Comprehensive Multibody AeroServoElastic Analysis of …home.aero.polimi.it/masarati/Publications/thesis.pdf · Comprehensive Multibody AeroServoElastic Analysis of Integrated Rotorcraft

ComprehensiveMultibody AeroServoElasticAnalysis

ofIntegratedRotorcraftActive Controls

Dipartimentodi IngegneriaAerospazialePolitecnicodi Milano

Dottoratodi Ricercain IngegneriaAerospaziale,XII Ciclo

PierangeloMasarati

Tutor: ProfessorPaoloMantegazza

Contents

Moti vation XIII

Intr oduction 1

I Rotorcraft modelling 3

1 Dynamicsof a fr eerigid body 51.1 Momentadefinition . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Equilibriumequations. . . . . . . . . . . . . . . . . . . . . . . . 61.3 Rigid-bodydynamicsasODEs . . . . . . . . . . . . . . . . . . . 6

2 Algebraic constraints 72.1 Reducedvs.redundantcoordinateset . . . . . . . . . . . . . . . 72.2 Constrainedrigid-bodydynamicsasDAEs . . . . . . . . . . . . . 8

3 Kinematics of finite rotations 113.1 Finite rotationproperties . . . . . . . . . . . . . . . . . . . . . . 113.2 Rotationparametrisation. . . . . . . . . . . . . . . . . . . . . . 12

3.2.1 Rotationvector . . . . . . . . . . . . . . . . . . . . . . . 133.2.2 Gibbs-Rodriguezparameters. . . . . . . . . . . . . . . . 13

3.3 Updatedapproach. . . . . . . . . . . . . . . . . . . . . . . . . . 143.4 Updated-updatedapproach . . . . . . . . . . . . . . . . . . . . . 15

4 Algorithmic implications 174.1 Unconditionalstability . . . . . . . . . . . . . . . . . . . . . . . 184.2 Integrationformula . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.2.1 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . 204.2.2 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.2.3 Linearisation . . . . . . . . . . . . . . . . . . . . . . . . 204.2.4 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

I

II CONTENTS

4.2.5 Implicit ODEs . . . . . . . . . . . . . . . . . . . . . . . 224.2.6 Algebraic-differentialequations . . . . . . . . . . . . . . 224.2.7 Noteon theclassificationof themethod . . . . . . . . . . 234.2.8 Higher-orderformulas . . . . . . . . . . . . . . . . . . . 244.2.9 Second-orderformula . . . . . . . . . . . . . . . . . . . 254.2.10 Furtherremarkson stability . . . . . . . . . . . . . . . . 27

4.3 Start-upof thesimulation . . . . . . . . . . . . . . . . . . . . . . 294.3.1 Initial assembly. . . . . . . . . . . . . . . . . . . . . . . 314.3.2 Derivativescomputation . . . . . . . . . . . . . . . . . . 324.3.3 Self-startingalgorithm . . . . . . . . . . . . . . . . . . . 334.3.4 Secondderivativeof theconstraints . . . . . . . . . . . . 34

5 Configuration-dependentinteractions 375.1 Lumpedflexible elements. . . . . . . . . . . . . . . . . . . . . . 385.2 Beammodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 405.2.2 Kinematicsof thebeam . . . . . . . . . . . . . . . . . . 405.2.3 Strainsandcurvatures . . . . . . . . . . . . . . . . . . . 415.2.4 Noteon thelinearisationof thecurvature . . . . . . . . . 425.2.5 Strainandcurvaturetime rates . . . . . . . . . . . . . . . 425.2.6 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.3 Beamsectioncharacterisation . . . . . . . . . . . . . . . . . . . 445.3.1 Kinematicsof thesection. . . . . . . . . . . . . . . . . . 455.3.2 Internalwork . . . . . . . . . . . . . . . . . . . . . . . . 475.3.3 Externalwork . . . . . . . . . . . . . . . . . . . . . . . . 485.3.4 Discretisation. . . . . . . . . . . . . . . . . . . . . . . . 505.3.5 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.3.6 Noteon thedeterminationof thewarping . . . . . . . . . 53

5.4 Finite volumebeamformulation . . . . . . . . . . . . . . . . . . 535.4.1 Finite equilibrium . . . . . . . . . . . . . . . . . . . . . 545.4.2 Constitutive law . . . . . . . . . . . . . . . . . . . . . . 545.4.3 Linearisation . . . . . . . . . . . . . . . . . . . . . . . . 555.4.4 Discretisation. . . . . . . . . . . . . . . . . . . . . . . . 555.4.5 Implementationnotes. . . . . . . . . . . . . . . . . . . . 57

5.5 Platemodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.5.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 585.5.2 Kinematicsof theplate . . . . . . . . . . . . . . . . . . . 595.5.3 Plateequilibrium . . . . . . . . . . . . . . . . . . . . . . 605.5.4 Singularityandcompatibility . . . . . . . . . . . . . . . . 60

5.6 Platefibre characterisation. . . . . . . . . . . . . . . . . . . . . 615.6.1 Kinematicsof thefibre . . . . . . . . . . . . . . . . . . . 61

CONTENTS III

5.6.2 Internalwork . . . . . . . . . . . . . . . . . . . . . . . . 625.6.3 Externalwork . . . . . . . . . . . . . . . . . . . . . . . . 635.6.4 Discretisation. . . . . . . . . . . . . . . . . . . . . . . . 655.6.5 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.6.6 Compatibilityenforcement. . . . . . . . . . . . . . . . . 675.6.7 Characterisationof thefibre . . . . . . . . . . . . . . . . 68

5.7 FiniteVolumePlate . . . . . . . . . . . . . . . . . . . . . . . . . 695.7.1 Finite Equilibrium . . . . . . . . . . . . . . . . . . . . . 695.7.2 Compatibility . . . . . . . . . . . . . . . . . . . . . . . . 715.7.3 Implementationnotes. . . . . . . . . . . . . . . . . . . . 72

5.8 Modal flexibility . . . . . . . . . . . . . . . . . . . . . . . . . . 725.9 Aerodynamicforces. . . . . . . . . . . . . . . . . . . . . . . . . 73

5.9.1 Strip-theory, quasi-steadyaerodynamicforces. . . . . . . 735.9.2 Inducedvelocity . . . . . . . . . . . . . . . . . . . . . . 75

II Control 77

6 Rotorcraft control 796.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 796.2 Trim tab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 806.3 Bladepitchcontrol . . . . . . . . . . . . . . . . . . . . . . . . . 826.4 Inducedtwist actuation . . . . . . . . . . . . . . . . . . . . . . . 83

7 DiscreteControl 877.1 DiscreteTimeEquation. . . . . . . . . . . . . . . . . . . . . . . 887.2 SystemIdentification(ID) . . . . . . . . . . . . . . . . . . . . . 89

7.2.1 Recursive Implementation . . . . . . . . . . . . . . . . . 907.2.2 Stabilisationof theParameterEstimates. . . . . . . . . . 917.2.3 Adaptive forgettingfactor . . . . . . . . . . . . . . . . . 92

7.3 PredictiveControl . . . . . . . . . . . . . . . . . . . . . . . . . . 937.3.1 GeneralisedPredictiveControl . . . . . . . . . . . . . . . 947.3.2 Interpretationof thePredictiveControl . . . . . . . . . . 957.3.3 Temporalweighting . . . . . . . . . . . . . . . . . . . . 96

8 Multidisciplinary problems 998.1 Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 998.2 Actuators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

8.2.1 Swashplate . . . . . . . . . . . . . . . . . . . . . . . . . 1008.2.2 Piezoelectricbeam . . . . . . . . . . . . . . . . . . . . . 1018.2.3 Otheractuationmeans . . . . . . . . . . . . . . . . . . . 101

IV CONTENTS

8.3 Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1028.3.1 Accelerometers. . . . . . . . . . . . . . . . . . . . . . . 1038.3.2 Straingages. . . . . . . . . . . . . . . . . . . . . . . . . 1038.3.3 Piezoelectricbeams. . . . . . . . . . . . . . . . . . . . . 1038.3.4 Direct unknown measure. . . . . . . . . . . . . . . . . . 103

8.4 Networking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1048.5 Generalpurpose. . . . . . . . . . . . . . . . . . . . . . . . . . . 1058.6 Otherproblems . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

9 Piezoelectricbeamanalysis 1079.1 Piezoelectricbeamsection . . . . . . . . . . . . . . . . . . . . . 107

9.1.1 Electricfield . . . . . . . . . . . . . . . . . . . . . . . . 1079.1.2 Internalwork . . . . . . . . . . . . . . . . . . . . . . . . 1089.1.3 Externalwork . . . . . . . . . . . . . . . . . . . . . . . . 1109.1.4 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 1119.1.5 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 112

9.2 Finite volumepiezoelectricbeam. . . . . . . . . . . . . . . . . . 1129.2.1 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . 1129.2.2 Chargebalance . . . . . . . . . . . . . . . . . . . . . . . 1139.2.3 Discreteform . . . . . . . . . . . . . . . . . . . . . . . . 113

III Applications 115

10 Preliminary studies 11710.1 Rigid bodymechanisms. . . . . . . . . . . . . . . . . . . . . . . 117

10.1.1 Pendulum. . . . . . . . . . . . . . . . . . . . . . . . . . 11710.1.2 Spinningtop . . . . . . . . . . . . . . . . . . . . . . . . 11810.1.3 Bipendulum. . . . . . . . . . . . . . . . . . . . . . . . . 119

10.2 Flexible elements . . . . . . . . . . . . . . . . . . . . . . . . . . 12210.2.1 Flexible pendulum . . . . . . . . . . . . . . . . . . . . . 12210.2.2 Bucklingof axially compressedbeam . . . . . . . . . . . 12710.2.3 Rotorblademodalanalysis. . . . . . . . . . . . . . . . . 12910.2.4 Flexible leverage . . . . . . . . . . . . . . . . . . . . . . 131

10.3 GeneralisedPredictiveControl . . . . . . . . . . . . . . . . . . . 13310.3.1 Threemassessystem . . . . . . . . . . . . . . . . . . . . 133

11 Tiltr otor modelanalysis 14311.1 Tiltrotor submodels. . . . . . . . . . . . . . . . . . . . . . . . . 146

11.1.1 Blade . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14711.1.2 Gimbal . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

CONTENTS V

11.1.3 Swashplate . . . . . . . . . . . . . . . . . . . . . . . . . 14911.1.4 Wing-Pylon . . . . . . . . . . . . . . . . . . . . . . . . . 149

11.2 Preliminaryconsiderations. . . . . . . . . . . . . . . . . . . . . 15111.3 Rotormodels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15311.4 Wing model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15411.5 Wing-rotormodels . . . . . . . . . . . . . . . . . . . . . . . . . 15611.6 Testcases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

11.6.1 Responseto controls . . . . . . . . . . . . . . . . . . . . 15611.6.2 Conversionmaneuvre. . . . . . . . . . . . . . . . . . . . 15811.6.3 Gustresponse. . . . . . . . . . . . . . . . . . . . . . . . 160

11.7 Computationalnotes . . . . . . . . . . . . . . . . . . . . . . . . 160

12 Tiltr otor vibration control 16512.1 Hover— harmonicexcitation. . . . . . . . . . . . . . . . . . . . 16512.2 Forwardflight — harmonicexcitation . . . . . . . . . . . . . . . 16712.3 Forwardflight — gustresponse. . . . . . . . . . . . . . . . . . . 16912.4 Forwardflight — flutter suppression. . . . . . . . . . . . . . . . 172

13 ActiveTwist Rotor analysis 18113.1 Materialcharacterisation. . . . . . . . . . . . . . . . . . . . . . 18113.2 Bladesectioncharacterisation . . . . . . . . . . . . . . . . . . . 18313.3 ActiveTwist Rotormodeldescription . . . . . . . . . . . . . . . 18813.4 Model validation . . . . . . . . . . . . . . . . . . . . . . . . . . 18913.5 Hoverharmonicactuation. . . . . . . . . . . . . . . . . . . . . . 19113.6 Forwardflight open-loopcontrol . . . . . . . . . . . . . . . . . . 192

14 Conclusionsand futur e research 195

A Rigid body momenta 197

B Integration formulas 199B.1 Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199B.2 Numericalintegration . . . . . . . . . . . . . . . . . . . . . . . . 199B.3 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

B.3.1 Cubicinterpolation . . . . . . . . . . . . . . . . . . . . . 200B.3.2 Parabolicinterpolation . . . . . . . . . . . . . . . . . . . 201B.3.3 Linearinterpolation. . . . . . . . . . . . . . . . . . . . . 203

B.4 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204B.4.1 Cubicinterpolation . . . . . . . . . . . . . . . . . . . . . 204B.4.2 Parabolicinterpolation . . . . . . . . . . . . . . . . . . . 205B.4.3 Linearinterpolation. . . . . . . . . . . . . . . . . . . . . 206

VI CONTENTS

B.5 TunableAlgorithmic Damping . . . . . . . . . . . . . . . . . . . 207B.6 Concludingremarks. . . . . . . . . . . . . . . . . . . . . . . . . 209

C Self-starting algorithm: convergence 217

D Constraints 221D.1 Kinematicconstraints. . . . . . . . . . . . . . . . . . . . . . . . 221

D.1.1 Coincidence. . . . . . . . . . . . . . . . . . . . . . . . . 221D.1.2 Orthogonality. . . . . . . . . . . . . . . . . . . . . . . . 222

D.2 DynamicConstraints . . . . . . . . . . . . . . . . . . . . . . . . 223D.2.1 Rods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223D.2.2 Springs . . . . . . . . . . . . . . . . . . . . . . . . . . . 225D.2.3 Rotationalsprings . . . . . . . . . . . . . . . . . . . . . 225D.2.4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 226

E Beamsectionanalysis 227E.1 Internalwork perunit volume . . . . . . . . . . . . . . . . . . . 227E.2 Discretisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228E.3 Finite Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . 229E.4 Distributedexternalloads . . . . . . . . . . . . . . . . . . . . . . 229

F Plate fibre analysis 233F.1 Internalwork perunit volume . . . . . . . . . . . . . . . . . . . 233F.2 Discretisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

G Piezoelectricbeamanalysis 235G.1 Internalwork perunit volume . . . . . . . . . . . . . . . . . . . 235G.2 Discretisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

H Implementation notes 239H.1 ObjectOrientedprogramming . . . . . . . . . . . . . . . . . . . 239

H.1.1 Drives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239H.1.2 Templateconstitutive laws . . . . . . . . . . . . . . . . . 240

H.2 Genericprogramming. . . . . . . . . . . . . . . . . . . . . . . . 240H.3 Reuseof code . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

H.3.1 Matrix Handling . . . . . . . . . . . . . . . . . . . . . . 241H.3.2 LinearAlgebra . . . . . . . . . . . . . . . . . . . . . . . 241H.3.3 Three-dimensionaldatastructures . . . . . . . . . . . . . 242H.3.4 Inputhandling . . . . . . . . . . . . . . . . . . . . . . . 243

H.4 Debug/releaseapproach. . . . . . . . . . . . . . . . . . . . . . . 243H.5 Safe-pointerprogramming . . . . . . . . . . . . . . . . . . . . . 244H.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

List of Figures

4.1 Integrationformulas— spectralradii, ρ. . . . . . . . . . . . . . 274.2 Integrationformulas— dampingerror, εξ (undampedsystem). . . 284.3 Integration formulas— phaseerror order, log2

εϕ 2h εϕ h

(slightly dampedsystem). . . . . . . . . . . . . . . . . . . . . . 28

5.1 Finitevolumethree-nodebeamelement. . . . . . . . . . . . . . 565.2 Transverseshearstressdistribution— compatiblevs.non-compatible

andself-balancedsolutions. . . . . . . . . . . . . . . . . . . . . 68

6.1 Rotorbladeactuationtechniques. . . . . . . . . . . . . . . . . . 816.2 ActiveFibreCompositeswith Inter-DigitatedElectrodes.. . . . . 84

