4 0.6 0.8 1−1500

−1000

−500

0

500


Sp

anw

ise

Dra

g S

hea

r (l

bf)

ψ = 135 deg

0.2 0.4 0.6 0.8 1−1500

−1000

−500

0

500


Sp

anw

ise

Dra

g S

hea

r (l

bf)

ψ = 225 deg

0.2 0.4 0.6 0.8 1−1500

−1000

−500

0

500


Sp

anw

ise

Dra

g S

hea

r (l

bf)

ψ = 315 deg

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


213

0.2 0.4 0.6 0.8 1−1500

−1000

−500

0

500


Sp

anw

ise

Dra

g S

hea

r (l

bf)

ψ = 45 deg

0.2 0.4 0.6 0.8 1−1500

−1000

−500

0

500


Sp

anw

ise

Dra

g S

hea

r (l

bf)

ψ = 135 deg

0.2 0.4 0.6 0.8 1−1500

−1000

−500

0

500


Sp

anw

ise

Dra

g S

hea

r (l

bf)

ψ = 225 deg

0.2 0.4 0.6 0.8 1−1500

−1000

−500

0

500


Sp

anw

ise

Dra

g S

hea

r (l

bf)

ψ = 315 deg

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


0.2 0.4 0.6 0.8 1−1500

−1000

−500

0

500


Sp

anw

ise

Dra

g S

hea

r (l

bf)

ψ = 45 deg

0.2 0.4 0.6 0.8 1−1500

−1000

−500

0

500


Sp

anw

ise

Dra

g S

hea

r (l

bf)

ψ = 135 deg

0.2 0.4 0.6 0.8 1−1500

−1000

−500

0

500


Sp

anw

ise

Dra

g S

hea

r (l

bf)

ψ = 225 deg

0.2 0.4 0.6 0.8 1−1500

−1000

−500

0

500


Sp

anw

ise

Dra

g S

hea

r (l

bf)

ψ = 315 deg

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


214

0.2 0.4 0.6 0.8 10

500

1000

1500

2000

2500


Sp

anw

ise

Ver

tica

l Sh

ear

(lb

f)

ψ = 45 deg

0.2 0.4 0.6 0.8 10

500

1000

1500

2000

2500


Sp

anw

ise

Ver

tica

l Sh

ear

(lb

f)

ψ = 135 deg

0.2 0.4 0.6 0.8 10

500

1000

1500

2000

2500


Sp

anw

ise

Ver

tica

l Sh

ear

(lb

f)

ψ = 225 deg

0.2 0.4 0.6 0.8 10

500

1000

1500

2000

2500

Spanwise Position (r/R)S

pan

wis

e V

erti

cal S

hea

r (l

bf)

ψ = 315 deg

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05

Figure 5-78: Spanwise vertical shear ( 30a .= and 30a .=ζ )

0.2 0.4 0.6 0.8 10

500

1000

1500

2000

2500


Sp

anw

ise

Ver

tica

l Sh

ear

(lb

f)

ψ = 45 deg

0.2 0.4 0.6 0.8 10

500

1000

1500

2000

2500


Sp

anw

ise

Ver

tica

l Sh

ear

(lb

f)

ψ = 135 deg

0.2 0.4 0.6 0.8 10

500

1000

1500

2000

2500


Sp

anw

ise

Ver

tica

l Sh

ear

(lb

f)

ψ = 225 deg

0.2 0.4 0.6 0.8 10

500

1000

1500

2000

2500


Sp

anw

ise

Ver

tica

l Sh

ear

(lb

f)

ψ = 315 deg

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


215

0.2 0.4 0.6 0.8 10

500

1000

1500

2000

2500


Sp

anw

ise

Ver

tica

l Sh

ear

(lb

f)

ψ = 45 deg

0.2 0.4 0.6 0.8 10

500

1000

1500

2000

2500


Sp

anw

ise

Ver

tica

l Sh

ear

(lb

f)

ψ = 135 deg

0.2 0.4 0.6 0.8 10

500

1000

1500

2000

2500


Sp

anw

ise

Ver

tica

l Sh

ear

(lb

f)

ψ = 225 deg

0.2 0.4 0.6 0.8 10

500

1000

1500

2000

2500


pan

wis

e V

erti

cal S

hea

r (l

bf)

ψ = 315 deg

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


0.2 0.4 0.6 0.8 10

500

1000

1500

2000

2500


Sp

anw

ise

Ver

tica

l Sh

ear

(lb

f)

ψ = 45 deg

0.2 0.4 0.6 0.8 10

500

1000

1500

2000

2500


Sp

anw

ise

Ver

tica

l Sh

ear

(lb

f)

ψ = 135 deg

0.2 0.4 0.6 0.8 10

500

1000

1500

2000

2500


Sp

anw

ise

Ver

tica

l Sh

ear

(lb

f)

ψ = 225 deg

0.2 0.4 0.6 0.8 10

500

1000

1500

2000

2500


Sp

anw

ise

Ver

tica

l Sh

ear

(lb

f)

ψ = 315 deg

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


216

0.2 0.4 0.6 0.8 10

500

1000

1500

2000

2500


Sp

anw

ise

Ver

tica

l Sh

ear

(lb

f)

ψ = 45 deg

0.2 0.4 0.6 0.8 10

500

1000

1500

2000

2500


Sp

anw

ise

Ver

tica

l Sh

ear

(lb

f)

ψ = 135 deg

0.2 0.4 0.6 0.8 10

500

1000

1500

2000

2500


Sp

anw

ise

Ver

tica

l Sh

ear

(lb

f)

ψ = 225 deg

0.2 0.4 0.6 0.8 10

500

1000

1500

2000

2500


pan

wis

e V

erti

cal S

hea

r (l

bf)

ψ = 315 deg

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


0.2 0.4 0.6 0.8 10

500

1000

1500

2000

2500


Sp

anw

ise

Ver

tica

l Sh

ear

(lb

f)

ψ = 45 deg

0.2 0.4 0.6 0.8 10

500

1000

1500

2000

2500


Sp

anw

ise

Ver

tica

l Sh

ear

(lb

f)

ψ = 135 deg

0.2 0.4 0.6 0.8 10

500

1000

1500

2000

2500


Sp

anw

ise

Ver

tica

l Sh

ear

(lb

f)

ψ = 225 deg

0.2 0.4 0.6 0.8 10

500

1000

1500

2000

2500


Sp

anw

ise

Ver

tica

l Sh

ear

(lb

f)

ψ = 315 deg

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


217

0.2 0.4 0.6 0.8 10

500

1000

1500

2000

2500


Sp

anw

ise

Ver

tica

l Sh

ear

(lb

f)

ψ = 45 deg

0.2 0.4 0.6 0.8 10

500

1000

1500

2000

2500


Sp

anw

ise

Ver

tica

l Sh

ear

(lb

f)

ψ = 135 deg

0.2 0.4 0.6 0.8 10

500

1000

1500

2000

2500


Sp

anw

ise

Ver

tica

l Sh

ear

(lb

f)

ψ = 225 deg

0.2 0.4 0.6 0.8 10

500

1000

1500

2000

2500


pan

wis

e V

erti

cal S

hea

r (l

bf)

ψ = 315 deg

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


0.2 0.4 0.6 0.8 10

500

1000

1500

2000

2500


Sp

anw

ise

Ver

tica

l Sh

ear

(lb

f)

ψ = 45 deg

0.2 0.4 0.6 0.8 10

500

1000

1500

2000

2500


Sp

anw

ise

Ver

tica

l Sh

ear

(lb

f)

ψ = 135 deg

0.2 0.4 0.6 0.8 10

500

1000

1500

2000

2500


Sp

anw

ise

Ver

tica

l Sh

ear

(lb

f)

ψ = 225 deg

0.2 0.4 0.6 0.8 10

500

1000

1500

2000

2500


Sp

anw

ise

Ver

tica

l Sh

ear

(lb

f)

ψ = 315 deg

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


218

0.2 0.4 0.6 0.8 10

500

1000

1500

2000

2500


Sp

anw

ise

Ver

tica

l Sh

ear

(lb

f)

ψ = 45 deg

0.2 0.4 0.6 0.8 10

500

1000

1500

2000

2500


Sp

anw

ise

Ver

tica

l Sh

ear

(lb

f)

ψ = 135 deg

0.2 0.4 0.6 0.8 10

500

1000

1500

2000

2500


Sp

anw

ise

Ver

tica

l Sh

ear

(lb

f)

ψ = 225 deg

0.2 0.4 0.6 0.8 10

500

1000

1500

2000

2500


pan

wis

e V

erti

cal S

hea

r (l

bf)

ψ = 315 deg

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3x 10

4


Sp

anw

ise

Rad

ial S

hea

r (l

bf)

ψ = 45 deg

0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3x 10

4


Sp

anw

ise

Rad

ial S

hea

r (l

bf)

ψ = 135 deg

0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3x 10

4


Sp

anw

ise

Rad

ial S

hea

r (l

bf)

ψ = 225 deg

0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3x 10

4


Sp

anw

ise

Rad

ial S

hea

r (l

bf)

ψ = 315 deg

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05

Figure 5-87: Spanwise radial shear ( 30a .= and 30a .=ζ )

219

0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3x 10

4


Sp

anw

ise

Rad

ial S

hea

r (l

bf)

ψ = 45 deg

0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3x 10

4


Sp

anw

ise

Rad

ial S

hea

r (l

bf)

ψ = 135 deg

0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3x 10

4


Sp

anw

ise

Rad

ial S

hea

r (l

bf)

ψ = 225 deg

0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3x 10

4


pan

wis

e R

adia

l Sh

ear

(lb

f)

ψ = 315 deg

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3x 10

4


Sp

anw

ise

Rad

ial S

hea

r (l

bf)

ψ = 45 deg

0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3x 10

4


Sp

anw

ise

Rad

ial S

hea

r (l

bf)

ψ = 135 deg

0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3x 10

4


Sp

anw

ise

Rad

ial S

hea

r (l

bf)

ψ = 225 deg

0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3x 10

4


Sp

anw

ise

Rad

ial S

hea

r (l

bf)

ψ = 315 deg

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


220

0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3x 10

4


Sp

anw

ise

Rad

ial S

hea

r (l

bf)

ψ = 45 deg

0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3x 10

4


Sp

anw

ise

Rad

ial S

hea

r (l

bf)

ψ = 135 deg

0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3x 10

4


Sp

anw

ise

Rad

ial S

hea

r (l

bf)

ψ = 225 deg

0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3x 10

4


pan

wis

e R

adia

l Sh

ear

(lb

f)

ψ = 315 deg

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3x 10

4


Sp

anw

ise

Rad

ial S

hea

r (l

bf)

ψ = 45 deg

0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3x 10

4


Sp

anw

ise

Rad

ial S

hea

r (l

bf)

ψ = 135 deg

0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3x 10

4


Sp

anw

ise

Rad

ial S

hea

r (l

bf)

ψ = 225 deg

0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3x 10

4


Sp

anw

ise

Rad

ial S

hea

r (l

bf)

ψ = 315 deg

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


221

0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3x 10

4


Sp

anw

ise

Rad

ial S

hea

r (l

bf)

ψ = 45 deg

0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3x 10

4


Sp

anw

ise

Rad

ial S

hea

r (l

bf)

ψ = 135 deg

0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3x 10

4


Sp

anw

ise

Rad

ial S

hea

r (l

bf)

ψ = 225 deg

0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3x 10

4


pan

wis

e R

adia

l Sh

ear

(lb

f)

ψ = 315 deg

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3x 10

4


Sp

anw

ise

Rad

ial S

hea

r (l

bf)

ψ = 45 deg

0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3x 10

4


Sp

anw

ise

Rad

ial S

hea

r (l

bf)

ψ = 135 deg

0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3x 10

4


Sp

anw

ise

Rad

ial S

hea

r (l

bf)

ψ = 225 deg

0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3x 10

4


Sp

anw

ise

Rad

ial S

hea

r (l

bf)

ψ = 315 deg

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


222

0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3x 10

4


Sp

anw

ise

Rad

ial S

hea

r (l

bf)