8.1 Pitchcontrolscheme . . . . . . . . . . . . . . . . . . . . . . . . 102

9.1 Piezoelectricbeamsection. . . . . . . . . . . . . . . . . . . . . 109

10.1 Spintop— case1 . . . . . . . . . . . . . . . . . . . . . . . . . 12010.2 Spintop— case2 . . . . . . . . . . . . . . . . . . . . . . . . . 12010.3 Spintop— case3 . . . . . . . . . . . . . . . . . . . . . . . . . 12110.4 Spintop— convergence . . . . . . . . . . . . . . . . . . . . . . 12110.5 Bipendulum— xCG vs.yCG . . . . . . . . . . . . . . . . . . . . 12210.6 Bipendulum— xCG vs.zCG . . . . . . . . . . . . . . . . . . . . 12310.7 Bipendulum— yCG vs.zCG . . . . . . . . . . . . . . . . . . . . 12310.8 Bipendulum— z reaction . . . . . . . . . . . . . . . . . . . . . 12410.9 Bipendulum— total reaction . . . . . . . . . . . . . . . . . . . . 12410.10Bathependulum— angularposition. . . . . . . . . . . . . . . . 12610.11Deformablependulum— elongation. . . . . . . . . . . . . . . . 12810.12Deformablependulum— correctinitial elongation. . . . . . . . . 12810.13Deformedshapesof a 4 three-nodeelementbeamunder1/2, 1,

3/2and2 timesthecritical buckling load . . . . . . . . . . . . . 13010.14Internalforcesdueto twice thecritical buckling load,at theeval-

uationpointsof thefour beamelementmodel . . . . . . . . . . . 13010.15Sketchof thediscretisedhelicopterblade . . . . . . . . . . . . . 132

VII

VIII LIST OFFIGURES

10.16Flexible Leverage— scheme. . . . . . . . . . . . . . . . . . . . 13410.17Flexible Leverage— axial forcecloseto theleft endof thebeam. 13510.18Flexible Leverage— zoomof Figure10.17. . . . . . . . . . . . . 13510.19Flexible Leverage— transverseshearforce closeto the left end

of thebeam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13610.20Flexible Leverage— bendingmomentcloseto theleft endof the

beam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13610.21Flexible Leverage— left link x reaction. . . . . . . . . . . . . . 13710.22Flexible Leverage— intermediatenode1 path. . . . . . . . . . . 13710.23Flexible Leverage— mid-nodepath. . . . . . . . . . . . . . . . 13810.24Flexible Leverage— intermediatenode2 path. . . . . . . . . . . 13810.25Flexible Leverage— nodepathsof the4 beammodel. . . . . . . 13910.26Flexible Leverage— frequency contentof the bendingmoment

closeto theleft endof thebeam. . . . . . . . . . . . . . . . . . . 13910.27Threemasses— scheme . . . . . . . . . . . . . . . . . . . . . . 14010.28Threemasses— controlsignals . . . . . . . . . . . . . . . . . . 14110.29Threemasses— displacements. . . . . . . . . . . . . . . . . . . 14110.30Threemasses— stabiliseddisplacements . . . . . . . . . . . . . 142

11.1 WRATS Modelat Langley’sTransonicDynamicsTunnel . . . . 14411.2 Pictorialhistoryof proprotor/tiltrotorinvestigationat Langley. . . 14511.3 Fromtheleft, PierangeloMasarati andDr. Mark W. Nixon in the

TDT duringAugust‘98 testcampaign529. . . . . . . . . . . . . 14611.4 AnalyticalModel . . . . . . . . . . . . . . . . . . . . . . . . . . 14711.5 Cantileveredbladefrequencies,Hz . . . . . . . . . . . . . . . . 14811.6 Pitch-flapcouplingasfunctionof thecollectivepitch θ75% . . . . 15011.7 Controlstiffnessasfunctionof thecollectivepitch θ75% . . . . . 15011.8 Wing frequencies,Hz — downstopoff (top)andon(bottom). . . 15211.9 Pitch-conecouplingasfunctionof thecollectivepitch θ75% . . . 15311.10Internalmomentsatthewing rootduringa10deg.collectivepitch

maneuvre,flexible blademodel . . . . . . . . . . . . . . . . . . 15711.11Blade#1pitchduringa10deg.collectivepitchmaneuvre,flexible

blademodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15811.12Internalmomentsat thewing root duringa 5 deg. fore/aft cyclic

pitchmaneuvre,flexible blademodel . . . . . . . . . . . . . . . 15911.13Blade#1 control link axial force during a 5 deg. fore/aft cyclic

pitchmaneuvre,flexible blademodel . . . . . . . . . . . . . . . 16011.14Internalmomentsat thewing root during theconversionmaneu-

vre,flexible blademodel . . . . . . . . . . . . . . . . . . . . . . 16111.15Gust— off-downstop:wing bending . . . . . . . . . . . . . . . . 16211.16Gust— on-downstop:wingbending . . . . . . . . . . . . . . . . 162

LIST OFFIGURES IX

12.1 Hoverbendingmoment,strainmeasure . . . . . . . . . . . . . . 16612.2 Hoverbendingmoment,accelerationmeasure. . . . . . . . . . . 16612.3 Hovercollective,strainmeasure . . . . . . . . . . . . . . . . . . 16712.4 Hovercollective,accelerationmeasure . . . . . . . . . . . . . . 16812.5 Hoveraccelerometersignal . . . . . . . . . . . . . . . . . . . . 16812.6 Forwardflight bending . . . . . . . . . . . . . . . . . . . . . . . 16912.7 Forwardflight controlsignals . . . . . . . . . . . . . . . . . . . 17012.8 Gustbending . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17112.9 Gustcontrolsignals . . . . . . . . . . . . . . . . . . . . . . . . 17112.10Baseline/persistentexcitation;wing rootbending,top(lb-ft); wing

tip acceleration,bottom(ft/s2). . . . . . . . . . . . . . . . . . . . 17312.11Baseline/non-adaptivecontrol;wing rootbending,top(lb-ft); wing

tip acceleration,bottom(ft/s2). . . . . . . . . . . . . . . . . . . . 17412.12Baseline/adaptivecontrol;wing rootbending,top (lb-ft); wing tip

acceleration,bottom(ft/s2). . . . . . . . . . . . . . . . . . . . . 17512.13Baseline/mixedstrain-accelerationcontrol;wing rootbending,top

(lb-ft); wing tip acceleration,bottom(ft/s2). . . . . . . . . . . . . 17612.14mixedstrain-accelerationcontrol:fluttersuppression;internalcou-

plesat thewing root. . . . . . . . . . . . . . . . . . . . . . . . . 17812.15mixedstrain-accelerationcontrol:fluttersuppression;pylon trans-

verseacceleration. . . . . . . . . . . . . . . . . . . . . . . . . . 17912.16mixedstrain-accelerationcontrol: flutter suppression;pitch con-

trol signals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

13.1 FEmodelsof thetwo specimen . . . . . . . . . . . . . . . . . . 18213.2 VonMisesstress(top)andelectricfield (bottom)dueto axialstrain

in thesquarefibre. . . . . . . . . . . . . . . . . . . . . . . . . . 18413.3 Von Misesstress(top) andelectricfield (bottom)dueto electric

voltagebetweentheelectrodes. . . . . . . . . . . . . . . . . . . 18513.4 Sketchof thebladesection,with detailof thepliesin thespar . . 18613.5 Warpingsdueto (a) out-of-planebending, (b) torsion,(c) electric

tensionthatgivesinducedtwist . . . . . . . . . . . . . . . . . . 18713.6 Frequency responseof root-to-tiptwist androotflapat maximum

controlvoltage . . . . . . . . . . . . . . . . . . . . . . . . . . . 19113.7 Bendingmomentsaboutx andy axesat themast;advancingratio

µ 0 20; actuatingtension:100V (40%of allowable)at 2/rev.;131o phaseshift . . . . . . . . . . . . . . . . . . . . . . . . . . 192

13.8 Bendingmomentsaboutx andy axesat themast;advancingratioµ 0 25;actuatingtension:137.5V (55%of allowable)at2/rev.;128o phaseshift . . . . . . . . . . . . . . . . . . . . . . . . . . 193

X LIST OFFIGURES

13.9 Bendingmomentsaboutx andy axesat themast;advancingratioµ 0 30;actuatingtension:187.5V (75%of allowable)at2/rev.;127o phaseshift . . . . . . . . . . . . . . . . . . . . . . . . . . 193

B.1 Phaseerrororder, log2

εϕ 2h εϕ h (undampedsystem) . . . 211

B.2 Phaseerror, εϕ (undampedsystem) . . . . . . . . . . . . . . . . 211B.3 Phaseerror, εϕ (slightly dampedsystem) . . . . . . . . . . . . . 212B.4 Phaseerror, εϕ, at h T 0 01(undampedsystem) . . . . . . . . 212B.5 Dampingerror, εξ (dampedsystem) . . . . . . . . . . . . . . . . 213B.6 Dampingerror, εξ, at h T 0 01(undampedsystem) . . . . . . 213B.7 Dampingerrororder, log2

εξ 2h εξ h (undampedsystem) . . 214

B.8 Dampingerrororder, log2

εξ 2h εξ h (dampedsystem) . . . 214

B.9 Spectralradii, ρ, polarplot (undampedsystem) . . . . . . . . . . 215

List of Tables

10.1 Pendulumaccuracy (ρ∞ 0 0). . . . . . . . . . . . . . . . . . . 11810.2 Pendulumaccuracy (ρ∞ 0 6). . . . . . . . . . . . . . . . . . . 11810.3 Spintopproperties(Ref. [25]) . . . . . . . . . . . . . . . . . . . 11910.4 Spintop initial conditions(Ref. [25]) . . . . . . . . . . . . . . . 11910.5 Bipendulumproperties(Ref. [25]) . . . . . . . . . . . . . . . . . 12210.6 Bathependulumproperties(Ref. [5]). . . . . . . . . . . . . . . . 12610.7 Deformablependulumproperties. . . . . . . . . . . . . . . . . . 12710.8 Frequenciesof ahelicopterrotor blade . . . . . . . . . . . . . . 13110.9 Flexible leverageproperties. . . . . . . . . . . . . . . . . . . . . 134

11.1 Cantileveredbladefrequencies,Hz . . . . . . . . . . . . . . . . 14811.2 Wing frequencies,Hz . . . . . . . . . . . . . . . . . . . . . . . 15111.3 Singlebladewith flexbeam(lockedgimbal),non-rotating,Hz . . 15411.4 Full rotor (freegimbal),non-rotating,Hz . . . . . . . . . . . . . 15411.5 Rotatingfrequencies,888rpm,θ75% 3 deg.,Hz . . . . . . . . 15511.6 Rotatingfrequencies,742rpm,θ75% 55 deg.,Hz . . . . . . . . 155

13.1 Equivalenthomogeneouspiezoelectricmaterialproperties . . . . 18313.2 Bladestiffnesspropertieswith rectangularfibre. . . . . . . . . . 18613.3 Sectionstiffnessandpiezoelectricmatrices,ref.25%chord;units:

lb, ft andV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18813.4 ActiveTwist Rotorgeometricanddynamicproperties(Ref. [90]) 18913.5 Bladesectionmaterials . . . . . . . . . . . . . . . . . . . . . . . 18913.6 Comparisonof in-vacuorotatingfrequencies[1/rev] . . . . . . . . 19013.7 Hoversimulationswith differentstiffnessproperties . . . . . . . 190

B.1 Summaryof propertiesof someintegrationformulas . . . . . . . 210

C.1 Crank-Nicholson/BackwardDifferencesolutionof constrainedprob-lem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

XI

XII LIST OFTABLES

Moti vation

This work originatesfrom the needto modelthe dynamicsof a rotorcraft. Theproblemof thesimulationof rotorcraft dynamicshasbeendiscussedfor years,asshownbythelargeamountof literatureon thesubjectand,moreover, by thelargenumberof formulationsandcomputercodesavailablenowadays.But it is not awell establishedtopic yet, as provenby the discussionson the suitability of thevariousapproaches,aswell asontheveryfirstprinciplestheformulationsshouldrely on. A very importantquestionis: do weneeda dedicatedanalysisformula-tion, or shouldweusea general purposemechanicalanalysistool? Thispoint isnot trivial sinceit encompassesefficiencyaswell asversatility andcodereusabil-ity considerations.Anotherquestionis: shouldweusean approach that is awareof the nature of the problem,or shouldwe give up any assumption?In the firstcase, by assumingfor instancethat weare dealingwith a mechanicalsystem,wemight take advantage of theconservationproperties,say, of themomentaratherthanof thetotal energy, by usingdedicatedalgorithms,while in thesecondcasemore general algorithmswouldgivemore freedomin formulating, say, multidis-ciplinary problemswith a reducedoverheadboth from the implementationandthecomputationstandpoint.Surely there is no right recipefor this problem,andmanyof the solutionsthat havebeenpresentedsubstantiallymeettheir goal, aswehopetheonewearepresentingwill. Maybeweshouldbetterclarify thepoint:this work is not intendedto present“the” solutionto the problemof the analy-sisof thedynamicsof a rotorcraft; it ratherproposes“a” meansthat representsa good trade-off betweenall the possiblechoicesfor the modellingof general,multidisciplinarymechanismswith a high level of refinementin thedetailsof thekinematics,with a goodcompromisebetweenefficiency, flexibility andaccuracyin themodellingof structural components,andwith sufficientflexibility to allowto considermultidisciplinary, in this casecontrol-related,models.Theformula-tion presentedherein resultsfrom years of investigationat the DipartimentodiIngegneriaAerospazialeof the Politecnicodi Milano, underthe coordination ofProfessorPaoloMantegazza. Asa consequence, someof thework shownin thefollowing chapters representssomeoneelse’s contribution to the project, and isincludedonly to easethereadabilityandto providea basisfor theresults.In the

XIII

XIV INTRODUCTION

middle-ages,philosophers,talkingabouttheir role in sciencecomparedto thatofthegeniusesof theancienttimes,usedto saythat “we aredwarfs,but becausewearestandingongiants’ shoulders,wecanseefarther”.I wantto acknowledge thecooperation,to differentextents,of manypeople, start-ing fromPaoloMantegazza, mytutor, whogavemethemostinvaluablehelp: herarely interfered with my work. Theadviceof Gian Luca Ghiringhelli wasalsoinvaluable, expeciallyfor all thestructural andpiezoelectricanalysispart. Mas-similiano Lanz andMarco Borri were sourcesof fine suggestionsandof impor-tantbibliographicreferences.MarcoBorri representeda veryspecialcounterpartbecausehis views on the subjecthavea remarkabletheoretical and algorithmicorientation,sohis critics to the foundationsof this work were very importantinhighlightingflawsand inconsistencies.MassimilianoLanzandGabrieleGilardihada substantialrole in the formulationof the integration scheme, basedon anidea of Paolo Mantegazza, while StefanoMarazzanicooperatedin the develop-mentof thefinite volumebeamformulation.MarcoMorandinideservesa specialplacedueto his invaluablesuggestionsbothfromthetheoretical andthecompu-tational standpoint.Finally, Mark W. Nixon, fromNASALangley, not only madeall thecorrelationswith theV-22wind-tunnelmodelpossible, but healsohonoredmewith his friendship,that lastsafter theendof myexperienceat LaRC.

Intr oduction

The dissertationis basicallydivided in threeparts,reflectingto someextent thechronological,not only thelogical sequencein which theresearchwork hasbeencarriedon.

Theapproachto themodellingof thedynamicsof a rotorcraftthathasbeenusedin this work is presentedfirst. Theframework of thefirst principlesequationsofmotionof a systemof constrainedbodiesis described;detailson theformulationfor thehandlingof finite rotationsaregiven. Algorithmic implicationsof these-lectedapproachandimplementationnotesarediscussed;theoriginal approachtothe modellingof flexible bodies,significantly the characterisationof beamsec-tionsandanoriginaldiscretebeammodel,is presented.A platemodel,developedin analogywith thebeammodel,is alsoformulatedanddiscussed,with a caveat:it hasnotbeenimplementednor usedin thework.

Thesecondpart refersto theactive controlof the rotorcraftsystem.TheGener-alisedPredictiveControlformulationthathasbeenadoptedis presentedfirst, in adiscretetime form thathasbeenpreferredsinceit is respectfulof theway digitalreal-timecontrol is practicallyperformed.Thesuggestionof thepredictive con-trol camefrom a shortcourseI attendedto, heldby ProfessorsBolzern, BittantiandColaneriat theDipartimentodi Elettronicaof thePolitecnicodi Milano. Atthat time, I wascollectinginformationon anything that couldbe helpful for mywork; I found the predictive control really suitablefor the problemat hand,soI madesometestsandput it aside,becauseI wasdeeplyinvolvedwith the me-chanicalmodellingat that time. The following year, at NASA Langley, I foundout that therewerepeopleworking exactly on predictive control for the tiltrotorwind tunnelmodel,following thesuggestionsof Jer-NanJuang. Thanksto veryenlighteningtalkswith ProfessorJuang, to whommy acknowledgegoes,I got alittle moreconfidentin suchtopic,andI decidedto catchup with it.

The last part is dedicatedto the applicationof the structuralmodelling and ofthe control to the simulationof rotorcrafts.The mostimportantexampleis rep-

1

2 INTRODUCTION

resentedby the analysisandthe active control of the wind-tunnelmodelof theV-22 tiltrotor thatis currentlyunderinvestigationat Langley’sTransonicDynam-ics Tunnel (TDT). This work becamepossiblethanksto the cooperationof theArmy ResearchLaboratoryattheLangley ResearchCenter, andremarkablyto theinvauablehelp of Dr. Mark W. Nixon. Otherapplicationsinvolve the modellingof a “smart” rotor, with inducedtwist actuationof the rotor blades.A modelofthis kind is underinvestigationat theActiveMaterial andStructuresLaboratoryof MIT by ProfessorHagood, anda cooperative effort is underway with NASALangley, underthecoordinationof Dr. Wilkie, but nowind tunneltestinghasbeenperformeduntil recently. In this caseonly open-loopcontrol hasbeeninvesti-gated,the major taskstill beingthe constructionof the smartbladesandof thetestingapparatus.

Thenomenclatureusedthroughouttheworksometimesmightappearnon-homogeneous,atafirst glance.This is becauseanattempthasbeenmadeto preservetheoriginalnomenclatureusedin formulatingeachpart. Wherenon“official” nomenclaturewasestablished,or whererequiredby the needto link differentformulationsina tight manner, an effort to using a uniform nomenclaturehasbeenmade. Toavoid any ambiguity, everychapterdefinesits own setof symbolswith abrief de-scription. Thecontext andbrief noteswherethesymbolsareusedshouldclarifywhethera symbolis “local”, andthusdefinedin thenomenclatureof thechapter,or it is “global”, andthusshouldbesoughtfor in apreceedingchapter’snomencla-ture.Thebibliographyis unique;brief bibliographicnotesarereportedthroughoutthetext whererequiredby thesubject.

Part I

Rotorcraft modelling

3

Chapter 1

Dynamicsof a fr eerigid body

The equationsof motion of a rigid body result from Newton’s first principles;they basicallystatethat the rateof changeof the linear andangularmomentaisproportionalto theappliedforcesandcouples,respectively.

x positionof a referencepointv velocityof a referencepointR rotationmatrixω angularvelocityβ linearmomentumγP angularmomentumreferredto polePm massassociatedto abodyS first orderinertiamomentJ secondorderinertiamoment

1.1 Momenta definition

The linear and angularmomenta,β and γ, of a body of massm and inertia J,attachedto a moving point x at somerelative distanceresultingin a first-orderinertiamomentSwith respectto x, are:

β mx ω Sγ S x Jω (1.1)

wherex andω axRRT respectively arethevelocityandtheangularvelocity1

of x. The determinationof the inertia propertiesof a rigid body is detailedinAppedixA.

1 Theoperatorax extractsthe“axial” of theoperand,a matrix; v ax m is thevectorthatcontainstheskew-symmetricpartof matrix m. Its partial inverseis representedby operator ,yieldingax m v skw m . Thenatureof theangularvelocity is discussedin Chapter3.

5

6 CHAPTER1. DYNAMICS OFA FREERIGID BODY

1.2 Equilibrium equations

The total angularmomentum,referredto a fixed pole O, is γO γ x β, andits time derivative resultsin γO γ x β x β. By subtractingthe momentdue to the derivative of the linear momentum,x β, the time derivative of theangularmomentumis referredagainto the moving pole, yielding γO x β γ x β. The usualmoving pole angularequilibrium resultsfrom equatingthemoving pole angularmomentumrate to the appliedcouples. Togetherwith theforce equilibrium, obtainedby equatingthe derivative of the momentumto theappliedforces,it representsthe inertial contribution to theequilibriumof a rigidbody:

β F γ x β M;

(1.2)

F, M areexternal forcesandcouplesthat may dependon the configuration,asdiscussedin Chapter5.

1.3 Rigid-body dynamicsasODEs

The dynamicsproblemis statedin termsof a systemof first-orderdifferentialequations. It is important to notice that the left-handside of the equilibriumequations,Eq. 1.2 is linear in the derivativesof the momentaregardlessof theparametrisationthat is usedfor the kinematicunknowns, i.e. the displacementsandsignificantlytherotations2. Thedynamicsof themodelaredescribedby thecollectionof Equations1.1and1.2,written for eachbodythat is usedin thedis-cretisation.Theequationsarewritten in thegeneral,implicit form

f y y t 0 (1.3)

being y the array of the unknowns and t the time. Equation1.3 representsanimplicit OrdinaryDifferentialEquation(ODE); togetherwith a setof initial con-ditionsit yieldsanInitial Value(IV) problem,whoseintegration,to someextent,is a well-establishedtopic andcanbe carriedout by meansof consolidatednu-merical techniques.The addition of kinematicconstraintequationsmakes theproblema little moredifficult, aswill beshown in thenext section.

2Therotations,hererepresentedby therotationmatrixR, aretherealunknownsof theproblem.After aparametrisationis chosen,therotationparametersbecomethepracticalunknowns.

Chapter 2

Algebraic constraints

Thesystemof differentialequationsgivenby Equation1.3describesthedynam-ics of thebodiesthatareusedto discretisetheproblem.TheexternalforcesandcouplesF andM, shown in Eq. 1.2, maycontainconfigurationdependentinter-actionforcesthat areusedto representthe elasticityandgenerallythe dynamicinteractionbetweenbodies.But thereis animportantclassof interactionsthatarepurelykinematicandthatrequirespecialtreatment.

2.1 Reducedvs. redundantcoordinateset

In fact,whena multibodymodelis considered,greatadvantagesboth in the im-plementationandin the numericaltreatmentof the problemcanbe achieved byusingaRedundantCoordinateSet formulation.It consistsin writing thecompletedynamicsof eachbodyandby explicitly constrainingtheir kinematicdegreesoffreedomto obtain the desiredkinematicbehaviour of the system. Considerforinstancea revolute joint, namelytheassemblyof two bodieswhoserelative mo-tion is representedby an axial rotation. The systemhas6 rigid body degreesoffreedomplus one relative rotation degreeof freedom. If suchreducedsystemis directly considered,only 7 differentialequationsneedbe written to describethe dynamicsof the system,in the spirit of the ReducedCoordinate Set ap-proach. This approach,with different formulations,hasbeenhistorically usedfor constraineddynamicsanalysissinceit allows largeproblemsto bedescribedby meansof comparatively smallsystemsof purelydifferentialequations.More-over, efficient explicit integrationschemescanbeused,with stepsizecontrol toensurethe numericalstability of the integration. On the otherhand,the writingof the motion in termsof relative degreesof freedommay representa problem,andgenerallyleadsto complicated,cumbersomeformulationsandimplementa-tions,andrequiressophisticatedautomaticreductiontechniques.TheRedundant

7

8 CHAPTER2. ALGEBRAIC CONSTRAINTS

CoordinateSet approach,onthecontrary, allowsto write thefree-bodydynamicsequationsfor eachbody in theglobal referenceframe,with theadditionof alge-braic constraintequations.They contribute to the equilibrium of the bodiesbymeansof algebraicunknownsthatrepresentthereactionforcesandcouples.Thesystemmaybecomevery large if comparedto thereducedsetcase;considerforinstancea rigid modelof an articulatedhelicopterrotor: a singlebladehastwoindependentdegreesof freedom,e.g. the flap andthe lag angles,the featheringbeing constrainedby the pitch link. A reducedset approachwould result in asystemof first-orderdifferentialequationsof order4. A redundantsetapproachmight resultin dozensof degreesof freedom:onebody is requiredto modelthehub,oneis requiredafter theflap hinge,oneafter the lag hinge,andfinally one,representingtherigid blade,afterthefeatheringhinge.Thesefour bodiesrequire24 degreesof freedom.Six degreesof freedomarerequiredby therevolutejointthatgroundsthehubandenforcestherotation,5 by eachof therevolutejointsthatmodelthethreehinges,andoneby thedistancejoint thatmodelsthepitchlink, re-sultingin 22reactionunknowns.Becausethedynamicsof thebodies,Eqs.1.1and1.2,arewrittenasasystemof first-orderequations,thebodiesactuallyrequire12degreesof freedomeach,sotheproblemat handwould have order70, comparedto order4 of thereducedsetcase.Of coursetheredundantsetapproachallows avery simple,generalformulation,resultingin easeandflexibility of implementa-tion, andin verysparsesystemmatrices;by efficiently handlingthesparsitywithspecialisedlinearalgebrasolvers,verygoodperformanceshavebeenobtained.

2.2 Constrained rigid-body dynamicsasDAEs

A systemof Differential-Algebraic Equations(DAE) resultsby addingthe al-gebraicconstraintsto Eqs1.1 and1.2. Dif ferential-algrebraicequationsrequirespecialtreatmentto besolved;theproblemhascapturedtheattentionof theinves-tigatorsduring the lastdecades[17]. Froma heuristicpoint of view, differentialalgebraicequationsmaybeconsideredverystiff differentialequations,with van-ishinginertiaterms;they will bediscussedwith moredetail in Chapter4. Thedi-rect,simultaneousintegrationof suchequationsrequiresimplicit, unconditionallystableintegrationformulas,astheonethatis proposedin thiswork in Section4.2.Equation1.3 is modifiedby theadditionof thedependenceon thealgebraicun-knownsv, while thealgebraicconstraintequationsg areadded,yielding

f y y v t 0 g y y t 0 (2.1)

2.2. CONSTRAINEDRIGID-BODY DYNAMICS AS DAES 9

Whenaconstrainedmechanicsproblemis considered,Eq.2.1assumestheform1 A y t y z 0 z GTv Q y y t 0

Φ y 0 (2.2)

which is theEulerequationof aproblemknown asLagrangianof thefirst kind; itrepresentsa Differential-AlgebraicEquationof index three2. In Eq. 2.2,A is theinertiamatrix, Q arearbitraryexternalforcesandcouples,andG ∂Φ ∂y is thederivativeof theholonomicconstraints3 with respectto thekinematicunknowns.Theconstraintmatricesresultingfrom the linearisationof Eq. 2.2aresymmetricwhentheproblemis formulatedby applyinga variationalprinciple; in this workthesymmetryis lost,at leastfor theangularequilibriumequations,thatareconju-gatedto theperturbationof therotationangles,ratherthanto theperturbationoftherotationparameters;this is discussedin Section3.2. Thesameconsiderationmayhold for thegeneralequationsthatwill beintroducedin Chapter8.

1Themomentumdifferentialunknownsin Eq.2.2arerepresentedby z, thesymboly referringto thekinematicunknownsonly.

2Quotingfrom [17], Definition 2.2.2,p. 17: “The minimum numberof timesthat all or partof the DAE F y! y ! t " 0 must be differentiatedwith respectto t in order to determiney as acontinuousfunctionof y, t is theindex of theDAE”.

3Non-holonomicconstraintsrequirespecialtreatment,that is not illustratedsinceit is not rel-evantto thediscussionpresentedhere.

10 CHAPTER2. ALGEBRAIC CONSTRAINTS

Chapter 3

Kinematics of finite rotations

Three-dimensional,finite rotationsarenot additive; in fact the configurationre-sulting from two successive finite rotationsis givenby the multiplication of thetwo relative rotationmatrices,namelyR0# 2 R1# 2R0# 1 andin general,regard-lessof theparametrisation,thetotal rotationcannotbeexpressedastheadditionof thetwo partialrotations.Thehandlingof finite rotationsis key to theeffective-nessof a multibody formulation. In fact, the fundamentalideaof themultibodyapproachto thestructuraldynamicsis to expressthekinematicsin aglobalframe,in orderto simplify thewriting of thecontributionof theinertiato theequilibrium,at thecostof amorecomplicateddescriptionof interactionalforces,eitherrelatedto flexibility or to kinematicconstraints,whichhavelittle need,if any, for thetimederivativesof thekinematicunknowns.

3.1 Finite rotation properties

A rotationbetweentwo referenceframesis describedby meansof anorthonormalrotationmatrix R. Theorthonormalitypropertydescendsfrom theconsiderationthatthescalar, or inner, or dot productbetweentwo vectorsis independentof thereferenceframethevectorsarereferredto,namelya $ b a $ b. Thus,beinga Raandb Rb, asa result

R $ R RTR RRT I (3.1)

holds,implying thatRT R% 1. This importantpropertyhasdeepimplicationsonthemathematicalnatureof therotations.Therelationshipbetweentheconfigura-tion matrixRandits perturbationθd canbedeterminedby differentiatingavectorv Rv that is constantin a local referenceframe, namelydv 0, resultingindv dRv; by consideringEq. 3.1, thedifferentiationof v resultsin dv dRRTv.

11

12 CHAPTER3. KINEMATICSOFFINITE ROTATIONS

Matrix dRRT is skew-symmetric,andthuscanbewritten asa crossproductma-trix, namely

θd & dRRT ; (3.2)

operator $'( , appliedto a vectora, givesthematrix that,multiplied by anothervectorb, yieldsthevectorproducta b, asdescribedin footnote1, page5. Whenin Eq.3.2adifferentiationwith respectto time is considered,theangularvelocityis obtained,namely

ω & RRT (3.3)

By rearrangingEq.3.3asfollows,

R ω R (3.4)

theexponentialnatureof finite rotationsis emphasized.Thesolutionof Eq.3.4 is

R e) ωdt * (3.5)

Theexponentis representedby a skew-symmetricmatrix, whoseeigenvaluesarezeroand + j , - ωdt , , somatrix R hasconstant,unit spectralradius,theimaginaryeigenvaluesgiving a pure rotation. Becausematrix R dependson the rotationparametersp only, the differentiationof a vectorv that is constantin the localframecanbeexpressedasdv dRRTv W p dp v, resultingin

θd p dp. ax W p dp/ G p dp (3.6)

Matrix G is requiredto differentiatetherotations,in orderto yield rotationveloc-ities,curvatures,andto linearisetheequations,soits efficientcomputationcanbevery important,aswill beshown in Section3.4

3.2 Rotation parametrisation

Many differentparametrisationshave beenproposedin the literature;a compre-hensivereview is nottheaimof thiswork. Thefundamentaldistinctionis betweeninvertibleandsingularparametrisations,the latterusuallybeingthree-parameter,while the former requirefour parameters,like the well-known quaternion for-mulation,andthe Euler parameters. The four-parameterformulasarerequiredwhena total rotationapproachis used,andthesingularitiesmustbe intrinsicallyavoided. But therearemany reasonsrelatedto theefficiency of the implementa-tion to preferanupdatedrotationapproach.Whenanupdatedapproachis used,thethree-parameterformulationsaremoreappealingbecausethey requirereducedcomputationaleffort. The risk of incurring in singularitiesis null whenconsid-ering that comparatively small rotationsbetweentwo successive time stepsarerequiredanyway to obtainreasonableaccuracy.