ψ = 45 deg

0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3x 10

4


Sp

anw

ise

Rad

ial S

hea

r (l

bf)

ψ = 135 deg

0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3x 10

4


Sp

anw

ise

Rad

ial S

hea

r (l

bf)

ψ = 225 deg

0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3x 10

4


pan

wis

e R

adia

l Sh

ear

(lb

f)

ψ = 315 deg

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3x 10

4


Sp

anw

ise

Rad

ial S

hea

r (l

bf)

ψ = 45 deg

0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3x 10

4


Sp

anw

ise

Rad

ial S

hea

r (l

bf)

ψ = 135 deg

0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3x 10

4


Sp

anw

ise

Rad

ial S

hea

r (l

bf)

ψ = 225 deg

0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3x 10

4


Sp

anw

ise

Rad

ial S

hea

r (l

bf)

ψ = 315 deg

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


223

Fz Fy Fz Mx My Mz−6000

−4000

−2000

0

2000

4000

6000

Ste

ady

Hu

b L

oad

s (F

orc

es −

lbf,

Mo

men

ts −

ft−

lbf)

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05

Figure 5-96: Steady hub loads ( 30a .= and 30a .=ζ )


−4000

−2000

0

2000

4000

6000

Ste

ady

Hu

b L

oad

s (F

orc

es −

lbf,

Mo

men

ts −

ft−

lbf)

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


224


−4000

−2000

0

2000

4000

6000

Ste

ady

Hu

b L

oad

s (F

orc

es −

lbf,

Mo

men

ts −

ft−

lbf)

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05



−4000

−2000

0

2000

4000

6000

Ste

ady

Hu

b L

oad

s (F

orc

es −

lbf,

Mo

men

ts −

ft−

lbf)

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


225


−4000

−2000

0

2000

4000

6000

Ste

ady

Hu

b L

oad

s (F

orc

es −

lbf,

Mo

men

ts −

ft−

lbf)

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05



−4000

−2000

0

2000

4000

6000

Ste

ady

Hu

b L

oad

s (F

orc

es −

lbf,

Mo

men

ts −

ft−

lbf)

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


226


−4000

−2000

0

2000

4000

6000

Ste

ady

Hu

b L

oad

s (F

orc

es −

lbf,

Mo

men

ts −

ft−

lbf)

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05



−4000

−2000

0

2000

4000

6000

Ste

ady

Hu

b L

oad

s (F

orc

es −

lbf,

Mo

men

ts −

ft−

lbf)

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


227


−4000

−2000

0

2000

4000

6000

Ste

ady

Hu

b L

oad

s (F

orc

es −

lbf,

Mo

men

ts −

ft−

lbf)

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


Fz Fy Fz Mx My Mz0

100

200

300

400

500

600

Vib

rato

ry H

ub

Lo

ads

(Fo

rces

− lb

f, M

om

ents

− f

t−lb

f)

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05

Figure 5-105: 4/rev vibratory hub loads ( 30a .= and 30a .=ζ )

228

Fz Fy Fz Mx My Mz0

100

200

300

400

500

600

Vib

rato

ry H

ub

Lo

ads

(Fo

rces

− lb

f, M

om

ents

− f

t−lb

f)

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


Fz Fy Fz Mx My Mz0

100

200

300

400

500

600

Vib

rato

ry H

ub

Lo

ads

(Fo

rces

− lb

f, M

om

ents

− f

t−lb

f)

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


229

Fz Fy Fz Mx My Mz0

100

200

300

400

500

600

Vib

rato

ry H

ub

Lo

ads

(Fo

rces

− lb

f, M

om

ents

− f

t−lb

f)

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


Fz Fy Fz Mx My Mz0

100

200

300

400

500

600

Vib

rato

ry H

ub

Lo

ads

(Fo

rces

− lb

f, M

om

ents

− f

t−lb

f)

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


230

Fz Fy Fz Mx My Mz0

100

200

300

400

500

600

Vib

rato

ry H

ub

Lo

ads

(Fo

rces

− lb

f, M

om

ents

− f

t−lb

f)

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


Fz Fy Fz Mx My Mz0

100

200

300

400

500

600

Vib

rato

ry H

ub

Lo

ads

(Fo

rces

− lb

f, M

om

ents

− f

t−lb

f)

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


231

Fz Fy Fz Mx My Mz0

100

200

300

400

500

600

Vib

rato

ry H

ub

Lo

ads

(Fo

rces

− lb

f, M

om

ents

− f

t−lb

f)

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


Fz Fy Fz Mx My Mz0

100

200

300

400

500

600

Vib

rato

ry H

ub

Lo

ads

(Fo

rces

− lb

f, M

om

ents

− f

t−lb

f)

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05


232

0 45 90 135 180 225 270 315 360−10

−5

0

5

10

15

20


Bla

de

Tip

Fla

p a

nd

Lag

Res

po

nse

(%

R)

Flap

Lag

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05

Figure 5-114: Blade tip flap and lag responses (HART rotor)

0 45 90 135 180 225 270 315 3601.5

2

2.5

3

3.5


Ab

sorb

er R

esp

on

se (

%R

)

αm

= 0.01α

m = 0.03

αm

= 0.05

Figure 5-115: Absorber response (HART rotor)

233

0 45 90 135 180 225 270 315 360−700

−600

−500

−400

−300

−200

−100

Bla

de

Ro

ot

Dra

g S

hea

r (l

bf)


Baselineα

m = 0.01

αm

= 0.03α

m = 0.05

Figure 5-116: Blade root drag shear (HART Rotor)

0 45 90 135 180 225 270 315 360−300

−250

−200

−150

−100

−50

0

50

100

Bla

de

Ro

ot

Lag

Ben

din

g M

om

ent

(ft−

lbf)


Baselineα

m = 0.01

αm

= 0.03α

m = 0.05

Figure 5-117: Blade root lag bending moment (HART rotor)

234

0 45 90 135 180 225 270 315 360−1000

−500

0

500

1000

1500

2000

Bla

de

Ro

ot

Ver

tica

l Sh

ear

(lb

f)


Baselineα

m = 0.01

αm

= 0.03α

m = 0.05

Figure 5-118: Blade root vertical shear (HART rotor)

0 45 90 135 180 225 270 315 360−150

−100

−50

0

50

100

150

200

250

300

350

Bla

de

Ro

ot

Fla

p B

end

ing

Mo

men

t (f

t−lb

f)


Baselineα

m = 0.01

αm

= 0.03α

m = 0.05

Figure 5-119: Blade root flap bending moment (HART rotor)

235

0 45 90 135 180 225 270 315 3602.3

2.31

2.32

2.33

2.34

2.35

2.36

2.37

2.38

2.39

2.4x 10

4

Bla

de

Ro

ot

Rad

ial S

hea

r (l

bf)


Baselineα

m = 0.01

αm

= 0.03α

m = 0.05

Figure 5-120: Blade root radial shear (HART rotor)

0.2 0.4 0.6 0.8 1−1000

−500

0

500


Sp

anw

ise

Dra

g S

hea

r (l

bf)

ψ = 45 deg

0.2 0.4 0.6 0.8 1−1000

−500

0

500


Sp

anw

ise

Dra

g S

hea

r (l

bf)

ψ = 135 deg

0.2 0.4 0.6 0.8 1−1000

−500

0

500


Sp

anw

ise

Dra

g S

hea

r (l

bf)

ψ = 225 deg

0.2 0.4 0.6 0.8 1−1000

−500

0

500


Sp

anw

ise

Dra

g S

hea

r (l

bf)

ψ = 315 deg

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05

Figure 5-121: Spanwise drag shear (HART rotor)

236

0.2 0.4 0.6 0.8 1−500

0

500

1000

1500

2000


Sp

anw

ise

Ver

tica

l Sh

ear

(lb

f)

ψ = 45 deg

0.2 0.4 0.6 0.8 1−500

0

500

1000

1500

2000


Sp

anw

ise

Ver

tica

l Sh

ear

(lb

f)

ψ = 135 deg

0.2 0.4 0.6 0.8 1−500

0

500

1000

1500

2000


Sp

anw

ise

Ver

tica

l Sh

ear

(lb

f)

ψ = 225 deg

0.2 0.4 0.6 0.8 1−500

0

500

1000

1500

2000


pan

wis

e V

erti

cal S

hea

r (l

bf)

ψ = 315 deg

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05

Figure 5-122: Spanwise vertical shear (HART rotor)

0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5x 10

4


Sp

anw

ise

Rad

ial S

hea

r (l

bf)

ψ = 45 degψ = 45 deg

0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5x 10

4


Sp

anw

ise

Rad

ial S

hea

r (l

bf)

ψ = 135 degψ = 135 deg

0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5x 10

4


Sp

anw

ise

Rad

ial S

hea

r (l

bf)

ψ = 225 degψ = 225 deg

0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5x 10

4


Sp

anw

ise

Rad

ial S

hea

r (l

bf)

ψ = 315 degψ = 315 deg

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05

Figure 5-123: Spanwise radial shear (HART rotor)

237


−500

0

500

1000

1500

2000

2500

3000

3500

Ste

ady

Hu

b L

oad

s (F

orc

es −

lbf,

Mo

men

ts −

ft−

lbf)

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05

Figure 5-124: Steady hub loads (HART rotor)

Fz Fy Fz Mx My Mz0

50

100

150

Vib

rato

ry H

ub

Lo

ads

(Fo

rces

− lb

f, M

om

ents

− f

t−lb

f)

Baselineα

m = 0.01

αm

= 0.03α

m = 0.05

Figure 5-125: 4/rev vibratory hub loads (HART rotor)

Chapter 6

Device Concepts

As described previously, the absorber will be required to operate in a high

centrifugal force field. This type of extremely large static loading dictates that the radial

absorber system will not be able to be constructed with a simple spring that has linear

stiffness characteristics. The system will require either some type of frequency-

dependent stiffness material, or a material with nonlinear spring stiffness characteristics,

or perhaps a combination of both. This chapter addresses some possible concepts that

could possess this type of stiffness characteristics and be built for use in the radial

vibration absorber system.

6.1 Fluid Elastic Absorber

Currently, the embedded chordwise inertial damper is designed to overcome the

centrifugal force and resulting large static displacement by means of a fluid elastic device

[44]. The chordwise fluid elastic inertial damper system consists of a mass which is

rigidly connected to a fluid vessel, on an elastomeric spring embedded within the rotor

blade. The design of the fluid elastic damper allows for a large static stiffness, enabling it

to withstand the centrifugal force with a relatively small static displacement, yet still have

a low dynamic stiffness, which allows it to have a tuned dynamic frequency to achieve

the levels of lag damping required for the aeromechanical stability of the rotor system.

239

The suitability of the fluid elastic concept for use with the radial absorber system

is examined in this section. As in Chapter 2, the rotor blade is modeled as a rigid body

undergoing lead-lag rotations about a spring-restrained hinge near the root with no hinge

offset and no elastic bending deformations considered. The absorber is assumed to be

embedded within the rotor blade at a distance a from the hub. The embedded absorber is

modeled as a fluid elastic damper moving radially within the blade. The fluid elastic

device is modeled in the same fashion as in [43-45], with the conceptual design of the

device shown in Figure 6-1. The mechanical equivalent of the fluid elastic damper [26,

41-44] is used to derive the equations of motion of the system and is shown in Figure 6-2.

The primary mass of the absorber is represented by apm , and the fluid mass is represented

by atm . The tuning port area ratio, G , is represented by the ratio of the lengths of arms

a and b : abG = . While the fluid elastic damper consists of two masses, it is still a

single degree of freedom system, with the fluid motion directly linked to the motion of

the primary mass through the kinematic constraint:

where tx is the motion of the tuning mass in the radial direction, and rx is the motion of

the primary mass, also in the radial direction. By design, when the primary mass moves

in the radial direction a distance rx , the tuning mass moves in the opposite direction a

distance tx .