3.2. ROTATION PARAMETRISATION 13

3.2.1 Rotation vector

Among the three-parameterformulations,the rotationvector parametrisationisvery important,becauseit directly representsthegeometricrotation.It consistsina vectorΦ ϕn whosedirectionn is therotationaxisandwhosemagnitudeϕ isthetotal rotationangle.Theparametrisedrotationmatrix is

R I sin ϕ n 0 1 cos ϕ n n .ThecorrespondingdifferentiationmatrixG, givenby Eq.3.6,is

G I 1 cos ϕ ϕ

n 132 1 sin ϕ ϕ 4 n n .

Thedeterminantof thematrix is

det G5 21 cos ϕ

ϕ2 ;

it is singular1, of course,thesingularitylying in theuncertaintyon thedirectionof the rotationwhena 2π angleis considered.Onemajor drawback,from a nu-mericalstandpoint,is thatcomputationallyinefficient trigonometricfunctionsarerequiredto computethe two matrices. It is interestingto notice that in caseofplaneproblems,therotationdegeneratesinto a scalarrotationabouttheaxisnor-mal to theplane,andthusit becomesadditive,while ϕ coincideswith theangularvelocityω, thusmakingsuchparametrisationveryattractive.

3.2.2 Gibbs-Rodriguezparameters

TheGibbs-Rodriguezparametersarerelatedto therotationvectorby

g 2 tan 6 ϕ2 7 n

Theform with theparametersmultiplied by 2 is preferredto themoretraditionalonebecauseit yields linearisedexpressionsidentical to thoseobtainedwith therotationvector. It is apparentthattheparametersaresingularwhentheargumentof thetangentis + π 2 dueto thetangentfunction.TherotationmatrixR is:

R I 44 gTg

2 g 8 12

g g 4 andmatrixG is:

G 44 gTg

2 I 12

g 4 Notice that now no trigonometricfunction evaluation is required,resulting inhighercomputationalefficiency.

1The singularity occursonly at ϕ 2π 9 2kπ, with k positive integer, while for ϕ 0 thedeterminanthasunit value.


3.3 Updatedapproach

The updatedapproachthat is requiredto allow the useof three-parameterfor-mulasrelieson theconsiderationthata changein configurationfrom R0 to R1 isrepresentedby

R1 R0# 1R0 (3.7)

Whensolving an initial valueproblem,R0 representsthe solutionat time t0 andremainsconstantfor therestof the integration,theunknown configurationbeingR1. So,by parametrisingonly the relativerotationmatrix, R0# 1, onecanmaketherotationparameterssmall,at leastconfinedto thedesiredrange,by changingthetime step.In fact, therotationparametersthatdescribea relative rotationareof the orderof magnitudeof ,ω∆t , ; by theway, this roughestimatecanbe usedto control the time stepto avoid the singularitiesrelatedto the three-parameterformulation. When the solutionat a given time stepis obtained,the referencematrix R0 is updatedby meansof the rule expressedby Eq. 3.7 andthe rotationparametersarereset.Theunknown rotationmatrixat time ti becomes

Ri R∆Ri % 1 andtheangularvelocitybecomes

ωi & R∆RT∆

becausethereferencerotationmatrixRi % 1 is constant,andthus

ωi G∆g The linearisationof the rotation-relatedquantitieswith respectto the unknownrotationparametersyields

∆Ri G∆∆g: Ri ∆ωi G∆∆g H∆ g g ∆g

with

H∆ ; 24 gTg

g 8 G∆ggT

3.4. UPDATED-UPDATED APPROACH 15

3.4 Updated-updatedapproach

Theupdatedapproachfixestheproblemof thepossiblesingularityof therotationparameters,but doesnot improvethehandlingof therotations.In facttherotationmatrix,R, andtherotationderivativematrix,G, dependin a nonlinearmannerontherotationparameters,thusmakingthelinearisationof therotation-relatedquan-tities very complicate,time consuminganderror pronefrom the implementor’sstandpoint.As will beshown in Section4.2, theintegrationof thesolutionis ob-tainedby collocatingthenonlinearsystemof equationsat theendof thetimestepandsolvingiteratively. Thetentativesolutionatsuchtime is predicted,andthefi-nal solutionis determinedby correctingthepredictedvaluein aNewton-Raphsonway, or correspondingmodifiedalgorithm. Thekey ideaof theupdated-updatedapproachconsistsin referringtherotationunknownsto thepredicted configura-tion at theendof thecurrenttime stepratherthanto thecorrected configurationat thebeginningof thetime step,resultingfrom thesolutionat theprevioustimestep.In this manner, therotationunknownsrepresenttherotationparametersre-latedto thecorrectionphaseonly, resultingin really “small” andthusquasi-linearvariables.In fact, sincethe accuracy of the predictionformulasis quadraticfora linear problem,asshown in Section4.2, the unknown rotationparametersareexpectedto have orderof magnitudeo

∆t2 , regardlessof the amplitudeof the

angularvelocity. Theunknown rotationmatrixRi now becomes

Ri RδR< 0=i whereR< 0=i is thepredictedconfigurationat time ti, andthelower caseδ referstoaperturbationof rotation.Theangularvelocitybecomes

ωi > RδRTδ RδR< 0=i R< 0=i

TRT

δ or

ωi ωδ ω ? 0@i wheretheperturbationof angularvelocityωδ Gδg is addedto thereference,pre-

dictedangularvelocity, ω ? 0@i Rδω < 0=i , rotatedinto thecorrectedreferenceframeby matrixRδ. Thelinearisationof matrix Randof theangularvelocity ω yields

∆Ri Gδ∆gA Ri (3.8)

∆ωi ∆ GδgB Gδ∆gC ω ? 0@i (3.9)

A first importantresult is that when the rotationparametersare reset,after theprediction,matricesR and G both degenerateinto the identity matrix, so that


Eqs.3.8,3.9become

∆Ri ∆g R< 0=i (3.10)

∆ωi ∆g ∆g ω < 0=i (3.11)

Successive iterationsrequireto considerEqs.3.8,3.9 insteadof Eqs.3.10,3.11;however, sincethecorrectionparametersg, g areexpectedto besmall,Eqs.3.10,3.11representa valid approximationwhencomputingtheJacobianmatrix. Suchapproximationcanbeviewedasasortof intrinsicallymodifiedNewton-Raphsoncorrectionalgorithm,whoseaccuracy remainsquasi-quadraticwith appreciablesavings in computationaltime. The sameidea,of course,canbe appliedto anyrotationparametrisation.

Chapter 4

Algorithmic implications

The solution of an initial value problemdescribedby a systemof differential-algebraicequationsrequiresspecialattention,dueto the singularityof the alge-braicequationsif treatedin thesamemannerof thedifferentialones.Consideradifferentialequationof theform of Eq.1.3,hererewritten for clarity:

f y y t . 0 If the equationis differentiablein a neighborhoodof the solution,andprovidedthederivativewith respectto y is non-singular, it canbewritten in explicit form:

∆y ∂ f∂y

% 1 2 f ∂ f∂y

∆y4 (4.1)

Thesystembecomesalgebraic-differentialwhenthis operationis no longerpos-sible, becausef D y is structurallysingular1, as in the casewe are interestedin.Of courseit would bepossible,afterseparatingthesystemin its differentialandalgebraicparts,

f y y zE 0 g y 0 (4.2)

wherenow f refersonly to the differentialpart of the problem,to rewrite it inordinarydifferentialform, by first differentiatingbothequations,

f D y∆y f D y∆y f D z∆z f 0 gD y∆y g 0;

(4.3)

1Thereareproblemsin which f is not invertibleonly in a limited numberof singularpoints,that representbifurcationsor locking pointsof mechanisms.This is not thecaseof theproblemwe areconsidering;we aredealingwith a problemthathasa structuralsingularityof somerankr that remainsconstantirrespective of time, the possibleextra singularpointsstill representingbifurcationconditionsfor thedifferential-algebraicproblem.

17

18 CHAPTER4. ALGORITHMIC IMPLICATIONS

thesecond,algebraicequationis thendifferentiatedwith respectto time,

gD y∆y gD yt gD yyy ∆y gD t gD yy 0 (4.4)

while y is explicitatedfrom thefirst of Eqs.4.3andprojectedorthogonallyto theconstraintby substitutingit into Eq. 4.4. This allows to staticallydeterminethealgebraicunknown ∆z asfunctionof ∆y,

gD y f % 1D y f D z∆z gD t gD yy gD y f % 1D y f F6 gD yt gD yyy gD y f % 1D y f D y 7 ∆ywhich,substitutedin thefirst of Eqs.4.3, transformsit in anordinarydifferentialequationthatimplicitly satisfiesthealgebraicconstraintof g.The differential-algebraicproblemcanbe written in a simpler, more redundantway by consideringa new unknow w, whosetime derivative is thealgebraicun-known z, namelyw z, andby maintainingtheredundantform of two blocksofequations,thatbecomeordinarydifferential,namelyG

f D y f D zgD y 0 H ∆y

∆w I G f D ygD yt gD yyy H ∆y f

gD t gD yy IThe differentialandthe algebraicunknownsaredeterminedsimultaneously, thesystembeingnon-singulardueto thetime differentiationof equationg. It is im-portanttonoticethatthesystemthatisactuallyintegratedisnolonger f 0 g 0,but rather f 0 g 0. This hasimportantimplicationsin theproblemsat handbecause,dueto numericalerrorsin theintegration,thesolutionmaydrift, andtheoriginal algebraiccostraintg 0 maybeviolated;noticethatthepossibleoccur-renceof suchviolation is relatedto numericalreasonsonly. Dif ferenttechniqueshave beenproposedin the literatureto solve this problem2; the point is still de-bated.Thedifferential-algebraicformulationof theproblemcancuretheproblem,becausetheconstraintequationis directly satisfied;this approachis followed inthepresentwork.

4.1 Unconditional stability

Theabovepresentedproceduressomehow collide with thekey ideaof theredun-dantcoordinatesetapproach,becausethey addunnecessarycomplexity, namely

2Theinterestedreadershouldconsult[17], pp.150–157for abrief presentationof thestabiliza-tion schemes,thereferencesreportedtherefor furtherdetails,or [20] for apresentationfocusedonstructuraldynamicsproblems;any goodtextbookon thesolutionof constraineddynamicsprob-lemsshouldsuffice.

4.1. UNCONDITIONAL STABILITY 19

theneedto differentiateequationg with respectto time3, to aproblemthatcanbeeasilyandefficiently solvedin a directmanner, with a dedicatedbut still generalsolutionscheme.Thefirst requirementfor suchschemeis to beunconditionallystable,to beableto integratethe infinitely stiff algebraicequations.Quotingfrom [17], p. 77, anintegrationalgorithm

“is A-stableif limn# ∞ yn 0 for all Re λ KJ 0 anda fixedpositiveh[the timestep]whenappliedto theproblemy λy.”

To solve a differential-algebraicproblem,an A-stablealgorithmis required,butthis is not enough. In fact, sincean unconditionallystablealgorithmcan inte-gratesystemswhosecharacteristicperiodis quiteshorterthanthetimesteph, andthis is surelythecaseof theproblemat hand,thedynamicsrelatedto suchhighfrequency motionsarelikely to becapturedwith pooraccuracy, if any. As a con-sequence,spuriousoscillationsandmisleadingaliasedhigh frequency behaviourmayappearin theresults.Thesecondrequirementfor the integrationschemeis to beableto filter thehighfrequency, algorithmic oscillationswhile preservinggood accuracy in integrat-ing the low frequencieswe aremainly interestedin. Again from [17], p. 77, anintegrationalgorithm

“is L-stableif it is A-stableand,in addition,limRe< λ = h# % ∞ , ynL 1 yn ,M 0”.

Thismeansthat,beingthetimesteph alwayspositive,if thesystemis stable,andthusRe λ (J 0,whenthetimestepgetslargerandlarger, andthusRe λ h NE ∞,the freeoscillationsof thesystem,i.e. the ratio , ynL 1 yn , , mustvanish.Suchre-quirement,expeciallyin low orderintegrationschemes,maybetoostrong,result-ing in a considerablelossin accuracy for comparatively smallvaluesof Im λ h,so it will be relaxed by imposingthat limRe< λ = h# % ∞ , ynL 1 yn ,O ρ∞, whereρ∞,suchthat 0 P ρ∞ P 1, is calledthe asymptoticspectralradiusof the integrationscheme.

3Strictly speaking,the time differentiationof the constraintequationsis requiredto bring toindex two an arbitrarily high index differential-algebraicsystemof equations.The satisfactionof the derivativesof the constraintsshouldbe implicit when the constraintequationis directlyconsidered,but practically it is obtainedonly within the accuracy of the solutionmethod. As aconsequence,the presentedapproachmay suffer from stability problemsin critical conditions,whenlong time stepsareusedto integratestiff problems.However theseconsiderationsdo notappreciablyaffect resultsin practicalcases,whenthestepsizeis drivenby accuracy.


4.2 Integration formula

A family of integrationschemes,basedon a polynomialinterpolationof theun-knowns and on the numericalsolution of the fundamentaltheoremof integralcalculus,is presented.The algorithmis introducedanddiscussedfor the solu-tion of ordinarydifferentialequations;it is subsequentlygeneralisedto implicitdifferentialequationsandto algebraic-differentialequationsaswell.

4.2.1 Inter polation

Consideramultistepinterpolationbasedon theunknownsandtheir derivativesatr 1 timesteps,

y ξ / r

∑i Q 0 mi ξ yn % i hni ξ yn % i R (4.5)

whereξ t tn h is thenon-dimensionaltime,m, n areshapefunctionsandris thenumberof stepintervalsof thescheme.Theunknownsareyn, yn, at theendof thetimestep.

4.2.2 Solution

Thesolutionisobtainedfromthefundamentaltheoremof integralcalculus,namely

yn yn % r TS tn

tn U r y dt which,afterdiscretisation,becomes

yn V yn % r h∑i

wi y ξi R (4.6)

Internalpointsξi andweightswi dependon thenumericalintegrationscheme.Acomprehensivedescriptionof somepossibleschemesis reportedin AppendixB.

4.2.3 Linearisation

Consideranexplicit ODEfirst, in theform

y f y t ;thenumericalsolutionbecomes

yn yn % r h∑i

wi f y ξi R t ξi WB

4.2. INTEGRATION FORMULA 21

thelinearisationof whichyields

∆yn h∑i

wi f D y XX ξi∆y ξi . yn yn % r R h∑

iwi f , ξi

Theperturbationof y at ξi canbeexpressedasfunctionof theunknown, ∆yn, bymeansof theinterpolationformula,Eq.4.5,yielding

∆y ξi . 6 m0 ξi I hn0 ξi f D y XX ξ Q 0 7 ∆yn (4.7)

SothesolutionisYI h∑

iwi f D y XX ξi

6 m0 ξi I hn0 ξi f D y XX ξ Q 0 7[Z ∆yn yn yn % r R h∑i

wi f , ξi (4.8)

It is importantto noticethatEq.4.8requiresthecomputationof multipleJacobianmatrices,aswell assomeJacobianmultiplications,which arecostly operationsandleadto lossof matrix sparsity.