The linearized equations of motion are derived to be

rt x1Gx )( −= 6.1

240

For the radial vibration absorber, the Coriolis coupling between the absorber and the lag

motion of the rotor blade is the mechanism which transfers damping to the lag mode. As

can be seen in the damping matrix of Eq. 6.2, the addition of the tuning mass decreases

the Coriolis coupling terms, which decreases the amount of damping transferred to the

lag mode; i.e., in that when the velocity of the primary mass is in one direction, the

velocity of the tuning mass is in the opposite direction. Preliminary simulation results

indicate that the maximum achievable damping ratio for a radial fluid elastic damper

using the same absorber parameters as in Chapter 1, while allowing broad variation in the

tuning mass and tuning port area ratio, is only approximately 1.6% critical damping.

6.2 Nonlinear Spring

Another possible concept that could be used to overcome the large centrifugal

force and still allow for the dynamic tuning of the absorber is a nonlinear spring. The

force vs. displacement of a piecewise nonlinear spring is shown in Figure 6-3. This type

of spring would have a large stiffness over a given displacement of the spring to

( )( ) ⎭

⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡

−+++

r2

atap

2atap

x1Gmm00ammI

&&

&&ζζ

( )[ ]

( )[ ] ⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡−−Ω

−−Ω−+

raatap

atap

xc1Gmma21Gmma20

&

&ζ

( )[ ] ( )[ ]⎭⎬⎫

⎩⎨⎧

−−Ω=

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡Ω−+−

+1Gmma

Mx1Gmmk0

0k

atap2

r22

atapa

ζζ ζ

6.2

241

withstand the large centrifugal force, and then the stiffness would decrease for the

absorber to be dynamically tuned at larger spring displacements.

6.2.1 Nonlinear Spring Simulation

To first determine if a nonlinear spring can withstand the large centrifugal force,

even when the stiffness is decreased to tune the absorber, the same two-degree-of-

freedom system from Chapter 2 is analyzed, except instead of using a spring with a

constant stiffness, a piecewise nonlinear spring is considered. The linearized equations of

motion are as derived in Chapter 2, except they are left in dimensional form, and the

absorber spring stiffness is now a function of the absorber displacement:

The piecewise stiffness of the absorber is constructed in the following manner.

First, a desired static displacement of the absorber is chosen, and the stiffness of the

absorber is determined using the desired static displacement in Eq. 6.4.

At the chosen static displacement, the stiffness of the absorber changes from the stiffness

calculated in Eq. 6.4 to the stiffness determined from the tuning requirements of the

system. This stiffness is determined such that the natural frequency of the absorber is

equal to the lag natural frequency at the desired RPM, using Eq. 6.5.

⎭⎬⎫

⎩⎨⎧

Ω=

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡Ω−

+

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡Ω

Ω−+

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡ +

2ar

2aa

raa

a

ra

2a

amM

xmxk00k

xcam2am20

xm00amI

r

ζζ

ζ

ζ

ζζ

)(

&

&

&&

&&

6.3

( )static

static2

aa x

xamk +Ω= 6.4

242

This piecewise stiffness is shown in Figure 6-3 for the BO-105-type rotor described in

Appendix C. The absorber is tuned to the lag natural frequency at the operating RPM,

with a mass ratio of 0.05, a static offset of 0.3R, and a damping ratio of 0.3.

While it is important for the radial absorber to have a small static displacement

due to the centrifugal force, it must also be capable of transferring a sufficient amount of

damping to the lag mode, regardless of construction. Therefore, the system must be

analyzed to determine how much damping can be transferred to the lag mode using this

type of piecewise spring stiffness. Since the stiffness of the absorber is a function of the

displacement of the absorber, a linear eigenanalysis is not performed. In this case, the

transient response is solved for using an ODE solver in Matlab®. The damping ratio is

then calculated using the log decrement method. In order to highlight the transient

response of the system, yet still determine the static displacement of the absorber, the

forcing on the right hand side of Eq. 6.3 is modified slightly. The centrifugal force term

in the absorber equation of motion, 2aam Ω , which determines the static displacement of

the absorber is retained for the solution. The aerodynamic moment, ζM , in the lag

equation of motion is discarded since it primarily affects the steady-state response of the

system, and an impulsive forcing term is substituted instead, with the impulse chosen to

act on the system at approximately five seconds into the simulation. This ensures any

transient response due to the centrifugal force on the absorber has died out. All initial

displacements and velocities are set to zero.

( )22aa mk Ω+= ζω 6.5

243

Responses of the absorber and lag motion are shown in Figures 6-4 and Figure 6-

5. From the transient lag response, a lag damping ratio of approximately 12% critical

damping is calculated, which is similar to that predicted from the linear eigenanalysis in

Chapter 2 (refer to Figure 2-4a). Additionally, the static displacement of the absorber is

approximately 0.017R, or 3.3 inches, which is significantly less than the unrealistic value

of 9.9 feet predicted for an absorber with a constant, dynamically tuned spring stiffness.

6.2.2 Buckling Beam

One concept that could potentially be used in a radial absorber system with a

nonlinear spring is that of a buckling beam. This type of nonlinear spring is analyzed for

use in a vertical vibration isolation system [63-65], which needs a soft spring for low

resonant frequency and good isolation, but must overcome be able to support the mass in

the presence of gravity. A linear spring experiences significant extension as a result of

this static load due to the gravitational force. Additionally, the dynamic amplitude of

vibration for these systems is very small compared with initial static extension.

Therefore, a beam in a post-buckled configuration designed to operate as a spring (also

called an Euler spring), is ideal for this type of isolation system, which must support a

large static load with small dynamic amplitudes of oscillation. The beam can withstand a

large static axial force, and then its stiffness decreases significantly in its post-buckled

state.

While axially loaded beams exhibit nonlinear characteristics beyond the buckling

point, it is possible to design a structure that possesses a considerable amount of post-

244

buckled strength [66, 67]. The exact solution for large lateral deflections in this type of

structure involves elliptic integrals. However, for small deformations beyond the critical

load, the structure exhibits an almost linear force vs. displacement behavior [64] (see

Figure 6-6).

The radial absorber system could be designed such that the critical buckling load

of the beam or Euler spring is set near the centrifugal force acting on the absorber mass.

The structure is able to support such a high static load, yet in a post-buckled state exhibits

the softer spring stiffness desired to match the lag frequency of the rotor under dynamic

loading. The critical buckling load of a beam depends on the stiffness and length of the

beam, as well as the boundary conditions. Depending on the absorber parameters, as well

as the parameters of the Euler spring, the steady-state oscillations of the absorber mass

could be such that the force-displacement behavior is within the approximate linear

region of the Euler spring.

6.2.3 Aeromechanical Stability with Buckling Beam

One issue that must be overcome with a nonlinear spring is that there must be

sufficient damping in the lag mode over a range of rotor speeds; i.e., ground resonance

must be prevented not only at the operating RPM, but also during the rotor spin-up when

the rotor speed is increased from idle to full operating RPM. With a linear spring, the

absorber natural frequency decreases with increasing RPM; the absorber is only able to

be tuned to the lag frequency at a specific RPM. However, the results generated in

245

Chapters 2 and 3 indicate that there is sufficient damping transferred to the lag mode over

a range of RPM to alleviate ground resonance with a linear spring.

As the rotor speed increases, the centrifugal force on the absorber mass also

increases. If the Euler spring is designed for operation at full operating RPM, when at a

lower rotor speed and lower centrifugal force, the critical buckling load would not be

reached, and the spring would be too stiff to transfer enough damping to the lag mode.

Therefore, the system may have to be designed such that the critical buckling load is

smaller at lower RPM and passively increases (either continuously or perhaps

incrementally) as the rotor speed increases to compensate for the increasing centrifugal

force on the absorber mass. A general force-displacement curve for a device whose

critical buckling load increases with increasing RPM may look like that in Figure 6-7.

The critical buckling load for a pinned-pinned beam loaded axially is shown in

Eq. 6.6.

It depends on not only the beam stiffness, EI , but also the length of the beam, L .

Additionally the critical buckling load depends on the type of boundary conditions. For a

clamped-clamped beam, the critical buckling load is four times greater than that shown in

Eq. 6.6. While the beam stiffness may be difficult to change, it may be possible to

change the effective length of the beam or the boundary conditions or a combination of

both. For example, the centrifugal force itself may be used as a tool to alter the effective

beam length; to increase the critical load, the length must decrease. Figure 6-8 shows one

possible method of decreasing the effective length of the beam with increasing rotor

2

2

cr LEIP π

= 6.6

246

speed. In this concept, the beam is attached to a very stiff spring on one end. As the

RPM increases, the centrifugal force on the absorber causes the spring to compress, thus

effectively shortening the beam. The length may have to be changed incrementally with

increasing RPM; i.e., there may have to be stops on the beam and/or mass that engage

and disengage as the rotor reaches certain speeds. These could be calibrated to coincide

with rotor speeds where the regressing lag mode of the rotor couples with the fuselage

roll mode, which are the points where ground resonance is likely to occur.

6.3 Summary

Some possibilities for designing an embedded radial absorber have been

investigated. It appears that, while the fluid elastic damper is a promising device for use

in a chordwise inertial damper system, it may not be a feasible concept for use in a radial

vibration absorber system.

However, a nonlinear (softening) spring seems to be a valid means to overcome

the problem of the large centrifugal force. With a large initial stiffness, the nonlinear

spring is able to resist the centrifugal force. As the spring is extended, it softens, and the

dynamic oscillations of the absorber would occur within the dynamic stiffness range of

the spring, allowing it to be tuned to a low dynamic frequency. Such a system could be

constructed using Euler springs. Additionally, it appears that it could be possible to

design a system such that its critical buckling load increases as the rotor speed increases,

thus providing sufficient lag damping over a range of rotor speeds.

247

Damper Amplitude

Primary Mass

Elastomer

Tuning Port

Inner CylinderOuter Cylinder

Fluid Chamber

Outer Housing Attached to Rotor Blade

Damper Amplitude

Primary Mass

Elastomer

Tuning Port


Fluid Chamber

Damper Amplitude

Primary Mass

Elastomer

Tuning Port


Fluid Chamber

Outer Housing Attached to Rotor Blade

Figure 6-1: Conceptual design of a fluid elastic damper [redrawn from 44]

b

a

mat

map

caka

xr

xt

b

a

mat

map

caka

xr

xt

Figure 6-2: Mechanical analogy of a fluid elastic damper [redrawn from 44]

248

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

200

400

600

800

1000

1200

1400

Absorber Displacement (ft)

Sp

rin

g F

orc

e (l

bf)

ka (dynamic)

ka (static)

xstatic

Figure 6-3: Nonlinear piecewise spring stiffness

0 1 2 3 4 5 6 7 8 9 10−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Time(s)

Lag

Res

po

nse

(d

eg)

Figure 6-4: Lag response with nonlinear spring

249

0 1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

Time (s)

Ab

sorb

er R

esp

on

se (

in)

Figure 6-5: Absorber response with nonlinear spring

Figure 6-6: Force-displacement and frequency characteristics of ideal buckling [64]

250

Figure 6-7: Nonlinear spring stiffness - critical buckling load changing as RPM increases

Euler springAbsorber mass Very stiff spring

Effective length, L, changesas spring compresses

Centrifugal force

L

Blade root

Blade tip

Figure 6-8: Schematic for changing effective length of Euler spring

Chapter 7

Conclusions and Recommendations

An embedded radial vibration absorber has been examined to determine its

feasibility for use to augment rotor lag damping. First, a two-degree-of-freedom model

has been developed to investigate the levels of damping that can be introduced into the

lag mode through the Coriolis coupling between the radial motion of the absorber and the

lag motion of the rotor blade. Second, a rigid blade ground resonance model with an

embedded radial absorber has been developed to explore the effects of the radial absorber

on the aeromechanical stability of the rotor system. An aeroelastic analysis of a rotor

blade undergoing flap and lag bending with an embedded radial absorber has also been

conducted to investigate the effect the absorber has on the flap and lag responses, as well

as the blade and hub loads. Finally, implementation possibilities and device concepts for

the embedded radial vibration absorber have been considered.