4.2.4 Stability

Thestabilitypropertiesof thealgorithm,at leastin a linearcase,thatcorrespondsto a local stability, can be investigatedby consideringa problem in the formy λy, whereλ is anarbitrarycomplex parameter. By consideringthatin ahomo-geneousdifferenceequationthesolutionat time i canbeseenasy ti A ρy ti % 1 ,andby substitutingtheassumedvalueof y in Equation4.6,oneobtains

ρr 1 h∑i

wiλr

∑j Q 0 mj ξi ρr % j hnj ξi λρr % j :

which,by exchangingtheorderof summationsover i and j, yields

ρr 1 λhr

∑j Q 0ρr % j

Y∑i

wimj ξi \ λh∑i

win j ξi Z (4.9)

namelya polynomial in ρ of order r, which may be linear or quadraticin λh,dependingon the valueof theshapefunctionsn j . The formula is A-stableif allther rootssatisfytheinequality ,ρ ,]P 1, with Re λ ^J 0, for any λh. Theformulais L-stableif in additionthenumeratorof all the rootsis oneorderlessthanthedenominatorin λh. If they areof thesameorder, thentheratioof thehighestordertermsof the numeratorandthe denominatorrepresentsthe asymptoticvalueofthespectralradius;its normis comprisedbetween1 and0, andtheformulahasa


behaviour betweenA- andL-stable,becauseit addssomealgorithmicdissipation.Considerfor instancethesingle-stepcase:theroot is

ρ 1 λh ∑i wim1 ξi R λh∑i win1 ξi W1 λh ∑i wim0 ξi R λh∑i win0 ξi W

Thustheformula is L-stableprovided∑i win0 ξi K_ 0 and∑i win1 ξi C 0, or, incase∑i win0 ξi ` 0, for ∑i wim0 ξi A_ 0 andboth∑i win1 ξi ( 0, ∑i wim1 ξi a0. Similarconditionscanbeanalyticallyformulatedin caseof a two-stepmethod;higherstepnumbersareof little practicaluse.Figure4.1showsthespectralradiusof someintegratorsof thefamily.

4.2.5 Implicit ODEs

Thecaseof animplicit ODE, in theform

F y y t . 0 (4.10)

is straightforward,sincethelinearisationof theproblemyields

FD y∆y FD y∆y F 0 (4.11)

andtheperturbationof y canbeexplicitatedasshown in Eq.4.1; thenthesolutionprocedureis analogousto thatof the explicit ODEs,only the inversionof FD y isrequired.

4.2.6 Algebraic-differential equations

WhenFD y in Eq. 4.11 is not invertible, the systemis differential-algebraic,anda differentapproachmustbe used.Considera problemof the form of Eq. 4.10,whoselinearisationis presentedin Eq. 4.11. Considernow the approximatedform of thefundamentaltheoremof integralcalculus,Eq.4.6,whoseperturbationyields

∆yn h∑j

w j∆yξ j (4.12)

Theperturbationof y is requiredateachcollocationpointξi , sotheproblemmustbesolvedin all thosepoints.Theperturbationof y at time ξ canbeobtainedfromthe linearisationof the interpolationformula, Eq. 4.7, usingEq. 4.12 to expressthe perturbationof y at the endof the time stepin termsof the perturbationsofy. Sothesystemnow consistsof asmany equationsasthecollocationpointsare,plus the problemcollocatedat the endof the time step,if this is not oneof the


collocationpointsyet. Theunknownsarethe∆y at thepointsin which thesystemis solved,resultingin

FD y XX ξi

Ym0 ξi h∑

jw j∆yξ j : hn0 ξi ∆y 0 Z FD y XX ξi

∆y ξi E F , ξi(4.13)

for eachcollocationpoint.

4.2.7 Noteon the classificationof the method

Theformulationthathasbeenpresentedfor thesolutionof differential-algebraicequationsclearlyshows that theproposedmethodcloselyresemblesthe generalImplicit Runge-Kutta (IRK) methodfor the integrationof differentialequations.Thebasicformulasof IRK schemesarehererecalled;thesolutionof F y y t isobtainedby solvingthesystem

FYi Yi tn % 1 cih 0

at someintermediatepointsi, while thefinal solutionis

yn yn % 1 h∑j

b jYj ;

estimatesof theintermediatevaluesof y tn % 1 cih maybeobtainedas

Yi yn % 1 h∑j

ai jYj Theschemecanbesummarisedin thesocalledButcherdiagram

c bbT c1 a11 $$W$ a1M

......

. . ....

cM aM1 $$W$ aMM

b1 $$W$ bM

By consideringa single-stepscheme,the ci become1 ξi , the b j becometheweightsw j ; thereis no immediatecorrespondencefor theai j , becausetheinternalvaluesof y are interpolatedfrom the boundaryvaluesof y andof its derivativeratherthanfrom the internalvaluesof y. Only for specificcasesa direct equiv-alencemaybe found: being j 1 thebeginningof the time stepand j M theendof the time step,the generalexpressionis ai1 m0 ξi w1 n1 ξi , aiM m0 ξi wM n0 ξi , theothercasesbeingai j m0 ξi w j . Theproperty∑mi 1hasbeenexploitedto collectyn % 1.


Thesingle-step,cubicinterpolation,4th orderaccuratewith no algorithmicdissi-pationmethod,detailedin AppendixB, coincideswith theLobattoIIIA of order4, while theCrank-Nicholsonformula,asis well known, is theLobattoIIIA of or-der2; thesingle-step,parabolicinterpolation,3rd orderaccurate,L-stablemethodcoincideswith theRadauIIA methodof order3 (see[50], pp.72–77).Theadditionalrequirementsfor stifflyaccuratemethods,namelycM 1,aM j b j

for j 1 WW M and b beingnonsingular, in many casesaresatisfied:the firstby definition, sincethe proposedmethodnaturally requiresthe last evaluationpoint to be theendof thetime stepto allow theexplicitation of theproblem;thesecondresultsfrom thesatisfactionof thefirst, sincethesameformula,Eq. 4.6,is usedto evaluateboth the final andthe intermediatetentative valuesof y, andm0 0" 1 andn0 0c n1 0" 0hold. Thelastrequirementdependsonthenatureof theinterpolationfunctionm0 andon theorderandthenatureof thenumericalintegrationscheme.Theproposedschemesusuallysatisfythecriterionprovidedthey do not usethe beginning of the time-stepasan internalpoint4; this holdstrue for the formulasthat introducealgorithmicdissipation.No correspondencewith IRK exists in caser _ 1; in suchcasethis formula canbe interpretedasamultistepextensionof the IRK methods;only very high-ordermethodscan beformulated,which seemto be of little practicalusein mutibodyanalysisduetoexcessive computationaloverhead,but may representan interestingchoiceforspecific,accuracy demandingproblems.This formulation may help in putting the RK methodsin a new light, sinceitshowshow thosepopularintegrationformulasmaybecastin afinite element-likenumericalintegrationscheme.Moreover, the family of integration formulasisevenbroadersinceit operatesasortof unificationof RK andmultistepintegrationschemes.

4.2.8 Higher-order formulas

Higher-order formulashave beendeveloped,all of themrequiring multiple Ja-cobianmatrix computationsandJacobianmultiplications5. Whena single-step,third-orderHermitian polynomial interpolationis used,a fourth-orderaccurateformula with no algorithmicdissipation,or a third-orderaccurateformula withtunablealgorithmicdissipationareobtained.This approachhasbeenconsideredunfeasiblefor a multibody implementationbecauseof the overheaddue to themultiple Jacobianmanipulationand to the loss of matrix sparsitythat reducestheefficiency; only the two-step,second-orderformulahasbeenactuallyimple-mentedin a multibodycode.In AppendixB somehigher-orderformulasarede-

4In which casea1 j 0 by definition.5Or anincreaseof problemsize,resultingfromtheadditionof intermediatestatesasunknowns.


tailed, and their accuracy and stability propertiesare discussedalong with thesecond-orderformulasintroducedin the following section.Figures4.1, 4.2, 4.3show a comparisonof thespectralradii andof thedampingandphasepropertiesof someof theformulasthatresultfrom theproposedscheme.

4.2.9 Second-orderformula

Thesecond-orderformularesultsfrom a two-step,parabolicinterpolationfor themi , with ni 0. The trapezoidrule, with threepointscorrespondingto thestart-, the mid- and the end-pointof the consideredtime interval, is used. Sinceyis assumedto be parabolic,the integration of its derivative with the trapezoidrule strictly requirestwo points only, thus the three-pointintegration leavesanindeterminationon thevalueof theweights,yielding

yn yn % 2 h 2d2 12 δ 4 yn 1 2δ yn % 1 2 12 δ 4 yn % 2 4 (4.14)

A parabolicpredictionfor the yn, basedon thederivativesof thecubicHermitianpolynomialswith yn % i andyn % i , i 1 2, is used,namely

yn 12h

yn % 1 12h

yn % 2 8yn % 1 5yn % 2; (4.15)

the formula hasbeenimplementedalso with variabletime-step,not presentedherefor sake of conciseness.A second-orderaccurateCrank-Nicholsonformula,appliedto theinterval e tn % 2 tn % 1 f ,

yn % 1 yn % 2 h2 yn % 1 yn % 2 5 0

and multiplied by a weight 1 α, is addedto the right-handside of Eq. 4.14,yielding

yn 1 α yn % 1 αyn % 2 h 2g2 12 δ 4 yn 2 12 1 α " 2δ 4 yn % 1 F6 α2 δ 7 yn % 2 4 (4.16)

Theformulais intrinsically second-orderaccurate,asresultsfrom Figure4.3; theparametersα, δ areusedto tunethestability properties.They canbewritten intermsof thedesiredasymptoticspectralradiusas

α 4ρ2∞ 1 ρ∞ 2

4 1 ρ∞ 2 δ 1

2 1 ρ∞ 2

4 1 ρ∞ 2


whereEquation4.9hasbeensolvedfor coincident,real-valuedasymptoticroots.Noticethat,for ρ 1, resultingin α 1, δ 0, aCrank-Nicholsonlike formula,

yn yn % 2 h 2 12

yn yn % 1 12

yn % 2 4 is obtained,with a three-pointweightingof y in a formally single-stepformulawith 2h stepsize. In caseof ρ 0, with α h 1 3, δ 1 6, theL-stable,two-stepBackwardsDifferentiationFormula(BDF)6,

yn 43

yn % 1 13

yn % 2 23

hyn is obtained.It shouldbenoticedthattheBDF aresecond-orderaccurate,but inef-ficient in mostpracticalproblemsdueto excessivealgorithmicdissipationstartingatverysmallIm λh , virtually atzero.Thiswasthemainreasonthepresentedfor-mulahasbeeninvestigated,sinceit ensuresbetterperformanceswith little, if any,extracomputationaleffort with respectto BDF. Figure4.1showsacomparisonofthespectralradii that characterisesomeof the formulasof the family. Considernow theseriesexpansionof theresidualof theproposedformula,

E 2 12 1 α B 6δ 4 h3 O

h4 ;

the coefficient of the term in h3, when ρ∞ 1, is 1, while, for ρ∞ 0, it is 4 3. Thereis anextremumpoint, for anintermediateρ∞, which happensto be

atρ∞ i 4 Tj 21 k 0 58andyieldsanerrorE l6 3 mj 217 2 kn 0 79, whose

absolutevalueis minimum.Whenappliedto thedifferential-algebraicproblem,theformulasimply resultsinthesolutionof theproblemcollocatedat theendof thetimestepsince,beingn0 0, thereis no needfor Jacobianmultiplications,andtheintermediatevaluesof y,y areknown becausetheformulais collocatedexactlyat thetimestepboundaries.Theproblemis

FD y∆yn FD y∆yn F and∆y is expressedin termsof ∆y by perturbingEq.4.16,yielding

∆yn h 2 12 δ 4 ∆yn (4.17)

sotheproblembecomesalgebraicin ∆yn:2 FD y h 2 12 δ 4 FD y 4 ∆yn F (4.18)

6SeeRef. [17], pp.41–42.


0

0.2

0.4

0.6

0.8

1

1.2

0.01 0.1 1 10 100 1000 10000

rhoo

h/T

cubic, trapezoidcubic, trapezoid (rho=.6)

parabolic, trapezoid (z=-2/3)Crank-Nicholson

Implicit Eulermultistep (rho=.6)

two-step BDF

Figure4.1: Integrationformulas— spectralradii, ρ.

with Eq.4.17asupdaterule,while Eqs.4.15and4.16areusedto predicty andyrespectively. Notice that this is a specialcaseof Eq. 4.13,written at ξ 0 only,with m0 1, n0 0 andw 1 2 δ.

4.2.10 Further remarks on stability

Theproposedmethodis shown to betunablebetweenA- andL-stability, rangingfrom noneto total algorithmicdissipation.Theseconsiderationsinvolve the lin-earstability, that is local. The global stability of the integrationis a completelydifferentmatter, andit cannotbeensuredby theformulaalone.On thecontrary,theproblemandtheintegrationformulasmustbedesignedto intrinsicallyensureglobal stability in somesense.For purelymechanicalproblemsthe formulationcanberewrittento intrinsicallypreserve7 thetotalenergy andtheangularmomen-tumbalance.Thisapproachis surelymorerespectfulof thenatureof theproblem,but might lack in generalitywhenmultidisciplinary, multifield integratedprob-lemsareaddressed,asin this case.Surelysomepreservation law canbe drawnout of every discipline,e.g.chemicalreactions,electricnetworksandsoon, butsuchaneffort is consideredtoo expensive whenthemajor limitation of thegen-eral, “off-the-shelf” approachhereconsidered,is simply the needto reducethe

7Suchformulationis somewhatuseless,in factmostof theauthorsreferto energy dissipatingratherthanenergy preservingalgorithms.This is becauseexceptfor verysimplecases,“stif f ” andpracticalproblemsusuallyrequirethe dissipationof high frequency perturbationsthat originatefrom thenumericalapproximation.Seefor instance[6].


0

0.002

0.004

0.006

0.008

0.01

0 0.02 0.04 0.06 0.08 0.1

diss

ipat

ionp

h/T




two-step BDF

Figure4.2: Integrationformulas— dampingerror, εξ (undampedsystem).

0

1

2

3

4

0 0.02 0.04 0.06 0.08 0.1

log(

e(2h

))/lo

g(e(

h))q

h/T




two-step BDF

Figure 4.3: Integration formulas — phaseerror order, log2

εϕ 2h εϕ h

(slightly dampedsystem).

4.3. START-UP OFTHE SIMULATION 29

integrationtime stepto achieve satisfactorystability properties.In factaccuracyrequirementsalreadydemandfor a time stepsizethat in thevastmajority of theapplicationsprovidesfor globalstabilityof theintegration.Oneideathathasbeenproposedto try to catchup with energy preservingschemes,andthusto providesomeglobalstabilisation,is thatof addinga scalarconstrainton thetotal energyof thesystem8. Thisseemsto beunpracticalbecause,asemergedfrom atalk withProfessorBorri, it might inhibit the useof energy dissipatingformulas; in fact,theenergy athighfrequency, thatis dissipatedby thealgorithm,couldbepumpedbackinto thesystemat low frequenciesby theconstraint.Theproblemmight befixedby imposingtheconstrainton thetime derivativeof thetotal energy, sotheenergy itself wouldbefreeto drift, andits growth wouldbeaclearsignof lossofstability; however theseaspectshavenotbeeninvestigated.Theabovereportedvaluefor ρ∞ k 0 60provedto giveadequatedampingin mostapplicationswith very goodaccuracy, without any stability loss. Higher values,of theorderof 0 85 r 0 90, led to instabilitieswhenintegratingstiff mechanicalsystemsrotatingathighspeed( ,ωh ,Wk 0 1); in non-rotationalproblemsevenρ∞ 1 canbeused,at thecostof undesirablelack of algorithmicdissipation.Smallervalueshavebeenused,sometimeswith appreciablelossof accuracy.