7.1 Conclusions

7.1.1 Two-Degree-of-Freedom Model

In the two-degree-of-freedom model, the rotor blade is modeled as a rigid blade

undergoing lag motion, and the absorber is modeled as a spring-mass-damper system

moving radially within the blade. The simple two-degree-of-freedom model has shown

that the radial vibration absorber can introduce damping in the lag mode through the

252

strong Coriolis coupling between the lag motion of the blade and the radial motion of the

absorber. Even for mass ratios as low as 1% to 5% of the blade mass, a significant

amount of damping can be introduced into the rotor lag mode for most combinations of

absorber parameters, and up to one-half of the damping in the absorber mode can be

transferred to the lag mode. In general, as a and mα increase, the amount of damping

transferred to the lag mode also increases. While the amount of lag mode damping that

can be achieved is an important consideration of the radial absorber, another important

aspect of the absorber is the dynamic displacement amplitude that is required to achieve

these damping levels. The absorber amplitude at 1/rev periodic motion is analyzed using

the two-degree-of-freedom model and is found not to be excessively large for the

combinations of absorber parameters examined.

7.1.2 Aeromechanical Stability Analysis

A rotor system is modeled as a rigid system undergoing flap and lag deflections

with the same spring-mass-damper absorber system embedded in the rotor blade as the

two-degree-of-freedom model, and the rotor hub is assumed to undergo rigid body pitch,

yα , and roll, xα , motions. Like the flap and lag equations of motion, the absorber

equations of motion are transformed to the nonrotating frame using the Multiblade

Coordinate Transformation to give two cyclic absorber equations and a total of eight

degrees of freedom in the non-rotating coordinate system. From the results presented in

Chapter 3, it is shown that the radial vibration absorber has a significant effect on

helicopter aeromechanical stability. For the example rotor used in the simulations, it is

253

possible to eliminate the instability in the regressing lag mode, and in most cases, the

stability margins are significant. These results indicate that not only is a radial absorber

able to transfer damping to the lag mode via Coriolis coupling, the damping transferred is

also sufficient to prevent ground resonance over a range of rotor speeds, even though the

absorber is tuned to the fundamental lag natural frequency at a specific rotor speed.

7.1.3 Aeroelastic and Loads Analysis

Once it was determined that the radial vibration absorber was able to transfer

sufficient damping to the lag mode and prevent ground resonance instabilities, an

aeroelastic loads analysis is performed to determine how the absorber affects the rotor

blade loads, blade root loads, and hub loads. The rotor blade is modeled as a beam

undergoing flap and lag deflections, with the same single-degree-of-freedom spring-

mass-damper absorber system embedded within the blade. Using a quasi-steady

aerodynamic model with linear inflow, the rotor performance and loads are examined. It

is shown that no significant increases in the blade and hub loads are introduced due to the

addition of the absorber for most of the absorber parameters examined. In general, as the

magnitude of the absorber response (and the amount of damping transferred to the lag

mode) increases, the effect the absorber has on the blade and hub loads also increases.

254

7.1.4 Implementation Concepts

A nonlinear spring is shown to be a promising means to overcome the problem of

the large centrifugal force acting on the absorber mass. With a large initial stiffness, the

nonlinear spring is able to resist the centrifugal force. As the spring is extended, it

softens, and the dynamic oscillations of the absorber would occur within the dynamic

stiffness range of the spring, allowing it to be tuned to a low dynamic frequency. A

buckling beam (or Euler beam) model to implement this concept is briefly examined.

7.2 Recommendations for Future Work

The work described in this thesis is a first attempt to model and analyze the

performance of an embedded radial vibration absorber. Demonstrated results,

particularly the amount of damping that is able to be transferred to the lag mode through

the Coriolis coupling and corresponding improvements in aeromechanical stability were

promising. The fact that loads do not significantly increase while introducing damping to

the system is also promising. Suggestions for further investigation are described below.

7.2.1 Articulated Rotor

All helicopter models used in this analysis were hingeless rotors. While

articulated rotor systems also require lag damping augmentation, they present some

unique challenges. First, the fundamental lag frequency of an articulated rotor is much

lower than a hingeless rotor. As a result, in order to be dynamically tuned, the dynamic

255

stiffness of the absorber would have to be lower for a given absorber mass. Considering

a nonlinear spring described in Chapter 6, the difference between the two stiffnesses may

be a design challenge. Secondly, the variation of the fundamental lag frequency of an

articulated rotor with rotor speed is much larger than for a hingeless rotor. As noted in

Chapter 3, the range of rotor speeds over which the absorber provides sufficient lag

damping is quite large. Although the absorber is tuned to the lag natural frequency at a

specific RPM, since the lag frequency of the hingeless rotor does not vary significantly

with rotor speed, the absorber frequency remains relatively close to the lag frequency for

a large range of RPM. This problem should be analyzed in depth to determine if it is

conceivable for a radial vibration absorber to provide sufficient lag damping and prevent

ground resonance in an articulated rotor.

7.2.2 Vibration Reduction

The possibility exists that this type of system could not only be used as an energy

dissipation device, but also as a vibration reduction device. It may be possible to design

the system to reduce the vibratory components of the hub loads, while still maintaining

acceptable levels of damping in the lag mode. Up to this point, all analyses have been

conducted with radial vibration absorber tuned to the fundamental lag natural frequency,

and it has been shown to exhibit a sensitivity to vibration, depending on absorber location

and mass ratio. It may be possible to tune the absorber closer to the rotor forcing

frequency, Ω , and have the potential to reduce vibrations while still functioning

effectively as an absorber. This may be particularly true for a hingeless rotor where the

256

fundamental lag natural frequency is relatively close to Ω . It is conceivable that a

combination of absorber parameters could be optimized to not only sufficiently augment

the lag damping and aeromechanical stability of a particular rotor, but also reduce the

vibratory hub loads of the helicopter.

7.2.3 Energy Harvesting

Because this is a damped system, there is energy dissipated, and this energy is

usually dissipated as heat. The energy that is dissipated by the absorber may be able to

be harnessed for use in the rotor blade to power small actuators to drive flaps, power

Health and Usage Monitoring Systems (HUMS) measurement devices, etc. Having this

type of system embedded within the rotor blade would be a great convenience over the

current situation of having to send power from the nonrotating system to the rotating

system of the helicopter. The amount of energy dissipated depends on the absorber

parameters and the response amplitude of the absorber.

A quick estimate of the amount of energy that could be available for harvesting

can be made using the two-degree-of-freedom lag-absorber model. The energy dissipated

per cycle can be determined by integrating the product of the force due to the damper and

its displacement over the cycle [37], as shown in Eq. 7.1:

where ω in this case is the rotor speed, Ω , and one cycle is one rotation around the rotor

azimuth. For the BO-105-type rotor undergoing typical lag motion, ζ , of ± 1 degree,

∫∫∫ ===ωπωπ // 2

0

2ra

2

0E dtxcdtxFdxFD && 7.1

257

with an absorber located at 0.7R, mass ratio of 0.05, damping ratio of 30% critical

damping, and tuned to the lag natural frequency, the energy dissipated per cycle is

approximately 120 ft-lbs, which for a rotor operating at 6.6 Hz, equates to slightly more

than 1000 watts or 1.3 horsepower. Of course, this is a simple estimate for a two-degree-

of-freedom system with its damping modeled as a linear viscous damper. It is expected

that not all the energy dissipated would be available for harvesting, and a percentage of

the energy would be unrecoverable due to losses in the system. The method of harvesting

the dissipated energy would be determined largely by the type of device used for the

radial absorber system. However, for a moving radial mass, the moving mass could be a

permanent magnet which moves inside a coil, thereby inducing a voltage in the coil.

Additionally, if the resistance in the coil can be controlled, the level of damping in the

absorber system could potentially be actively changed.

7.2.4 Device Design and Experimental Investigation

Finally, experimental verification is necessary to reaffirm (or disprove) the

effectiveness of the embedded radial vibration absorber. Experimental setup and

verification of a two-degree-of-freedom system would seem to be a straightforward first

step. However, the construction of a device that is able to withstand the large centrifugal

force, yet still be tuned to the lag natural frequency at a particular rotor speed and small

enough to fit within a small model rotor blade may be a somewhat difficult task. As

described in Chapter 6, a nonlinear spring is a possible solution to this problem.

Modeling and analysis of a system with a nonlinear spring, followed by construction and

258

testing of a device would be a logical step in the analysis process of the embedded radial

vibration absorber.

7.2.5 Other

There are many other possibilities associated with the embedded radial vibration

absorber that can be explored. Additional suggestions for future research are described

below:

1. Examine the susceptibility to limit cycle oscillations during transient conditions

such as rotor spin-up.

2. If the absorber system is designed with a tunable spring stiffness, it may be

possible to vary the stiffness of the absorber for each individual rotor blade. This

would change the static displacement of the absorber mass and in turn change the

radial center of gravity of the rotor blade which could be used for balancing the

rotor system.

3. Conduct a failure mode analysis for the radial vibration absorber. There are

several failure possibilities, including spring failure and catastrophic absorber

mass separation from one or more rotor blades, which would result in not only a

loss of lag damping in that particular rotor blade, but also in a unbalanced rotor

system.

4. Examine how coupled rotor-fuselage vibration would affect the response and

effectiveness of the radial vibration absorber.

259

5. Examine the time history of the rotor center of gravity with and without the radial

vibration absorber at a rotor speed when the regressing lag mode couples with a

fuselage mode and the rotor system has the potential to become unstable to

illustrate how the absorber reduces the tendency toward instability.

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Appendix A

Mass, Stiffness, and Damping Terms in Ground Resonance Analysis

The terms in the mass, stiffness, and damping matrices used in the ground

resonance equations of motion are listed in this appendix.

A.1 Mass Matrix

2m11 a31M α+=

( )( )[ ]0m2

m18 hea3Sa31M βαα β ++++−= *

2m22 a31M α+=

( )( )[ ]0m2

m27 hea3Sa31M βαα β ++++= *

2m33 a31M α+=

( ) ( )[ ]2m0m37 a31a3ShM αβαβ +++= *

2m44 a31M α+=

( ) ( )[ ]2m0m48 a31a3ShM αβαβ +++= *

1M 55 =

( )heM 058 −−= β

1M 66 =

( )heM 067 −= β

268

( )( )[ ]0m2

m72 hea3Sa31M βαα β ++++= *

( ) ( )[ ]2m0m73 a31a3ShM αβαβ +++= *

( )heSMM 0bm76 −= βα β**

( )

( ) ( ) ⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

++++

++++=

a3S2h2a3Se2

1SMh2a3

2N

I

I1

M

m0m

mb22

mb

x

77

αβα

αα

ββ

β

β

**

**

( )( )[ ]0m2

m81 hea3Sa31M βαα β ++++−= *

( ) ( )[ ]2m0m84 a31a3ShM αβαβ +++= *

( )heSMM 0bm85 −= βα β**

( )

( ) ( ) ⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

++++

++++=

a3S2h2a3Se2

1SMh2a3

2N

I

I1

M

m0m

mb22

mb

y

88

αβα

αα

ββ

β

β

**

**

A.2 Damping Matrix

A.2.1 Inertial Terms

( )2m12 a312C α+=

( )2m013 a312C αβ +−=

( ) ( )[ ]eaa3Se12C m17 +++= αβ*

( )2m21 a312C α+−=

269

( )2m024 a312C αβ +−=

( ) ( )[ ]eaa3Se12C m28 +++= αβ*

( )2m031 a312C αβ +=

Ω=

β

ζ

IC

C33

( )2m34 a312C α+=

a6C m35 α−=

( )2m042 a312C αβ +=

( )2m43 a312C α+−=

Ω=

β

ζ

IC

C44

a6C m46 α−=

( )ea2C53 −=

ζναζ fa55 2C =

2C56 =

057 a2C β=

( )ea2C64 −=

2C65 −=

ζναζ fa66 2C =

068 a2C β=

270

( ) ( )[ ]a3Sea312C m2

m71 αα β +++−= *

( )a3SMe2C b0m75 +−= **βββα

2N

I

CC

b2

x77

Ω=

β

( ) ( )a3Se4a312C m2

m78 αα β +++= *

)]()[( * a3Sea312C m2

m82 αα β +++−=

( )a3SMe2C b0m86 +−= **βββα

( ) ( )a3Se4a312C m2

m87 αα β +−+−= *

2N

I

CC

b2

y88

Ω=

β

A.2.2 Aerodynamic Terms

8Ca11

γ=

⎟⎠⎞

⎜⎝⎛ −=

34Ca13

λθγ

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −=

6443hCa 017

λθβλθγ

⎟⎟⎠

⎞⎜⎜⎝

⎛+−= 018 6

h81Ca βγ

8Ca22

γ=

271

⎟⎠⎞

⎜⎝⎛ −=

34Ca24

λθγ

⎟⎟⎠

⎞⎜⎜⎝

⎛+= 027 6

h81Ca βγ

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −=

6443hCa 028

λθβλθγ

⎟⎠⎞

⎜⎝⎛ −−=

38Ca31

λθγ

⎟⎠⎞

⎜⎝⎛ +=

6aCd

41Ca33

θλγ

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ ++⎟

⎠⎞

⎜⎝⎛ +=

6aCd

41

4aCd

31hCa 037

θλβθλγ

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −=

26h

38Ca 038

λθβλθγ

⎟⎠⎞

⎜⎝⎛ −−=

38Ca42

λθγ

⎟⎠⎞

⎜⎝⎛ +=

6aCd

41Ca44

θλγ

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −−=

26h

38Ca 047

λθβλθγ

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ ++⎟

⎠⎞

⎜⎝⎛ +=

6aCd

41

4aCd

31hCa 048

θλβθλγ

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −−=

3826hCa 071

λθβλθγ

⎟⎠⎞

⎜⎝⎛ ++= h

61

6e

81Ca 072 βγ

272

⎥⎦

⎤⎢⎣

⎡+⎟

⎠⎞

⎜⎝⎛ ++⎟

⎠⎞

⎜⎝⎛ +=

aCmc

31

4aCd

31h

6aCd

41Ca 073

θλθλβγ

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −=

43h

43e

64Ca 074

λθβλθλθγ

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟

⎠⎞

⎜⎝⎛ ++

⎟⎠⎞

⎜⎝⎛ ++++⎟

⎠⎞

⎜⎝⎛ +

=]

32h

aCmc

2aCd

21h

e41

2aCd

32

31he

61

81

Ca02

0

77 βθλ

θλβγ

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −+

⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −

=

23

43h

43e

283

22he

43

2h

Ca

02

0

0

78 λθβλθβ

λθβλθλθ

γ

⎟⎠⎞

⎜⎝⎛ ++−= h

61

6e

81Ca 081 βγ

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −−=

3826hCa 082

λθβλθγ

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −−=

43h

43e

64Ca 083

λθβλθλθγ

⎥⎦

⎤⎢⎣

⎡+⎟

⎠⎞

⎜⎝⎛ ++⎟

⎠⎞

⎜⎝⎛ +=

aCmc

31

4aCd

31h

6aCd

41Ca 084

θλθλβγ

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −+

⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −

−=

23

43

02h

430e

283

022he

43

2h

Ca87 λθβλθβ

λθβλθλθ

γ

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+++⎟

⎠⎞

⎜⎝⎛ ++

⎟⎠⎞

⎜⎝⎛ ++++⎟

⎠⎞

⎜⎝⎛ +

=

30

2h

aCmc2h2

041

2aCd

212h

e41

2aCd

32

31

0he61

81

Ca88 ββθλ

θλβ

γ

273

A.3 Stiffness Matrix

A.3.1 Inertial and Elastic Terms

ea31K m2

11 αν β +−=

( )2m014 a312K αβ +−=

ea31K m2

22 αν β +−=

( )2m023 a312K αβ +=

( )2m032 a312K αβ +=

ea3a31K m2

m2

33 ααν β +−−=

Ω=

β

ζ

IC

K 34

a6K m36 α−=

( )2m041 a312K αβ +−=

Ω−=

β

ζ

IC

K 43

ea3a31K m2

m2

44 ααν β +−−=

a6K m45 α=

( )ea2K54 −=

1K 22f55 −= ζνα

ζναζ fa56 2K =

274

( )ea2K63 −−=

ζναζ fa65 2K −=

1K 22f66 −= ζνα

ah3K m072 αβ=

( )0bm0m76 ehSMa6K βαβα β ++−= **

2NI

kKb2

x77

Ω=

β

ah3K m081 αβ−=

( )0bm0m85 ehSMa6K βαβα β ++= **

2NI

kK

b2

y88

Ω=

β

A.3.2 Aerodynamic Terms

8Ka12

γ=

⎟⎠⎞

⎜⎝⎛ −=

34Ka23

λθγ

8Ka21

γ−=

⎟⎠⎞

⎜⎝⎛ −−=

34Ka23

λθγ

275

⎟⎠⎞

⎜⎝⎛ −−=

38Ka32

λθγ

⎟⎠⎞

⎜⎝⎛ +=

6aCd

41Ka34

θλγ

⎟⎠⎞

⎜⎝⎛ −=

38Ka41

λθγ

⎟⎠⎞

⎜⎝⎛ +−=

6aCd

41Ka43

θλγ

⎥⎦

⎤⎢⎣

⎡−++++=

46aCd

81h

61

6e

81Ka

2

071λθλβ

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −−=

3843

3hKa 072

λθβλθγ

⎟⎠⎞

⎜⎝⎛ ++−= h

61

6e

81Ka 073 βγ

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛++=

4aCd

81

2aCd

61h

aCm

6cKa

2

0

2

74λβλγ

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −=

3843

3hKa 081

λθβλθγ

⎥⎦

⎤⎢⎣

⎡−++++=

46aCd

81h

61

6e

81Ka

2

082λθλβ

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛++−=

4aCd

81

2aCd

61h

aCm

6cKa

2

0

2

83λβλγ

⎟⎠⎞

⎜⎝⎛ ++−= h

61

6e

81Ka 084 βγ

Appendix B

Aerodynamic Formulation

The formulation of the aerodynamic load vector is described in this appendix.

The calculation of the aerodynamic load vector begins with the blade section velocity,

which contains contributions from the blade velocity (motion of the blade relative to the

hub) and wind velocity (helicopter forward speed and rotor rotation). Blade airloads are

then calculated using quasi-static aerodynamics, which is based on two-dimensional thin

airfoil theory, and are a function of airfoil properties and blade section velocity. The

elemental aerodynamic loads are obtained by integrating the blade airloads along the

length of the element.

B.1 Resultant Velocity

The resultant velocity of a blade section consists of the blade velocity due to the

helicopter forward speed and rotor rotation, as well as the blade motion relative to the

rotor hub. The rotor inflow which is included in the blade velocity is determined using a

linear inflow model. After several substitutions and a transformation from the

undeformed frame to the deformed frame, the resultant velocity components are

expressed as follows:

277

The tangential and perpendicular components of the velocity are required for calculation

of the blade section lift and drag in the aerodynamic load formulation, while the radial

velocity is required for calculating the axial force using the force summation method

described in Chapter 5.

B.2 Inflow

For the current analysis, a linear induced inflow distribution is assumed. Linear

inflow is an extension of uniform inflow predicted by momentum theory to approximate

the radial and azimuthal variations in lift and thus inflow over the rotor disk in forward

flight. The inflow is approximated with Eq. B.2

( )( )

( )

( )( )

( )( ) ( )[ ]

( ) ( )( ) ( )

( )22

rrp

pR

p

2

p

prrP

pp

2

p

T

wvR21ww

vvwvwRwRRRxvvuu

RRxwvwRv

Rxv21R

wuxvvRvvwu

RRvwRxwvRwvw

Rxv21R

wuxvRvvu

′+′Ω+′+

′+′−′+Ω−′+Ω+

′Ω−Ω−+′Ω+Ω−=

Ω+Ω+Ω′′−+′Ω+Ω+⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+′Ω+Ω−

−+Ω−−′Ω−′Ω−+Ω+′Ω=

⎥⎦

⎤⎢⎣

⎡Ω+Ω+Ω++

Ω+Ω′′−Ω′+′Ω+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

Ω+Ω′−Ω+

−+Ω++Ω+′Ω=

ψμ

θηθηβλ

βψμψμ

θλψμβψμ

θψμψμ

βψμβηη

θλψβμβ

ψμψμ

θψμψμ

βψμ

cos

sincoscossin

cossincos

sinsinsin

cos

sincos

sincos

cossinsin

cos

&

&&&

&

&

&

&

B.1

( )ψκψκλμ

λ sincos xx12

Cyx22

Ti ++

+= B.2

278

The constants, xκ and yκ , are calculated using Drees model [1]:

B.3 Blade Loads

Once the velocities at a blade section are known, the airloads can then be

calculated. They are first determined in the blade section coordinates and then

transformed back to the undeformed frame for use in the finite element model. The blade

section airloads per unit length in the rotating, deformed frame can be calculated by

Eq. B.4:

where the lift and drag coefficients of the airfoil can be defined as

These loads are rotated to the blade coordinates to be the normal and chord force, i.e.,

forces acting along the flap, lag, and axial directions rather than relative to the angle of

attack. These forces can be placed directly into the force vector for the finite element

model. The angle Λ is the radial yaw angle due to the axial velocity RU .

( )⎥⎥

⎦

⎤

⎢⎢

⎣

⎡−⎟

⎠

⎞⎜⎝

⎛+−=

μλ

μλμκ

22

x 181134 .

μκ 2y −=

B.3

l2cCV

21L ρ=

d2cCV

21D ρ=

B.4

α10l ccC += 2

20d ddC α+= B.5

279

Using the following small angle assumptions,

the airloads can be expressed as

The airloads are transformed back to undeformed frame to be placed in finite element

force vector.

where

αα cossin DLLv −= αα sincos DLLw +=

Λ−= sinDLu B.6

αα ≈sin 1≈αcos

TuV ≈

T

P

uu−

≈α

T

R

uu

≈Λsin

B.7

( )[ ]2P21TP0

2T0v udcuucudc

21L −+−−= ρ

( )[ ]TP012Tow uudcucc

21L +−= ρ

[ ]TR0u uudc21L −= ρ

B.8

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧=

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

w

v

uTDU

Aw

Av

Au

LLL

TLLL

B.9

( )

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

⎟⎠⎞

⎜⎝⎛ ′−′′−⎟

⎠⎞

⎜⎝⎛ ′−−′−′

⎟⎠⎞

⎜⎝⎛ ′−′′−⎟

⎠⎞

⎜⎝⎛ ′−′+′−

′′′−′−

=

θθθθθ

θθθθθ

coscossincossin

sinsincossincos

22

22

22

DU

w211wvv

211wv

w211wvv

211wv

wvw21v

211

T

B.10

280

The linear terms in Eq. B.9 are used to construct elemental aerodynamic damping and

stiffness matrices, with the constant and nonlinear terms forming the elemental

aerodynamic load vector.

Appendix C

Rotor Data

C.1 AFDD Rotor Data

Table C-1: AFDD rotor data

Number of blades 3

Lock number, γ 7.37 Main Rotor Properties

Rotational speed, Ω 720 RPM

Blade radius, R 81.1 cm

Blade chord, c 4.19 cm

Hinge offset, e 8.51 cm

Blade mass, bM 209 g

Flap inertia βI 17.3 g-m2

Blade profile NACA 23012

Lift curve slope, a 5.73

Rotor Blade Properties

Profile drag coefficient, 0dC 0.0079

Pitch inertia, pitchI 633 g-m2

Roll inertia, rollI 183 g-m2

Pitch mode damping 3.2% critical

Roll mode damping 0.929% critical

Pitch frequency, pitchω 2 Hz

Roll frequency, rollω 4 Hz

Fuselage Properties

Rotor height above cg, h 24.1 cm

282

C.2 Full Scale BO-105-Type Rotor Data

Table C-2: Main rotor data

Number of blades 4

Lock number, γ 6.34

Solidity ratio, σ 0.1 Main Rotor Properties

Rotational speed, Ω 40.12 rad/s

Blade radius, R 16.2 ft

Blade chord, Rc / 0.08

Mass per unit length, om 0.135 slug/ft

Flap bending stiffness, 42oy RmEI Ω/ 0.008345

Lag bending stiffness 42oz RmEI Ω/ 0.023198

Lift curve slope, a 5.73

Skin friction drag coefficient, 0dC 0.0095

Rotor Blade Properties

Induced drag coefficient, 2dC 0.2

283

Table C-3: Tail and fuselage data

Number of blades, trN 4

Tail rotor radius, trR 3.24 ft

Solidity ratio, trσ 0.15

Rotor speed, trΩ Ω5

Lift curve slope, tra 6.0

Tail Rotor Properties

Tail rotor location, ( )RzRx trtr /,/ (1.2, 0.2)

Area, 2ht RS π/ 0.011

Lift curve slope, hta 6.0 Horizontal Tail Properties

Horizontal tail location, Rxht / 0.95

CG location, ( )cgcg yx , (0, 0)

Hub location, Rh / 0.2 Fuselage Properties

Net weight, W 5800 lbs

284

C.3 HART Rotor Data

Table C-4: General rotor properties

Property Symbol Value Dimension Number of blades N_b 4.00 Radius scaling factor S 2.46 Chord scaling factor S_c 2.23 Nominal speed Omega 109.00 rad/s Radius R 2.00 m Pitch arm length 0.06 m Radius of pitch bearing 0.08 m Radius of blade bolt r_H 0.15 m Root cutout r_a 0.44 m Radius of zero twist r_tw 1.50 m Chord c 0.12 m Tab length 0.01 m Tab thickness 0.00 m Blade area A_b 0.97 m^2 Rotor area A_R 12.57 m^2 Solidity sigma 0.08 Airfoil NACA23012mod Linear twist theta_tw -8.00 deg/R Precone beta_p 0 deg Blade mass m_b 2.24 kg Lock number gamma 8.06

285

Table C-5: Structural properties - 1

r/R TWISTI MASS XI XC KP2 chord (xi/R) (xc/R) (kp/R)**2 (c/R) - [degree] [kg/m] - - - - 0 4.2 3 0 0 0.000008 0.03 0.08005 4.2 3 0 0 0.000008 0.03 0.0958 4.2 1.57 0 0 2.45E-05 0.03 0.117 4.2 1.62 -0.0005 0 6.05E-05 0.03 0.13833 4.2 1.71 -0.0006 0 0.000117 0.03 0.1596 4.2 1.7 -0.001 0 0.000162 0.03 0.18088 4.2 1.63 -0.0012 0 0.000162 0.03 0.20215 4.2 1.51 0.002 0 0.000162 0.03 0.2164 4.2 1.33 0 0 0.000162 0.0605 0.24833 3.995 0.95 0.00275 0 1.71E-05 0.0605 0.30858 3.535 0.95 0.00275 0 1.71E-05 0.0605 0.37243 3.025 0.95 0.00275 0 1.71E-05 0.0605 0.43628 2.515 0.95 0.00275 0 1.71E-05 0.0605 0.50013 2.005 0.95 0.00275 0 1.71E-05 0.0605 0.56398 1.495 0.95 0.00275 0 1.71E-05 0.0605 0.62783 0.985 0.95 0.00275 0 1.71E-05 0.0605 0.69168 0.475 0.95 0.00275 0 1.71E-05 0.0605 0.75553 -0.035 0.95 0.00275 0 1.71E-05 0.0605 0.81938 -0.545 0.95 0.00275 0 1.71E-05 0.0605 0.88323 -1.055 0.95 0.00275 0 1.71E-05 0.0605 1 -2 0.95 0.00275 0 1.71E-05 0.0605

286

The abbreviations in Tables C-5 and C-6 are defined in Table C-7:

Table C-6: Structural properties - 2

r/R e0 EIZZ EIXX ITHETA GJ (e0/R) (FLAP) (LAG) - - [N*m*m] [N*m*m] [kg*m^2/m] [N*m*m] 0 0 6.85E+02 2.40E+03 0.00048 2.95E+02 0.08005 0 6.85E+02 2.40E+03 0.00048 2.95E+02 0.0958 0.0036 6.85E+02 2.40E+03 0.000404 2.95E+02 0.117 0.0016 6.85E+02 3.90E+03 0.000605 3.45E+02 0.13833 0 6.65E+02 4.60E+03 0.001679 3.95E+02 0.1596 -0.0016 5.25E+02 5.40E+03 0.00119 3.15E+02 0.18088 -0.0021 5.35E+02 5.40E+03 0.001142 2.25E+02 0.20215 -0.0026 3.55E+02 4.70E+03 0.001589 2.25E+02 0.2164 0.0026 3.55E+02 4.70E+03 0.001246 2.25E+02 0.24833 0.0026 2.35E+02 6.90E+03 0.000672 1.15E+02 0.30858 0.0026 2.25E+02 6.90E+03 0.000672 1.15E+02 0.37243 0.0026 2.25E+02 6.90E+03 0.000672 1.15E+02 0.43628 0.0026 2.25E+02 6.90E+03 0.000672 1.15E+02 0.50013 0.0026 2.25E+02 6.90E+03 0.000672 1.15E+02 0.56398 0.0026 2.25E+02 6.90E+03 0.000672 1.15E+02 0.62783 0.0026 2.25E+02 6.90E+03 0.000672 1.15E+02 0.69168 0.0026 2.25E+02 6.90E+03 0.000672 1.15E+02 0.75553 0.0026 2.25E+02 6.90E+03 0.000672 1.15E+02 0.81938 0.0026 2.25E+02 6.90E+03 0.000672 1.15E+02 0.88323 0.0026 2.25E+02 6.90E+03 0.000672 1.15E+02 1 0.0026 2.25E+02 6.90E+03 0.000672 1.15E+02

Table C-7: Property descriptions

TWISTI blade twist relative to 75% radius MASS blade mass distribution XI offset of center of gravity aft of elastic axis XC offset of tension center aft of elastic axis KP2 polar radius of gyration about elastic axis (area moment of inertia) EIZZ flapwise bending stiffness EIXX lagwise bending stiffness ITHETA distribution of pitch mass moment of inertia about elastic axis GJ torsional stiffness chord chord length e0 offset of pitch axis aft of elastic axis

Appendix D

Waterbed Effect

The waterbed effect is a standard limitation in control systems, even in passive

systems such as the rotor blade with a radial vibration absorber. The Bode integral

theorem states that if the amplitude of the frequency response of the system is reduced in

one part of the frequency spectrum, it may have to get larger in another frequency range.

This effect is often compared to a waterbed; when it is “pushed down” in one place it

“pops up” in another [68, 69]. The frequency response functions of the baseline system

and then the system with the radial vibration absorber added ( 70a .= , 30a .=ζ , and

050m .=α ) are shown in Figure D-1, with the lower frequency range shown further in

Figure D-2. As can be seen from the figures, the addition of the absorber increases the

magnitude of the response slightly in certain frequency ranges. However, the absorber

significantly decreases the magnitude of the response at the fundamental lag natural

frequency, as designed. Additionally, the magnitude at the fundamental flap frequency

also decreases, and the magnitude of the response at most of the other system’s

frequencies remain similar or decrease slightly when compared to the baseline rotor.

288

0 5 10 15 20 2510

−12

10−10

10−8

10−6

10−4

10−2

100

102

Frequency ( /rev)

Fre

qu

ency

Res

po

nse

Mag

nit

ud

e (l

og

)

Baseline

With Absorber

Figure D-1: Frequency response function with and without absorber

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 210

−6

10−5

10−4

10−3

10−2

10−1

100

Fre

qu

ency

Res

po

nse

Mag

nit

ud

e (l

og

)

Frequency ( /rev)

With Absorber

Baseline

Figure D-2: Frequency response function with and without absorber – fundamental lag

and flap frequencies shown

Appendix E

Elastic Blade Analysis – Lag Only

An elastic blade analysis is conducted to examine blade and hub loads due to the

addition of the absorber. As a first step to understand the process of including the

absorber in the model, the rotor blade is modeled as an elastic beam undergoing pure in-

plane bending, and the coupling terms due to other modes of motion are ignored. All

rotor blades are assumed to be identical. The equations of motion are derived using

Lagrange’s equation and are spatially discretized using the finite element method. The

blade is discretized into a number of beam-absorber elements. Each beam-absorber

element consists of four degrees of freedom describing the motion of the beam element

and one degree of freedom describing the absorber. Approaching the discretization in

this manner allows for the absorber to be embedded anywhere along the blade, as well as

allowing for the possibility of embedding more than one absorber in the blade, although

for this analysis, only one absorber is considered.

With only the lag and absorber motion modeled, the aerodynamic forcing terms

are not included in this model. Consequently, a blade tip response equivalent to that

obtained from an alternate model (RCAS) is achieved by applying a tip force. The blade

lag and absorber responses, as well as the blade root and hub loads, are then determined.

While this does not approximate the aerodynamic forcing seen by an actual rotor system,

it does allow for an absorber modeling process and a solution procedure to be developed

for future, more complete elastic blade analyses.

290

E.1 Coordinate Systems and Nondimensionalization

E.1.1 Coordinate Systems

The coordinate systems used in this analysis are shown in Figure E-1. The

inertial frame of reference is defined as the hub-fixed nonrotating coordinate system



upward, parallel to the rotor. The hub-fixed rotating coordinate system ( RRR zyx ,, ) with

unit vectors, RRR kji ˆ,ˆ,ˆ , is attached to the hub and rotates with the blades at an angular

velocity of Rk̂Ω , relative to the hub-fixed nonrotating coordinate system. This

coordinate system is also the undeformed blade coordinate system.

The deformed blade is characterized by the deformed blade coordinate system,

( ζηξ ,, ), with corresponding unit vectors ζηξ kji ˆ,ˆ,ˆ . The ξ axis is aligned radially with

the deformed blade, the η axis points toward the leading edge of the blade, and the ζ

axis is aligned vertically through the blade cross section. Since there is no flap or torsion

motion considered in this model, the ζ axis is parallel to HZ and Rz .

E.1.2 Nondimensionalization

All forces and moments are nondimensionalized for a more systematic and direct

comparison between helicopters of different sizes. The forces are nondimensionalized by

291 22

0 Rm Ω and all moments are nondimensionalized by 320 Rm Ω . The reference blade

mass, 0m , is defined as the mass per unit length of a uniform blade with the same inertia

as the blade being considered. It is calculated by

E.2 Velocity and Acceleration of Blade and Absorber

In order to apply Lagrange’s equation, the velocities of the blade and absorber

must be determined. These are similar to the velocities derived for the two-degree-of-

freedom model, but the blade undergoes in-plane bending, v , instead of a rigid rotation

about the hub, ζ . The accelerations of the blade and absorber are determined for use in

the force summation method of determining blade loads described in section E.8.

E.2.1 Blade

The position and velocity of the blade is determined by first determining the


The velocity can be determined by taking the first time derivative of the position

vector, as described in Eq. 2.3 where k̂Ω=ω . In this case, the elastic displacements,

u and v are also functions of time. The blade velocity is given by :

3

R

0

2

3b

0 R

drmr3

RI3m ∫== E.1

( ) jviurr bˆˆ ++= E.2

292

The blade acceleration is determined using Eq. 2.4 and is

The total extension, u , is broken up into elastic and kinematic components

The elastic component is neglected for this analysis, but the axial motion due to

foreshortening resulting from lag deflection is included, with the following velocity and

accelerations:

E.2.2 Absorber

As in Chapters 2 and 3, the absorber is modeled as an embedded spring-mass-

damper system that moves radially in the rotor blade. Its velocity is similarly determined

starting with its position vector:

and is given in Eq. E.8:

The acceleration of the absorber is then

( ) ( )[ ] jvurivuvbˆˆ && +Ω++Ω−= E.3

[ ] [ ] jvu2viurv2ua 22b

ˆˆ)( Ω−Ω++Ω+−Ω−= &&&&&& E.4

∫ ′−=x

0

2e dxv

21uu E.5

( )∫∫

′+′−=

′−=x

0

x

0

dxvvvvu

dxvvu

&&&&&&

&& E.6

( ) jvixar raˆˆ ++= E.7

( ) ( )[ ] jvxaivxv rraˆˆ && +Ω++Ω−= E.8

293

E.3 Derivation using Lagrange’s Equation

The differential equations of motion of the blade-absorber system are derived

using Lagrange’s equation (see section 2.3.3). In order to apply Lagrange’s equation, the

strain and kinetic energies of the blade and absorber must first be determined.