4.3 Start-up of the simulation

Thissectiondealswith practicalproblemsthatarisewhenintegratingadifferential-algebraicproblem. A differential-algebraicinitial-value problemconsistsin adifferential-algebraicequationwith appropriateinitial conditions. Theseinitialconditionsarerepresentedby thestateof thesystem,namelythepositionandthevelocityof thebodieswhena mechanicalproblemis considered.Theinitial stateof thesystemmustsatisfythealgebraicconstraints9 to beconsistent.Moreover,thevelocitiesmustsatisfythetimederivativeof theconstraintto allow aconsistentinitial motionof thesystem;theotherunknownsrequiresomeadditionalspecifi-cations.Thelinearandangularmomentacanbecomputedfrom their definitionsasfunctionsof thestateandthusaredetermined.Thederivativesof themomentaandtheconstraintreactions,on theotherhand,mustsatify theequilibriumat theinitial time,asthey resultfrom theproblem.Thepoint is thatthealgebraicpartoftheproblemallows multiple solutions,what is called,for linear, constantcoeffi-cientDAEs,a pencil10. This canbeclarifiedby consideringthat in a constrainedproblemtheaccelerationof a constrainedbody, for a givenstate,canassumeanyvalue,providedthecorrespondinginertiaforceis balancedby anappropriatecon-

8Suchconstraintneedsnot bea purepreservation,unlessthesystemis closed.9Thesameappliesto thevelocities,thatmustsatisfythenon-holonomicconstraints.

10SeeRef. [17], p. 18.


straintreaction.Thechoiceof theinitial valuesfor linearandangularmomentumderivatives, and for constraintreactions,requiresto decidewhich problemwewant to solve. The “right” problemis the one that compliesnot only with thealgebraicconstraint,but alsowith its time derivativesup to oneorder lessthanthe index of the DAE problem;up to secondorder for mechanicalsystems,toyield continuousvaluesfor thederivativesof the linearandangularmomentum.In suchcase,theinitial motionof thesystemwill accountfor thepresenceof theconstraintandyield a regular solution. If suchconditionon the satisfactionofthe highestderivative of the constraintis not fulfilled, the problemis still legal,but it is ill-posed,becauseasthe integrationstarts,theconstraintis violatedandtheconstraintreactionsmustrestorethecompatibility, yielding a solutionthat isdiscontinuousin the derivativesof the momentaandin the constraintreactions.However the above describedsolution is regular only in a local sense.In fact,a solutionrespective of the equilibrium andof the derivativesof the constraintscan still lead to a rough solution in someglobal sense.Considerthe practicalexampleof a flexible rotor bladewhoseintegrationstartswith thebladerotatingat constantangularvelocity but with no centrifugalloads.Thestateis correctintermsof positionsandvelocities;theconstraintat the root is satisfiedaswell asits derivativesup to secondorderare;nonetheless,anaxial loadwave propagatesfrom root to tip assoonastheinertiaalongtheblade“feels” thecentripetaleffectof theconstraint.Moreover, sincetheaxial stiffnessof thebladeis usuallyhigh,the celerity of the wave is high, so it cannotbe integratedaccuratelyat typicalintegrationtime steps,leadingto thenumericalproblemsdiscussedearlierin thischapter. This problemis well-posed,the initial conditionscomply with balanceandconstraintequationsto thedesiredorder, thesolutionis “regular”, but “stif f ”.In this casethe stiffnessof the problemis emphasizedby the initial conditions,asusuallyhappens.A globally regular solutionstartsfrom steadyinitial valuesof thestate;in thepreviously mentionedexample,therotor blademustbeappro-priatelystrainedto apply theright centripetaleffect to its inertia. A steadysetofinitial valuesresultsfrom thesolutionof aninitial trim problem;it is notsufficientfor thestateto berespectfulof equilibriumandcompatibility, it mustsatisfyalsothetrim condition.Steadyin somecasesmeans“static”; in dynamicsystems,likehelicopterrotors,it rathermeans“periodic”. In this work, thesteadyinitial val-uesarenot directly determined.On thecontrary, usuallytheintegrationdoesnotevenstartat locally regular initial values.A locally regular solutionis obtainedby integratingthesystem,duringtheveryfirst steps,by aproceduredetailedlater,while a globally regular, steadysolution,i.e. a trimmedcondition,is obtainedbyartificially cancelling,by meansof thealgorithmicdissipation,the transientdueto theinitial values,in absenceof externalperturbations.Thedifferentphasesofthestart-uparedetailedin thefollowing.


4.3.1 Initial assembly

In usualproblems,theinitial conditionsareavailableonly to someextent,andun-dersomeconditions.Thecanonicalinitial conditionsof an initial valueproblemarethevaluesof theunknownsat the initial time; the initial valueof thederiva-tivesis computedfrom theproblemitself, e.g. y f y t in caseof anordinarydifferentialproblem.Theinitial valuesmustbeavailableandmustsatisfytheal-gebraicpartof theproblem,i.e. Φ y0 t0 : 0. Moreover, they shouldalsosatisfyhigher-order time derivativesof the algebraicconstraint,which are implied bythealgebraicconstraintitself, but notnotautomaticallysatisfiedby thenumericalsolution,otherwisetheconstraintwouldbeviolatedastheintegrationstarts.In caseof asimplifiedmechanicalproblem,writtenin Lagrangianform of thefirstkind, i.e. Eq. 2.2, the initial conditionsarethe positionandthe velocity of eachbodyattheinitial time,while theinitial valueof thelinearandangularmomentum,as well as their time derivative, must be obtainedfrom the problem. Both theposition and the velocity of eachbody are not independent,so the satisfactionof the algebraicequationsmust be ensured. An assemblyprocedurehasbeendeveloped,basedon the idea that the differential part of the systemshouldbeleft free to changeits initial conditionsunderthealgebraicconstraints,sincethehighest-orderderivativesof the differential unknowns, as well as the algebraicunknowns, canaccomplishfor the satisfactionof the differentialequations.Analgebraicsystemis written, with the algebraicpart of the original problem,andwith fictitious springsthatkeepthedifferentialunknownscloseto their tentativeinitial position,namely

K∆y ΦTD y∆ζ K y y0 c ΦTD yζ Φ D y∆y Φ y t0 R

HereK is a fictitious stiffnessmatrix that may have the very simple form of ascaledidentity matrix, K kI , or it may be morearticulated,with independentdiagonalcoefficientsandevenelementstiffnessmatricesfor flexible elementsinafinite elementfashion,to allow thetuningof thecompatibility, i.e. to forcesomespecificdegreesof freedomto changetheir initial value insteadof others. Thelatter choicehasbeenmadein the implementation,so that the flexible elementscontributewith their stiffnessmatrices,andeachnodehasindependentdiagonalstiffnesscoefficients.Thefictitious algebraicunknownsζ have beenusedinsteadof zbecausetheirvalueis discardedaftertheassembly, sincethey donotrepresentthe actualalgebraicreactions,but simply a measure,in a Lagrangianmultipliersense,of theactivationof thealgebraicconstraints.Thesameprocedureis appliedto the time derivativesof the y; in this casethe time derivative of the constraintequationis added,togetherwith extrafictitiousreactionsthatenforcethesatisfac-tion of theextra constraints.Theresultis aninitial solutionthat is compatibleto


thefirst order. It is importantto understandthatthesolutionstronglydependsontheinitial values,which includetheinitial velocities.Thecompatibilityof theve-locitiesis not explicitly statedby thedifferential-algebraicproblem;nonetheless,it is stronglyrequiredbecauseof the index of the problem. If a non-compatibletentative initial solutionis used,the compatibleinitial solutionthat resultsfromtheassemblyproceduredependson thevaluesof thefictitious springs,andthusit might be quite different from the initial solutionactuallydesiredby the user,so this procedureshouldbe considereda sort of compatibility checkratherthana really “universal”startuptechnique.Imaginefor instancethepurelykinematicone-dimensionalproblemof two rigidly linked points, that aregiven a compat-ible initial relative position equal to the length of the joint but different, non-compatibleinitial velocities; the assemblyprocedurewill make the two initialvelocitiesequal,but their valuewill bethemeanof their incompatiblevelocities,weightedby thestiffnessof therespective fictitious springs.Thepoint is thereisno “right” solutionto suchproblem;on theotherhand,it is not easyto supplyathoroughlycompatiblesetof initial values.Thanksto thisprocedure,theusercan“suggest”the desiredbehaviour by meansof the stiffnessof the springs. In thepreviousexample,wheretheusermightbetrying to simplify theinputby supply-ing thevelocityof onepointonly, expectingtherestof thesystemto follow it, thedesiredbehaviour canbeobtainedby giving unit weightto thenodewhoseveloc-ity is given,andnull weightto theremainingnodes;asa result,themechanismispulledby thefirst nodeandacompatiblesetof initial velocitiesis obtained.

4.3.2 Derivativescomputation

After ensuringthe satisfactionof the algebraicequations,the differentialequa-tions must be satisfiedas well, to start from a balancedas well as compatibleinitial solution. In anexplicit differentialproblem,this operationis very simple,consistingin computingtheinitial derivativesas

y0 f y0 t0 ;in an implicit, differential-algebraicproblemthe sameoperationrequiressomecare. Consideragainthe exampleof a mechanicalsystem,after the compatibil-ity hasbeenassessedandrestoredif required. Supposethe initial positionandvelocity of the bodiesareknown, andthe algebraicequationsaresatisfied.Thelinearandangularmomentacanbecomputedfrom their definitionsincetheve-locities are known. The only uncertaintyis on the derivativesof the momentaandon thevalueof thealgebraicreactions.They shouldbecomputedby solvingthe differentialsystemmadeof the equilibrium equationsplus the second-ordertimederivativeof theconstraints,requiringadedicatedsolutionprocedure.In the


presentwork, a ratherdifferentstrategy hasbeeninvestigated,consistingin solv-ing theentiresystemin theusualform, thusresortingto thenormalroutinesforthecomputationof theJacobianandof theresidual,andin iteratingthesolutionat the initial time with a modifiedupdatingprocedure.In fact, whensolving atthe initial time for the derivativesonly, the idea is to integratethe systemwitha null time step,so that the updaterule of Eq. 4.17doesnot affect the valueofthe momenta,thatareexactly computedby meansof the initial velocities. Thisis not possiblebecausethesystembecomessingular:considertheapplicationofthesecond-orderformulaof Eq.4.18to a simplified,linearsystemin Lagrangianform of thefirst kind,st

M cI 0cK I ΦTD yΦ D y 0 0 uv ∆y

∆z∆v wxy My z

F Ky z ΦTD yv Φ c wxywherec is the coefficient of the updateformula of Eq. 4.17; the last block-row,containingthe algebraicequations,has beendivided by c for numericalpur-poses11. If a null time stepis considered,thematrix is singular, becausethecou-pling termsbetweenthedifferentialunknownsvanish,beingc k h. Soanarbitrarytime stepis used,but thecomputationof thesolutionis modifiedby updatingthederivativesof thedifferentialunknownsandthealgebraicunknownsonly, exceptfor thederivativesof thedisplacements,namelythey. They areleft untouched,asarethedifferentialunknowns: they, computedby meansof the initial assemblyprocedure,andthemomentaz, directly computedfrom their definitionby meansof thecompatibleinitial velocities.Dueto theirpeculiarstructure,themechanicalproblemsareusuallylinearin thederivativesof themomentaandin thealgebraicunknowns,so thesolutionrequiresvery little iterations,usuallyoneonly, to ob-taininitial derivativesthatsatisfytheequilibrium.Theveryimportantdrawbackisthatsuchsolutionis not respectfulof thesecond-orderderivativeof theconstraintequation,resultingin “non-compatible”derivativesof themomenta,sospecialat-tention,in Section4.3.4,will be dedicatedto theproblem. The main advantageis thatno dedicatedproceduresfor thedeterminationof the initial conditionsarerequired.

4.3.3 Self-starting algorithm

The integrationformulapresentedin Section4.2 is two-step,so it is not abletostartautomaticallywhentheinitial conditionsatonepointonly areprovided.Theproblemhasbeenovercomeby using,for thefirst step,a second-orderaccurate,

11As suggestedin [17], pp.144–148,to obtaina betterscalingof thematrix andthusto reduceroundoff errors.


single-stepCrank-Nicholsonformula. This formula is self-startingbecauseit issingle-step,andit hasthedesiredsecond-orderaccuracy, so thenumericalsolu-tion is guaranteedto remainconfinedin a neighborhoodof theexact solution12.TheCrank-Nicholsonrule is A-stable,with no algorithmicdissipation,sonumer-ical oscillationsmayarise;they canbedampedby introducingsomealgorithmicdissipationin thesubsequenttime stepsperformedwith theformulaproposedinSection4.2. The algorithmic dissipationmay be requiredalso for the reasonsaddressedin thefollowing section.

4.3.4 Secondderivativeof the constraints

Earlier in this chapter, theproblemof satisfyingthesecond-ordertime derivativeof theconstraintshasbeenintroduced.This problemis very important,asa verysimpleexamplewill show. Consider, for instance,a rigid pendulumperformingaplanemotionundertheeffectof gravity acceleration.In thespirit of themultibodyapproach,it is modelledby a rigid body, constrainedby a distancejoint, namelyx2 y2 l2, beingx andy the coordinatesof the massand l the lengthof thependulum.If we assumethat theinitial positionandvelocity respectively satisfytheconstraintequationandits derivative,i.e. thevelocityostangentto thecircularpath of the pendulum,and we computethe momentum,its derivative and thereactionforceasshown in Section4.3.2,weobtainaninitial solutionthatsatisfiesthe constraintand the equilibrium, but with constraintreactionandmomentumderivativesthat accountfor the gravitational effect only. So, as the integrationstarts,thesolutionviolatestheconstraint,becausethederivativeof themomentumdoesnot accountfor thecentripetaleffect of theconstraint,which descendsfromits second-orderderivative. The integrationdoesnot fail, but oscillationsin thereactionforcearise,andthesolutionis perturbed.Noticethatthiseffectresemblesthe behaviour of a flexible pendulum,the rigid onebeinga sort of limit caseofa truependulum,andthealgebraicconstraintbeinga sortof limit caseof a stiffspring[49]A procedureto overcomethiseffecthasbeendeveloped,consistingin performingasortof numericaldifferentiationof theconstraintby integratingthesystemwitha very short time step,to minimize the amplitudeof the oscillations,andwithhigh algorithmicdissipation,to completelydampenthenumericaloscillationsasquickly aspossible.After achieving agoodestimateof themomentumderivativesandof thereactionforces,thesystemis restartedwith thedesiredtimestep.Suchprocedureshowed a goodbehaviour in many problemsof differentcomplexity,

12As shown in [17], Section3.2,theBDF convergencepropertiesfor asemi-explicit index threeDAE requirethe startingconditionsto satisfy someaccuracy properties. SeeAppendix C forfurtherdetails.


including rotorcraft simulations. The main reasonfor its choiceis to avoid toimplementthesecond-orderderivativesof theconstraints,thatcanbevery cum-bersome.Of course,if veryspecialproblemsmustbeconsidered,whosesolutionrequiresa very accuratestart-up,it may be necessary, and even convenient,tomakesuchaneffort.