E.3.1 Strain Energy

Expressions for the total strain energy of the system are derived. The total strain

energy consists of contributions from the blade and the absorber.

The rotor blade is modeled as a long, slender beam undergoing in-plane bending.

Using Bernoulli-Euler beam theory, the strain energy is given by:

The strain energy of the absorber comes from the spring and can be written as

where ak is the absorber stiffness.

The total strain energy of the system is the sum of the strain energy components

from the blade and absorber.

[ ] [ ] jvx2vixav2xa 2r

2rra

ˆˆ)( Ω−Ω++Ω+−Ω−= &&&&&& E.9

drdx

vdEI21U

R

0 2

2

yyb ∫ ⎟⎟⎠

⎞⎜⎜⎝

⎛= E.10

2raa xk

21U = E.11

294

E.3.2 Kinetic Energy

The total kinetic energy consists of contributions from the blade and the absorber.

The kinetic energy of the blade is dependent on the blade velocity (Eq. E.3) and is

The kinetic energy of the absorber is also dependent on its velocity (Eq. E.8) and is

E.3.3 Rayleigh Dissipation Function

The damping in the absorber is captured in Lagrange’s equation through the use

of the Rayleigh dissipation function:

E.4 Finite Element Discretization

The energy expressions are spatially discretized using the finite element method.

The blade is represented by a number of beam-absorber elements. Each beam element

consists of four beam degrees of freedom, with two degrees of freedom at each element

boundary node, and one absorber degree of freedom. The beam deflections within each

element are expressed in terms of spatial shape functions and the element nodal

drvvm21T

R

0 bbb ∫ ⋅= E.12

aaaa vvm21T ⋅= E.13

2rad xc

21r &= E.14

295

displacements: ∑=

=4

1iii tqxHtxv )()(),( . The shape functions are Hermitian polynomials

and are defined as:

and the elemental nodal displacement vector is defined as

The discretized displacements are substituted into the expressions for kinetic and

strain energies, and Lagrange’s equation is applied. This yields the mass, stiffness, and

damping matrices for use in the finite element equations of motion. After obtaining the

elemental matrices, the global matrices are assembled, and compatibility of the global

degrees of freedom between adjacent elements is ensured. The absorber embedded in the

rotor blade adds an additional equation of motion to the system, which is added to the

global equations as they are assembled. The assembly of the global matrices is shown

graphically in Figure E-2, where, as indicated in the figure, the absorber is located at the

1i + element. The discretized equations of motion can be written in terms of the blade

equations and the absorber equation. The blade equations are identical to the in-plane

bending equations as outlined in [56], with the addition of absorber terms in the blade

el

2

el

3

el4

2

el

3

el3

elel

2

el

3

el2

2

el

3

el1

llx

lxH

lx3

lx2H

llx

lx2

lxH

1lx3

lx2H

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛=

+⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛=

E.15

[ ]bbaaTi vvvvq ′′= ,,, E.16

296

mass (inertial term) and stiffness (centrifugal force term) matrices, as well as additional

damping terms due to the absorber damping and the Coriolis coupling between the blade

and absorber:

The absorber equation is very similar to that derived for the two-degree-of-freedom

system:

Note there are Coriolis coupling terms present in Eq. E.17 and E.18 as expected, but this

coupling only appears at the element where the absorber is located. Therefore, the

coupling term in Eq. E.17 and all terms in Eq. E.18 are not summed over the number of

elements. The terms in the mass, stiffness, and damping matrices are given in the next

sections.

E.4.1 Blade Matrices

The mass matrix of the thi element can be expressed as:

The stiffness matrix of the thi element can be expressed as:

where ρρdmT 2lel

xΩ= ∫ .

( ) ( )[ ] FxCqKqM rba

Nel

1iiibbiibb =++∑

=

&&& E.17

aabraaraaraa FqCxKxCxM =+++ &&&& E.18

1T1alel

Tbb HHmHdxmHM += ∫ E.19

1T1

2alel

T

lel

T2

lel

Tbb HHmdxHHEIHdxHmdxHHTK Ω−′′′′+Ω−′′= ∫∫∫ E.20

297

The blade-absorber damping matrix can be expressed as

E.4.2 Absorber Terms

The absorber mass term is simply the absorber mass:

The absorber damping term is the viscous damping term from the absorber model:

The absorber stiffness term can be expressed as

The absorber-blade damping matrix can be expressed as:

In the above equations,

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

0100

H Ta and is used to properly place the blade-absorber and

absorber-blade terms within the elemental blade matrices, which are 4x4 matrices.

T1aba Hm2C Ω−= E.21

aaa mM = E.22

aaa cC = E.23

2aaaa mkK Ω−= E.24

aaab Hm2C Ω= E.25

298

E.5 Blade and Absorber Response Solution

It is possible for the set of discretized equations of motion to involve a significant

number of degrees of freedom, depending on the number of elements chosen to represent

the rotor blade. While this is not necessarily the case for lag bending only, it will become

more so as other modes of motion are added to the model. To reduce computational time,

the blade and absorber equations can be transformed into modal space using the

eigenvectors of Eq. E.26:

A number of modes are chosen to represent the rotor blade. Using the eigenvectors

associated with the selected modes, Φ , the blade and absorber equations of motion in

modal space can be written as:

where

[ ]{ } [ ]{ } { }0qKqM =+&& E.26

[ ]{ } [ ]{ } [ ]{ } { }FpKpCpM =++ &&& E.27

{ } { }pq Φ=

[ ] [ ]ΦΦ= MM T

[ ] [ ]ΦΦ= CC T

[ ] [ ]ΦΦ= KK T

and { } { }FF TΦ=

E.28

299

The modal response, p , is then solved for using the harmonic balance method,

and the physical response, q , is determined from the modal response using the

transformation pq Φ= .

E.6 Absorber Static Displacement

Up to this point, the absorber spring stiffness has been considered to be a

constant, and its value is determined by the tuning requirements of the system. A major

factor in the future design of the absorber is the large centrifugal force field in which the

absorber will be required to operate. The static displacement of the absorber due to the

centrifugal force is dependent on the rotor speed, the radial offset of the absorber from

the hub, the absorber mass, and the absorber spring stiffness. Using the spring stiffness

required to achieve the desired tuning frequency results in the static displacement of the

absorber essentially “pegged” at the end of the rotor blade. Therefore, a frequency-

dependent spring stiffness is required for the absorber, with a high static stiffness to

withstand the centrifugal force, yet a low enough dynamic stiffness to still achieve the

desired tuning frequency of the absorber.

E.7 Blade Root and Hub Loads

Once the blade response is known, the blade root loads can be determined. From

these, the hub loads can be calculated.

300

E.7.1 Blade Root Loads

The blade root shear forces and moments are calculated using the finite element

constraint equation method, where the unconstrained global matrices and displacement

vectors are used to determine the loads at the blade root.

From the previous section, the displacements (and velocities and accelerations)

are known. These displacements are used to calculate the reaction forces at the

constraint:

Only the rows of the mass, damping, and stiffness matrices corresponding to the

constrained degrees of freedom are necessary to calculate the reaction forces (and

moments) at the constraint. For the case of a hingeless rotor blade modeled as a

cantilever beam, this corresponds to the first two rows in the matrices. The forcing

vector, { }F , is included in Eq. E.29, although the entries corresponding to the

constrained degrees of freedom are generally zero, as there are usually no forces applied

to the constrained end of the root element.

Alternatively, the blade root loads can be determined using a force summation

method as outlined in Reference [57]:

where [ ]∗K is the modified stiffness matrix with all elastic strain energy-related terms

removed. { }S is the blade nodal load vector, which contains inertial loads for all degrees

of freedom ( v and v′ ) at every finite element node, as well as the inertial loads for the

{ } [ ]{ } [ ]{ } [ ]{ } { }FqKqCqMR −++= &&& E.29

{ } [ ]{ } [ ]{ } [ ]{ } { }FqKqCqMS −++= ∗&&& E.30

301

absorber degree of freedom ( x ) at the node where the absorber is located. The blade root

loads can be calculated by summing the contributions from the blade nodal forces and

moments along the blade span:

Since the in-plane displacement of the blade is the only displacement considered for this

analysis, only the in-plane shear force, xS , and bending moment, ζM , at the blade root

can be calculated with Eq. E.29 or Eq. E.31.

E.7.2 Rotor Hub Loads

Once the blade root shear forces and moments are known, the rotor hub loads can

then be calculated. Although only one rotor blade is used to determine the response, and

all other blades are assumed to have identical responses, all blades must be considered

when calculating the hub loads.

∑ ∑∑

∑ ∑∑

∑

∑

∑

=

−

==

=

−

==

=

=

=

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

=

=

=

nodesof

1k

1k

1n

nkx

nodesof

1k

k

nodesof

1k

1k

1n

nkz

nodesof

1k

k

nodesof

1k

k

nodesof

1k

kzz

nodesof

1k

kxx

lfmM

lfmM

mM

fS

fS

##

##

#

#

#

ζζ

ββ

φφ E.31

302

The general expression for the azimuthal position of each blade, iψ , is found by

Again, since only in-plane bending is considered, zF , xM , and yM are not

calculated in this analysis.

E.8 Shear Force and Moment Distributions Along the Blade Radius

Because the absorber is embedded within the rotor blade and can be located at any

point along the blade space, calculation of the shear force and moment distribution along

the blade is necessary to determine what impact the absorber has on these loads. The

blade loads can be calculated in two ways: reaction force method and force summation

method.

, ,

cossin

sincos

cossin

sincos

∑

∑

∑

∑

∑

∑

=

=

=

=

=

=

−=

−=

+=

=

−=

+=

b

b

b

b

b

b

N

1i

iHz

N

1i

iiiiHy

N

1i

iiiiHx

N

1i

iz

Hz

N

1i

iix

iir

Hy

N

1i

iix

iir

Hx

MM

MMM

MMM

SF

SSF

SSF

ζ

βφ

βφ

ψψ

ψψ

ψψ

ψψ

E.32

b

i

N21i πψψ )( −+= E.33

303

Using the reaction force method, the blade loads are calculated by solving the

finite element governing equation at the elemental level to determine the reactions at the

element endpoints [58]. The reaction forces on an element are calculated by:

where the terms in the above equation are the unmodified elemental matrices and vectors.

The elemental matrices must be used and not the global matrices, as the internal reactions

between elements sum to zero in the global system. This method can also be used to

determine the blade root forces and moments.

In order to calculate the forces and moments in a particular direction using the

reaction force method, a displacement in that same direction must be modeled in the

finite element model. In this analysis, elastic axial deformation is neglected, which

makes little difference in the rotor lag response because the axial deformations are small.

However, the axial forces are very large, primarily due to the centrifugal force, and these

forces can not be captured using the reaction force method. The radial distribution of the

forces and moments can also be calculated using the force summation method. In this

method, analytical expressions for the inertia and external forces are integrated along the

blade span to obtain the sectional forces, as well as the hub loads.

The force summation method uses Newton’s second law, ∑ = amF , where the

sum of the inertial loads must equal the applied loads. At each blade section, the inertial

loads are calculated using the accelerations of the blade and absorber. The applied loads

in this analysis are the elastic forces in the blade and the in-plane periodic force applied

{ } [ ]{ } [ ]{ } [ ]{ } { }elelelelelelel FqKqCqMQ −++= &&& E.34

304

to the blade tip. By calculating the inertial forces, the elastic forces can be determined,

since the applied force is known.