Chapter 5

Configuration-dependentinteractions

The bodiesthat build up a systemmay exchangerelative, or internal forcesofdifferent nature,e.g. elastic,gravitational, electromagnetic,contact,or frictionforces,andmany otherkindsof interactions.In many casesthoseforcesdependon theconfigurationof thesystem,i.e. on theabsoluteor relativeposition,veloc-ity andaccelerationof the bodies. In this sectionwe basicallydealwith elasticinteraction,but othersourcesof relative forcesmay be treatedin a similar way.It is importantto noticethat, regardlessof the mathematicalmodelthoseforcesarebasedon, theirapplicationto amultibodycontext will resultin discreteforcesthat act on discretebodies,and the configurationfield they arecomputedfromwill result from someinterpolationof the configurationsof the discretebodiesthat representthe multibody model. In fact, in the presentwork, the definition“multibody” hasthe currently acceptedmeaningof “nonlinear finite elements,with exactnodalkinematics”. As a consequence,a multibodymodelmay rangefrom a very simplecollectionof rigid bodiesjoinedby kinematicconstraints,toa rathersophisticatedmodel of a structuralelasticcomponent,suchas a rotorblade,or a completehelicopter, with flexible componentsmodelledwith a finiteelement-likediscretisationof a three-dimensionalcontinuum.

V3 spaceof vectorsin ℜ3

p V3 positionof apointv V3 velocityof a point ( p)R LIN

V3 rotationmatrix (from local to globalframe)

ω linV3 angularvelocity

x V3 positionof a referencepoint (node)f V3 offset(relativeposition, Rf )

37

38 CHAPTER5. CONFIGURATION-DEPENDENTINTERACTIONS

5.1 Lumped flexible elements

A lumpedflexible elementis representedby a discreteforceor momentthatactsbetweena pool of bodies,usuallytwo, dependingon their relative configuration,i.e. the distance,or the relative rotation,or both1. This allows to introducetheexpressionsfor therelativeentitiesthatcanbeusedto computetheactionsrelatedto flexible elements.The configurationof a body is expressedby meansof itsposition in the global frame, x, whosetime derivative is the global velocity ofthebody, x, thathasalreadybeenusedin thedefinitionof themomenta,Eq.1.1.Theorientationof thebody is expressedby meansof its rotationmatrix, R, thattransformsa vectorin the local referenceframe,usuallydenotedwith a tilde ˜ $z ,into its representationin the global frame. The time derivative of the rotationmatrix is relatedto theangularvelocity, ω; they havebeenbothintroducedin thediscussionon the finite rotationsin Chapter3. A point whoseposition is usedto computethe force may be offset from the nodeby a vector f , that is usuallyconstantin the local frame. As a result,thepositionof anarbitrarypoint rigidlyattachedto anodeis

p x f (5.1)

where f Rf is the offset in the global frame. The orientationat the point, ifrequired,is thatof the body, possiblycorrectedby a constantorientationmatrixRh, namely

Rp RRh Considerfor instancea simple spring, representedby a force acting along theline betweentwo pointsthatdependson their distance.Thedistanceis a vectord x2 f2 ( x1 f1 , whosenorm is the lengthof the spring, l ,d , . Theforce is F d l s ε , thescalarforceexpressedby theconstitutive law s arbi-trarily dependingon thenormalisedelongationof thespring,ε l l0 1, basedon someinitial measureof the length, l0. This force is appliedto both nodes1and2, with oppositesign,F1 h F, F2 F, in thepointswith offsets f1 and f2from the respective nodes,so the nodesthemselvesarealsosubjectto the mo-mentsM1 f1 F, M2 f2 F. Theintroductionof adependenceof s on therelativevelocityof thepointsleadsto a visco-elasticelement.Therelativeveloc-ity is relatedto the time derivative of the distanced, namely l d $ d l , whered x2 ω2 f2 a x1 ω1 f1 ; in otherwords, it representsthe projectionof d in direction d l . More sophisticatedmodels,basedon three-dimensional

1A finite elementbeam,for instance,falls into this category, but, ascommonlyaccepted,it istreateddifferentlysinceit is intendedto describea structuredone-dimensionalcontinuum.

5.2. BEAM MODEL 39

springs,with anisotropicconstitutive laws, or rotationalsprings,may be formu-lated,andhavebeenactuallyimplementedin thecode;theirdefinitionandimple-mentationis beyondthescopeof thissection,andwill betreatedin AppendixD.2.It is importantto remarkthatin suchcomplex lumpedelasticcomponents,thege-ometryof theactualcomponentmustbereproducedwith care,to ensurethefinitestrainsarecorrectlydescribedandthemodelis representative of thecomponent.An exampleof sucha complex lumpedviscoelasticcomponentis anelastomericbearing,currentlyusedin many commercialrotorcrafts.

5.2 Beammodel

A beamelementis agoodexampleof a rathersimpleflexible componentthatcanbe introducedin a multibody model in a finite elementstyle. It canbe usedtomodelvery importantstructuralcomponentswith an acceptabledegreeof accu-racy andrefinementexpeciallyin rotorcraftmodelling,whentheflexibility of thebladeis to beconsideredwithoutexcessivedetail.

B ℜ domainof thebeamS ℜ2 domainof thesections B S |N V3 positionof apointp B |N V3 positionof thereferencelineR B |N LIN

V3 rotationmatrix (from local to globalframe)

t S |N V3 offsetof apoint from thereferencelineei B |N V3 directionsof Rl B |N V3 slopeof thereferenceline ( pD ξ)ρ B |N lin

V3 curvatureof thebeam( RD ξRT)

ν B |N V3 generalisedstrainsof thereferencelineκ B |N lin

V3 generalisedelasticcurvatureof thesection

v B |N V3 velocityof thereferencelineω B |N lin

V3 angularvelocityof thesection

ψ B |N V6 generaliseddeformationsF B |N V3 internalforcesM B |N V3 internalmomentsϑ B |N V6 generalisedinternalforcesτ B |N V6 generalisedexternalforcesperunit lengthx V3 positionof anodef V3 offsetof thereferenceline from anode


5.2.1 Definitions

A beamis a structuralcomponentwhosegeometric,structuralandinertial prop-ertieschangevery smoothlyalong one direction, that is predominantover theothertwo. More detailedandspecialiseddefinitionsof the beammodelmay bemisleading,becausethey imply underlyingassumptionsthatrestrictsomehow themodel to specialcases.The restrictingassumptionsthat aremadein this workwill be presentedassoonasthey arerequired,to clearly indicatethe propertiesof the modelwe call “beam”. The predominantdirectionof the structuralcom-ponentnaturallyidentifiesa line, or bettertheslopeof a line, thatwill beusedasreference.A plane,nearlynormalto thepredominantdirectionof thebeam,willidentify asectionof thecomponent,andsomereferenceonthissectionwill allowustho choosetwo orthogonalaxesto definea local referencesystem.Thebeamis generatedby a rigid rototranslationof the sectionalonga predominantdirec-tion. A continuous,regularandlimited variationof thesectionmaybeaccepted,at somecostthatwill behighlightedlater2; a sharpchangeof sectionproperties,on the contrary, canbe handledby partitioning the domainof definition of thebeamin multiple, piecewiseuniform beams,with someinterfacerepresentedbycontinuityboundaryconditions.Thissuggeststhat,for practicalpurposes,afiniteelement-likestringof piecewiseuniformbeamscanbeusedto describea taperedbeam.Notice thatwe arenot defininga systembasedon somegeometric,struc-tural or inertial propertyof thestructuralcomponent,but we aresimply trying toput somereferencesonto a geometricentity to give a quantitative descriptionofits properties,undertheassumptionthata differentchoiceof references,with anappropriatereductionto a commonreferenceconfiguration,will be ableto givethesamedescription.Thereferenceline is calledp. Thedirectione1 of thesec-tion frameis closeto that of the slopel of the line p, being l pD ξ, whereξ isa local abscissa.In fact,thesectionreferenceframeneedsnot beexactly normalto the slopeof the referenceline; the only assumptionwe make is that e1 $ l bepositiveregardlessof thedeformationlevel.

5.2.2 Kinematics of the beam

An arbitrarypoint on the sectionis identifiedby a vector t in the frameof thesection.In referenceconditions,weassumeits componentin directione1 benull,sothatthereferencesectionis plane3. It maybenon-nullin adeformedcondition,

2Basically, a non-uniformbeamsectionrequiresto solve a variablecoefficient linear differ-ential systemof equations,wherea uniform beamsectionresultsin the solution of a constantcoefficientsystem.

3This assumptionis not strictly requiredandmayberelaxed,allowing warpedreferencesec-tionsto beconsidered,but, sincethepropertiesof thesectionarerequiredto beindependentof the

5.2. BEAM MODEL 41

to describeanout-of-planewarpingof thesection.Sothepositionof anarbitrarypointof thestructuralcomponent,in anarbitraryconfiguration,is s p t, wheret Rt is the relative positionof the point in the section,rotatedinto the globalframe.Therateof changeof thepositionof point s alongthereferenceline givesadeeperinsightinto thebeammodel.Considerfirst abeamwhosesectionis rigid,namelyvectort is constant.Thereferenceline andtheorientationof thesectionbothdependon thelocalabscissaξ, sotheaxialderivativeof syields

sD ξ pD ξ RD ξt By substitutingthepreviously definedslopeof p, andrecallingthepropertiesofthederivativeof a rotationmatrix (Eq.3.2),it becomes

sD ξ l ρ t (5.2)

whereρ axRD ξRT is thecurvatureof thebeam.Sothechangeof configuration

alongthe referenceline of thebeamcanbe describedby meansof two intrinsicvectorialquantities,theslopeof theline andthecurvatureof thereferenceorien-tationof thesection.

5.2.3 Strains and curvatures

Theelasticdeformationof thebeamis definedasthechangein theaxialrateof theposition,Eq.5.2,dueto achangein theconfigurationof thesystem.In orderto becomparable,the two positionratesmustbe rotatedbackto a commonreferenceframe,calledthematerialframe,which mayberepresentedby thesectionframewith nolossin generality. By denotingwith thesubscript $' 0 thereferenceentitiesandwith nosubscriptsthedeformedentities,thedifferencebetweentheaxialratesresultsin4

RTsD ξ RT0 s0D ξ RT l RT

0 l0 RTρ R RT0 ρ0 R0 t

or

RTsD ξ RT0 s0D ξ l l0 ` ρ . ρ0 t

Theelasticstrainsandcurvatures,in thematerialframe,are

ν l l0 (5.3)

κ ρ ρ0 (5.4)

yielding

RTsD ξ RT0 s0D ξ ν κ t

axial position,a referencesectioncanalwaysberedefinedin orderto beplane.4Rememberthat t t0 becausenull warpingis assumed.


5.2.4 Noteon the linearisation of the curvature

Thestrainsν do not requireany specialattention;on the contrary, thecurvatureκ, whenconsideredin view of theupdated-updatedapproachintroducedin Sec-tion 3.4,canbewrittenas

κ θ ~δ Rδθ ~ < 0= κδ κ ? 0@ whereθ ~ G p p~ regardlessof the rotationparametersp that areconsidered.The strainsare linearisedin the materialframe,becausethe constitutive law isnaturallyexpressedin suchframe.Thelinearisationof thecurvatureyields

∆κ RTθ∆ T κ ∆κ (5.5)

Noticethatthelinearisationof thecurvaturein theglobalframeyields

∆κ ∆κδ θ∆ κ ? 0@ andits substitutionin Eq.5.5simplyyields

∆κ RT ∆κδ κδ θ∆ RBy consideringthesimplificationsallowedby theupdated-updatedapproach,suchformulascanbeeffectively approximatedwith

∆θ ~ V ∆p~ ∆p θ ~ < 0= and

∆θ ~ V R< 0= T ∆p~ θ ~δ ∆p Thelatterexpressioncanbesimplifiedevenfurtherby recallingthat,accordingtotheupdated-updatedapproach,theperturbationof thecurvaturecanbediscardedwhenlinearising,namelyθ ~δ V 0, thusyielding

∆θ ~ V R< 0= T∆p~ 5.2.5 Strain and curvature time rates

Thetime ratesof strainandcurvaturemayberequiredto introducesomedepen-dency of the internalforceson thedynamicsof thebeam,i.e. to introducesomeviscoelaticeffectsin theinternalforcesandcouplesof thebeam.Thevelocity ofthearbitrarypoint s is definedas

s v ω t

5.2. BEAM MODEL 43

wherev p is thevelocity of thereferenceline andω is theangularvelocity ofthesection.Theaxialderivativeof thevelocityyields

ddξ

s vD ξ ω D ξ t ω ρ t while thetimederivativeof theaxial rateof s resultsin

ddt

sD ξ l ρ t ρ ω t;

it is apparentfrom Schwartz’ theoremthat pD ξ l ; by the sametheorem,therelationρ ω D ξ ω ρ canbe inferred. The time ratesof thestrainandof thecurvature,in thematerialframe,are

˙ν RTl ω l

˙κ RT ρ ω ρ RTheir linearisationis straightforward.

5.2.6 Equilibrium

Thedifferentialequilibriumequationsfor a rigid sectionbeamcanbeeasilyob-tainedby consideringthat the axial derivative of the work madeby the internalforcesof the beamsubjectto a rigid rototranslationmust be null. The virtualdisplacementof thereferencepoint of thebeamatabscissaxi, dueto arigid roto-translation,is δp ξ ` δp 0B δϕ 0: p ξ p 0 , while thevirtual rotationis δϕ ξ δϕ 0 . Being F ξ , M ξ the internal forcesand momentsof thebeamat pointξ, suchconditionresultsin

∂∂ξ W δp 0\ δϕ 0A p ξ " p 0WWc$ F ξ R δϕ 0"$ M ξ 5 0

which,aftersimplealgebramanipulation,yields

FD ξ 0 (5.6)

M D ξ l F 0 (5.7)

where l pD ξ hasbeenused,and the momentsequationhasbeenevaluatedinp 0 without any loss in generality. Theseequationswill be usedto write adiscreteform of theequilibriumof aportionof beamin Section5.4.


5.3 Beamsectioncharacterisation

The internal forcesof the beammustbe written asfunctionsof the generaliseddeformations,namelythe strainsand the curvatures,by meansof an adequateconstitutive law. Theconstitutive law is requiredto closethesystemof theequi-librium equations,in which the internal forcesand momentsare unknown, bywriting them in termsof the generalisedstrains,and thus of the configurationof the beam. This result can be obtainedby finding a compatiblesolution forthe problemof a samplesectionof the beamloadedby indicial internal forces.The constitutive law of the beamis not determinedfrom the first principlesofmicromechanicsof materials;the constitutive problemis simply scaleddown tothecontinuummechanics,by eliminatingthedependenceon thegeometryof thesectionthroughthe determinationof a setof solutionsthat arerespectfulof thekinematicandnaturalboundaryconditionsof the beamsection. Many authorshave beenworking on the characterisationof beamsections,a completereviewbeingoutof thescopeof thisdissertation;theinterestedreadershouldconsultthegoodonegivenby Hodges in 1990[56], andrefreshedin 1999[86]. Only abriefreview of thepreviousworksdoneattheDipartimentodi IngegneriaAerospazialeof thePolitecnicodi Milano is presented.Thevery first noteon thecharacterisa-tion of semi-monocoquesectionsappearedin 1977by Mantegazza[67], followedby acomprehensiveformulationof theproblemfor thelinearanalysisof arbitrarysections,presentedin 1983by Giavottoet al. [44], specialisedto theanalysisofrotor bladesby Borri andMantegazzain 1985[13] andsubsequentlyextendedandgeneralisedin a geometricallynonlinearframework by Borri and Merliniin 1986[14]. In 1992Borri et al. solved the problemof the characterisationofa twisted/curvedbeam[12], andin 1994Ghiringhelli andMantegazza showedthe possibility to usea commecialfinite elementcodeto characterisea straight,untwistedbeam[31]. In 1997Ghiringhelli andGhiringhelli et al. extendedtheformulationto the analysisof thermalloads[28] andto the electroelasticchar-acterisationof a compositebeamembeddingpiezoelectricmaterials[34]. Theextensionto theanalysisof piezoelectricmaterialsrepresentsa fundamentalpartof thegraduationthesisof theauthor[68]; it is reportedandgeneralisedin Sec-tion 9.1. Thenotationintroducedin [31] is basicallypreservedin this work, withsomedifferences,themostimportantbeingthechangein thedenominationof thereferenceaxes;in theoriginalwork, thebeamaxiswasz, with x andy lying in thesection,while herethebeamaxisis ξ, andη andζ defineaCartesianframeonthesection.