The axial and in-plane blade loads at any given point can be calculated by

Eq. E.35 (adapted from [58]):

where AL is the external forcing, which would normally be the aerodynamic loads, but in

this analysis, it is the periodic tip force. The inertial loads, IuL and I

vL are calculated

from respective components of the blade accelerations:

The inertial loads must also include the forces due to the absorber:

The forces in Eq. E.38 and E.39 are not integrated along the length of the blade; rather

they are added to the integrated forces at the absorber location along the blade. The blade

root loads can also be calculated using Eq. E.35, by substituting 0x0 = .

Iav

R

x

Av

Ivx

Iau

R

x

Au

Iur

LdxLLS

LdxLLS

0

0

++=

++=

∫

∫)(

)( E.35

( ) ⎥⎦⎤

⎢⎣⎡ ′+′−Ω−Ω−== ∫

x

0

2x

Iu dxvvvvv2rmmaL &&&&& E.36

⎥⎦⎤

⎢⎣⎡ ′Ω−Ω−== ∫

x

0

2y

Iv dxvv2vvmmaL &&& E.37

r2

aaraIau xmvm2xmL Ω−Ω−= &&& E.38

ra2

aaIav xm2vmvmL &&& Ω−Ω−= E.39

305

E.9 Results

All results were generated using the HART I (Higher Harmonic Control

Aeroacoustic Rotor Test) rotor data (see Appendix C), with the exception of the

comparison with the rigid blade, which used the rotor blade data from [57]. For the

analysis using the HART I rotor, a tip force of )sin(Ψ+= 21 FFF , was applied to the

rotor. This type of forcing very generally approximates the static and dominant 1/rev

periodic in-plane aerodynamic forces experienced by the rotor, which results in a general

approximation for the lag response of the rotor. All results presented for the HART I

rotor were generated with and without the absorber for comparison. The absorber

properties used in the simulation were chosen based on results generated in Chapter 2. A

miE-span location ( 50a .= ) and a low value of absorber damping ( 30a .=ζ ) were

selected, which results in approximately 15% critical damping transferred to the lag

mode, regardless of the mass of the absorber. The largest absorber mass evaluated in

Chapter 2 ( 050m .=α ) was selected, primarily to evaluate the upper limit of the effect of

the absorber mass on the blade root and hub loads. The dynamic spring stiffness of the

absorber, ak , was calculated to tune the absorber at the first lag natural frequency at the

operating RPM. The static spring stiffness of the absorber, staticak , was calculated to

obtain a 2.5%R static displacement of the absorber due to the static component of the

centrifugal force. The static spring stiffness was approximately 15 times larger than the

dynamic spring stiffness.

With just the lag motion modeled with the absorber, it is difficult to say how the

actual rotor blade with the absorber will respond. However, the method of constructing

306

the finite element equations with the absorber, as well as the methods of calculating the

blade loads, is validated with this simple model. Furthermore, some observations about

the system can be made that may carry over to a more complex model with additional

modes and aerodynamic forcing.

E.9.1 Comparison with Rigid Blade Response

The blade and absorber responses determined from the elastic blade analysis were

compared with the responses determined from the two-degree-of-freedom rigid blade

analysis to assist in determining the accuracy of the elastic blade model. In order to

compare the elastic blade with the rigid blade, the blade stiffness was increased

significantly to approximate a rigid blade, and the hub boundary conditions were

modified from a cantilever beam to that of a simply-supported beam with a hinge spring.

The blade tip response and absorber response from the two-degree-of-freedom and elastic

lag models are shown in Figure 4-1. The frequencies and damping ratios of the rigid and

elastic blades also compare favorably with the results obtained in Chapter 2.

E.9.2 Blade and Absorber Response

The first three lag natural frequencies were calculated for the HART I rotor and

compared with results from [60] (see Table E-1).

307

The first three lag mode shapes were plotted with and without the absorber in Figure E-4.

The mode shapes appear as expected, and the addition of the absorber does little to

change the mode shapes.

The blade tip response with and without the absorber is plotted in Figure E-5.

With the addition of the absorber, the blade tip response has a static offset increase with

respect to the original response. The static offset change is due to the increase in the

inertia force in the lag direction due to the absorber mass. The blade tip response with

the absorber also changes with azimuthal location. The dynamic change is due to the

sinusoidal variation in the absorber velocity, and hence the Coriolis damping force, which

is 90° out of phase with the absorber response. The absorber response is plotted along

with the blade tip response (with absorber) in Figure E-6.

E.9.3 Blade Root Loads

The blade root drag shear and moment were calculated using the three methods,

reaction force method, force summation method, and constraint equation method, to

verify each of the methods. The blade root drag shear force is plotted in Figure E-7, and

the blade root lag moment is plotted in Figure E-8. As can be seen in the two figures,

Table E-1: Comparison of lag mode frequencies with results generated by RCAS

Mode Frequency (Hz) RCAS frequency (Hz)

1st Lag 10.30 10.27 2nd Lag 75.99 75.52 3rd Lag 190.75 190.945

308

each of the three methods generates the same results for this simple model and blade.

The addition of the absorber slightly increases both the in-plane shear force and lag

moment at the blade root.

Since there is no axial degree of freedom in the model, the blade root axial shear

force could only be calculated using the force summation method. The results with and

without the absorber can be seen in Figure E-9. While the blade and absorber model and

results will change with increased model complexity, this simple model can capture much

of the effect the absorber has on the axial shear force due to the centrifugal force. There

is about a 6% increase in the axial force on the hub due to the addition of the absorber,

with the absorber parameters used in this simulation.

E.9.4 Radial Distribution of Blade Loads

The changes in the radial distribution of the in-plane shear and moment, as well as

the axial force are shown in Figures E-10 – E-12, with azimuthal locations of 0° and 180°

plotted in each figure. All blade loads show an increase at the absorber location, as

expected, and the effect of the absorber continues inboard to the hub. The radial

distribution of the in-plane shear and moment are the same at the two azimuthal locations

for the case without the absorber due to the sinusoidal variation of the applied tip force.

However, with the addition of the absorber, the change in the in-plane forces and

moments along the blade varies with azimuthal location as a result of the Coriolis

coupling. Since the axial force distribution depends largely on the centrifugal force,

309

which does not vary azimuthally, the axial force distribution at all azimuthal locations

show approximately the same result as seen in Figure E-10.

E.9.5 Blade Loads at Absorber Location

The increase in the blade loads at the absorber location discussed in the previous

section can also be seen in Figures E-13 and E-14. In these two figures, the in-plane

shear and axial forces at the absorber location are plotted to visualize how the blade

forces at the absorber location vary azimuthally. The effect of the absorber on the blade

loads is important to determine the structural requirements of the blade due to the

addition of the absorber.

E.10 Conclusions

Even though this simple model is by no means a complete analysis, it allowed for

several issues to be resolved, and these can now be addressed with confidence in a more

complex model. First, the process of understanding how to include the absorber in the

finite element model of the blade was an important step in the analysis. Second, the

methods of determining blade root loads, the radial distribution of blade loads, as well as

the hub loads were thoroughly examined. Finally, some of the questions concerning the

effect of the absorber on the blade and the rotor hub, particularly the effect of the

absorber on the axial forces, were addressed.

310

While this was an important step, it obviously cannot replace a more complex

analysis, which includes additional modes and aerodynamic forcing.

311

ψ

XH

YH

ZH, zR, ζ

xRv

ξψ

XH

YH

ZH, zR, ζ

xRv

ξψψ

XH

YH

ZH, zR, ζ

xRv

ξ

XH

YH

ZH, zR, ζ

xRv

ξξξ

Figure E-1: Blade coordinate system

av av′ bv bv′

av

av′

bv

bv′

Element i

Element i+1

Element i+2

Blade-absorber terms (4x1)

Absorber-blade terms (1x4) Absorber term (1x1)

av av′ bv bv′

av

av′

bv

bv′

Element i

Element i+1

Element i+2

Blade-absorber terms (4x1)

Absorber-blade terms (1x4) Absorber term (1x1)

Figure E-2: Global assembly of blade elemental matrices with absorber terms

312

0 45 90 135 180 225 270 315 360−3

−2

−1

0

1

2

3


Res

pons

e (in

)

Rigid blade tip responseRigid blade absorber responseElastic blade tip responseElastic blade absorber response

Figure E-3: Blade tip response comparison – rigid and elastic blade analyses

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1


Without absorber

With absorber

Figure E-4: First three lag mode shapes

313

0 45 90 135 180 225 270 315 3602.4

2.6

2.8

3

3.2

3.4

3.6

3.8

4


Bla

de T

ip R

espo

nse

(cm

)

Without absorber

With absorber

Figure E-5: Blade tip response – with and without absorber

0 45 90 135 180 225 270 315 3602.5

3

3.5

4

4.5

5

5.5

6


Res

pons

e (c

m)

Blade tip responseAbsorber response

Figure E-6: Blade tip and absorber responses

314

0 45 90 135 180 225 270 315 360−11

−10.5

−10

−9.5

−9

−8.5

−8

−7.5

−7

−6.5

x 10−3

Bla

de R

oot I

npla

ne S

hear


Without absorber

With absorber

Figure E-7: Blade root drag shear force – with and without absorber

0 45 90 135 180 225 270 315 360−2

−1.8

−1.6

−1.4

−1.2

−1x 10

−3

Bla

de R

oot L

ag B

endi

ng M

omen

t


Without absorber

With absorber

Figure E-8: Blade root lag moment – with and without absorber

315

0 45 90 135 180 225 270 315 360−0.56−0.56

−0.555

−0.55

−0.545

−0.54

−0.535

−0.53

−0.525

−0.52

−0.515

Bla

de R

oot A

xial

For

ce


Without absorber

With absorber

Figure E-9: Blade root axial force – with and without absorber

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0


Axi

al F

orce

Without absorber

With absorber

Figure E-10: Radial distribution of axial force – with and without absorber

316

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.01

−0.009

−0.008

−0.007

−0.006

−0.005

−0.004

−0.003

−0.002

−0.001

0


In−

plan

e S

hear

ψ = 0 degψ = 180

Without absorber

With absorber

Figure E-11: Radial distribution of drag shear force – with and without absorber

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1.8

−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0x 10

−3


Lag

Ben

ding

Mom

ent

ψ = 0 degψ = 180

Without absorber

With absorber

Figure E-12: Radial distribution of lag bending moment – with and without absorber

317

0 45 90 135 180 225 270 315 360−9

−8

−7

−6

−5x 10

−3


In−

plan

e S

hear

For

ce a

t Abs

orbe

r Lo

catio

n

Without absorber

With absorber

Figure E-13: Drag shear force at absorber location – with and without absorber

0 45 90 135 180 225 270 315 360−0.425

−0.42

−0.415

−0.41

−0.405

−0.4

−0.395

−0.39

−0.385

−0.38


Axi

al F

orce

at A

bsor

ber

Loca

tion

Without absorber

With absorber

Figure E-14: Axial force at absorber location – with and without absorber

VITA

Lynn Karen Byers

Education:

The Pennsylvania State University Ph.D. in Aerospace Engineering, August 2006 The Pennsylvania State University M.S. in Aerospace Engineering, May 1997 United States Military Academy B.S. in Mechanical Engineering (Aerospace), May 1987

Selected Publications:

Byers, L. and Gandhi, F., “Rotor Blade with Radial Absorber (Coriolis Damper) - Loads Evaluation,” Proceedings of the American Helicopter Society 62nd Annual Forum, Phoenix, AZ, May 9-11, 2006.

Byers, L. and Gandhi, F., “Helicopter Rotor Lag Damping Augmentation Based on a Radial Absorber and Coriolis Coupling,” Proceedings of the American Helicopter Society 61st Annual Forum, Grapevine, TX, June 1-3, 2005. Byers, L. and Gandhi, F., “Embedded Absorbers for Rotor Lag Damping,” To be presented at the 32nd European Rotorcraft Forum, Maastricht, The Netherlands, 12-14 September 2006.

Professional Position:

Lieutenant Colonel, US Army US Army Aviator, May 1987 - Present

helicopter rotor lag damping augmentation based …

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