5.3. BEAM SECTIONCHARACTERISATION 45, , , , , b stiffnessmatricesof thesection

U0, U1 B |N V3 * N indicial sectiondiscretewarpingP0, P1 B |N V3 * N indicial sectionnodalworkΨ0, Ψ1 B |N V6 indicial generaliseddeformationsΘ0, Θ1 B |N V6 indicial generalisedinternalforces

5.3.1 Kinematics of the section

Considerthepositions of a point on anarbitrarysection,discardingthepreviousassumptionof constantwarpingt. Thedifferentiationof snow requiresthethree-dimensionalgradient,definedas

∇ $z. $' D ξ $z D η $' D ζ (5.8)

Thegradientoperator, Eq.5.8,appliedto positions yields

∇s l 0 0 ( ρ 0 0 t R∇t andthelinear5 strainsat suchpoint arerepresentedby thesymmetricpartof ∇s,in which thedifferentiationwith respectto thelocal transversecoordinateson thesection, η ζ , is alsoconsidered,namely

d 12

∇s ∇sT

The elasticstrainsareobtainedasthe differencebetweenthe deformedandthereferencesymmetricpart of the gradientof the position, rotatedback into thematerialframe,namelyε RTd RT

0 d0; theelasticgradientis

RT∇s RT0 ∇s0 l l0 0 0 K ρ 0 0 tT ρ0 0 0 t0 ∇ t t0 ;

thegradientof thereferencewarping,∇t0, representsameasureof theaxesη andζ, sincethereferencewarpingt is assumedto beindependentof ξ andto havenullcomponentin directione1. By arrangingthestrainsasε γ23 ε22 ε33 ε11 γ12 γ13 Cthefollowing compactmatrixnotationcanbeused:

ε 0ν ρ t ρ0 t0 t D ξ I t t0

or, by recallingthatρ ρ0 κ, Eq.5.4,

ε 0ν κ t ρ0 t t0 B t D ξ I t t0

5A moregeneral,nonlinearformulationcanbeobtainedby consideringGreen’s straintensor,in matrix notation:d 1 2 ∇s 9 ∇sT 9 ∇sT ∇s.


where ν, κ are the generaliseddeformationsof the rigid sectionas definedinSection5.2, andoperator $z performsthe strain-relateddifferentiationin theplaneof thesection,i.e.

$'.st 0 $z D ζ $z D η

0 $' D η 00 0 $z D ζ0 0 0 $' D η 0 0 $' D ζ 0 0

u'vThegeneralcaseof a constantcurvature/pretwistbeamis presented.It hasbeenfirst studiedby Borri et al. in [12]; it is hereformulatedby consideringa lineari-sationabouta reference,warpedanddeformedcondition.Assumethatthegeometricandstructuralpropertiesof thesectionremainconstantalongtheaxis.Thisassumptionis requiredby theschemehereconsidered;it mayleadto a modelwith poor convergenceif a smoothlyaxially varying propertiesbeamis analysed[55]. An acceptablemodel in mostcasesmay be obtainedby“sampling” theactualbeamin a finite numberof sections,thatareanalysedeachundertheassumptionof constantaxialproperties6. Thestrainsbecome

ε 0ν κ t ρ0 t t0 \ t D ξ I t t0

andtheir perturbation,with respectto thereferenceconfiguration,yields

∆ε G 0I H ∆ν G 0 t H ∆κ G 0

I H ∆t D ξ32 G 0ρ H $' 4 ∆t

6Therearedifferentwaysof relaxingthis assumption;noneof themhasbeeninvestigatedyet,sincethe stepwiseconstantapproximationgave acceptableresultsin practicalapplications.Thesimplestpossibilityis to consideran“affine” changeof properties;in thiscase,thegeometryof thesectionis definedby meansof vectort, definedast Qt , beingt a referencepositionat someabscissaξ andQ Q ξ asymmetricprojectionmatrix that“scales”theposition.Thisapproachis very simpleandrequiresvery little changesto theformulation,but it is of little usein practicalcases,becausepracticaltaperedbeamsdo no taperin suchasmoothway; think, for instance,of asemi-monocoquetaperedwing, with piecewiseconstantthicknesspanelsandpiecewiseconstantsectionflangestringers.

5.3. BEAM SECTIONCHARACTERISATION 47

5.3.2 Inter nal work

Thestrainwork perunit lengthmadeby a virtual variationof strainsagainstthestressesin thesectionis

∂∂ξ

δLi SS

δεT σ JdS

whereJ det ∇s0 is themeasureof theintegrationvolume.Thevirtual variationof thestrainsis

δε G 0I H δν G 0 t H δκ G 0

I H δt D ξ 2 G 0ρ H $' 4 δt

sotheinternalwork perunit volumeis

δεT σ δt D ξδtδψ w xy T Σ tξ

Σ tΣ ψ w xy

The Σ representreferencegeneralisedstressesthat work againstthe differenttermsof thestrains;theasterisk $' meansthey areentitiesperunit volume,sotheir integrationover the domainS mustbe performedyet. They aredefinedinAppendixE. Theperturbationof theinternalwork yields

∂∂ξ

∆δLi SS

∆δεT σ dS S

S2 δεT ∂σ

∂ε∆ε ∆δε T σ 4 dS

being ∆δε T σ δtTσS ∆κ δκT σS ∆t whereσS e 0 I f σ arethe stresseson the sectionof the beam. The last definedcontribution to the strainwork accountsfor the nonlinear(quadratic)effect thatcouplesthewarpingt to thecurvatureκ. Whena linearelasticconstitutive law isconsidered,thestiffnessof thematerial∂σ ∂ε D canbeused.By considering,at theright-handside,someimposedstrainsεp andstressesσp, ageneralsolutioncan be found, e.g. accountingfor thermal[28] ratherthan piezoelectriceffects[34]. Thepiezoelectriccasehasbeenimplementedto characterisetheactive twistrotorblade;it is describedin Section9.1anddetailedin AppendixG. Theinternalwork perunit volumeresultsin

∆δεT σ δt D ξ

δtδψ w xy T st ¡

sym b¢ uv ∆t D ξ∆t∆ψ w xy

wherethe asteriskagainremarksthat the stiffnessmatricesareper unit volume.They aredefinedin AppendixE.


5.3.3 External work

Thework madeby a virtual variationof displacementagainstall thestressesthatacton thesectionon both sides,the stresseson the boundaryandthe forcesperunit volume can be divided in two main parts. The first is relatedto externalstresseswc thatacton theboundaryof thesectionandforcesperunit volumewV

insidethe section;it is a trueexternalwork becauseit is madeby externaldeadloads.Thework madeby thestressesthatacton thetwo sidesof thesection,onthe contrary, is relatedto the transmissionof loadsalong the beam,in form ofinternalforces.Thewholeexternalwork is thus

∂∂ξ

δLe ∂∂ξS

SδsTσS dS TS

SδsTwV dS TS

cδsTwc dc

Noticethattheexternalwork hasbeenwritten in theglobalframesincetheloadsandthedisplacementsarenaturallyknown in suchframe.

Transmissionwork

The termδsTσS D ξ requiresthe axial derivative of the virtual variationof the

displacement7,

δs δp δϕ t Rδt (5.9)

thatis

δsD ξ δpD ξ δϕ D ξ t δϕ ρ t δϕ Rt D ξ ρ Rδt Rδt D ξwhich, by consideringthat in analogywith the time derivative of the curvatureδϕ D ξ δρ ρ δϕ, andrecallingthat pD ξ l , resultsin

δsD ξ δl δρ t ρ δϕ t ρ Rδt δϕ Rt D ξ Rδt D ξ As aconsequence,andconsideringthatσSD ξ ρ σS RσSD ξ,

δsTσS D ξ δpTρ σS σSD ξ ` δlT σS δϕTt D ξ σS t σSD ξ ` δρT t σS (5.10) δtTσSD ξ δtTD ξσS;

7Noticethat,in Eq.5.9,thewarpingin thematerialframe,δt, hasbeenassumedasindependentwarpingvariable.


notice that the work per unit volume madeby the transmissionof the internalforceshasbeenwritten in thematerialframe.Recallingthedefinitionsof thegen-eralisedstrainsandcurvaturesof thebeam,Eqs.5.3and5.4,thevirtual variationof thegeneraliseddeformationsis

δν RTδϕ T l δl

δκ RTδϕ T ρ δρ

thusyielding

δl δν δϕ T l δρ δκ δϕ T ρ

By substitutingtheabove written expressionsin Eq. 5.10,thetransmissionworkperunit volumefinally becomes

δsTσS D ξ δpTρ σS σSD ξ : δνT σS δϕTl σS t D ξ σS t σSD ξ ρ t σS δκT t σS δtT σSD ξ δtTD ξσS;

By noting that p, ν, ϕ and κ do not dependon the position in the section,theinternalforcesresultfrom theintegrals

F SS

σS dSM S

St σS dS

Theintegralof thetransmissionwork thusresultsinSS

δsTσS D ξ dS δpTFD ξ δνTF δϕT

l F M D ξ δκTM (5.11) S

S6 δtTσSD ξ δtTD ξσS7 dS;

Notice that, by independentlyperturbingδp, the differential force equilibriumequationFD ξ 0 is obtained;at thesametime, by independentlyperturbingδϕ,themomentequilibriumequationl F M D ξ 0 is obtained.


Deadloadswork

Thework madeby thedeadloadsis a novel contribution to the formulationandit appearshere for the first time. Assumethe external loadswV , wc have theform w RW η ζ g ξ , so that they areknown in the form of a loadmodeW,dependingon the positionin the section,that multiplies a scalarfunction of theaxialposition,g. Theexternalwork becomesS

SδsTwV dS S

cδsTwc dc 2 δpTFe δϕTMe TS

SδtTWV dS TS

cδtTWc dc4 g ξ B

where

Fe R 2 SSWV dS S

cWc dc4

Me R 2 SSt WV dS S

ct Wc dc4

Theexternalwork contributionhasbeenwritten for straightbeamsonly, andwillbediscussedin AppendixE. It is intendedmainly for theaccuratestressrecoveryin caseof importantdistributedloadeffectson thestressdistributionof thebeam.Notice that theexternalloadsmaycontributealsoto thedifferentialequilibriumof thebeam,sincethey work againstthevirtual variationsof configurationof thereferenceline.

5.3.4 Discretisation

Theproblemof characterisingthe sectionresultsfrom equatingthe internalandthe externalwork, andby independentlyperturbingthe referenceconfiguration,the generaliseddeformationsandthe warping. In generalcases,a discretisationneedsbe appliedto the warping to resolve the integrals in the stiffnessmatrixandin the externalandthe transmissionwork. The warping t canbe describedby an arbitrary set of nodal unknowns u through an appropriateset of shapefunctions, i.e. t N η ζ u ξ . The differentiationof the warping thus yields∇t NuD ξ N u. Thedetailsof thediscretisationof theinternalwork arere-portedin AppendixE. It is importantto noticethatafterdiscretisationtheintegralform of theinternalandexternalwork canbeeasilywritten in termsof aconstantcoefficient lineardifferentialequation.In caseof non-uniform,regularly varyingbeamsection,a variablecoefficientequationwould result.An isoparametricinterpolationof the nodaldisplacementshasbeenusedto im-plementbrick finite elements.


5.3.5 Solution

A simplifiedconfigurationof thebeamis considered,to allow aneasyunderstand-ing of theformulationof thecurved/twistedcharacterisation.Let’sassumethatinreferenceconditiontheanglebetweentheslopeof thereferenceline andthesec-tion remainsconstant;this meansthat in thematerialframetheslopeof the linemustbe constant,namely

RT l D ξ 0, yielding l D ξ ρ l . Considernow the

curvature.It is tranformedfrom thelocal to theglobalframeby matrixR, namelyρ Rρ. Theaxialdifferentiationof thecurvatureyieldsρξ ρ Rρ Rρξ, whichonturnresultsin ρξ Rρξ, because,beinga anarbitraryvector, a a 0 by def-inition. Let’s assumethatthecurvatureis constantin thelocal frame,i.e. ρξ 0;asaresult,it mustbeconstantalsoin theglobalframe.Theseassumptionsleadtothesocalledhelicoidalbeammodel,discussedby Borri et al. in [10, 11] andin[12]. It owesthenameto thepropertythatits geometry, namelythereferencelineandthesectionorientation,is completelydescribedby ahelicoidalrototranslationof asection,with constantcurvature.Thiswork is notdirectlyconsideringaheli-coidalbeam;it is ratherassumingthatanarbitrarily curvedandtwistedbeamcanbelocally approximatedby ahelicoidalmodel.Thebalanceof thediscretisedinternalandtransmissionwork8 resultsinst

sym b uv ∆uD ξ∆u∆ψ w xy ∆P

∆PD ξ∆ϑ w xy

whereP arethenodalforcesthatwork for thewarpingshapefunctions,

P SS

NT σS dS

By consideringasetof concentratedloads,theperturbationsof theinternalforces,in thematerialframe,are

∆ϑ G I 0p I H RTF0

RTM0 I Herep p ξ is assumedasthepositionof thepoint wherethesectionis beinganalysed,in thereferenceconfiguration;without any lossin generality, theposi-tion wheretheforcesareappliedis assumedastheorigin for p, i.e. p 0( 0. As-sumetheunknownsaswell canbeexpressedasa combinationof constantterms,denotedby subscript $z 0, andof termslinearly dependingon p, denotedby sub-script $z 1, bothmultiplied by theunknown concentratedforcesandmomentsϑ0,

8Theexternal,deadloadsarenot consideredhereinbecausethey do not intervenein thechar-acterisationof thebeamsection.They aredescribedin AppendixE.


namely

∆u U0 p U1 ϑ0 ∆P P0 p P1 ϑ0 ∆ψ Ψ0 p Ψ1 ϑ0

wheretheoperationp U mustbeintendedasthecrossproductof p by theU ofeachnodeof thediscretisation;again,p ψ mustbeintendedasthecrossproductof p by both the strainsandthe curvaturein ψ £ ν κ . The derivativesof theunknownstake theform

∆uD ξ l U1 ϑ0 ∆PD ξ l P1 ϑ0

By assuminga local linearisationof p in theunknowns,andconsideringthat thep resultingfrom theinitial assumptionsis asinein ξ plushigher-orderterms,theyresultin

∆u V U0 lξ U1 ϑ0 ∆P V P0 lξ P1 ϑ0 ∆ψ V Ψ0 lξ ¤ Ψ1 ϑ0

sotheproblembecomesst sym b uv l U1

U0 ξl U1

Ψ0 ξl Ψ1 wxy P0 ξl P1

l P1

Θ0 ξΘ1 wxy Theproblemcanbesolvedby separatelyconsideringthe linearandtheconstantpartof thesolution;thelinearpart,aftersomemanipulation,becomesG

symb H l U1

l Ψ1 I 0Θ1 I

whosesolutionis substitutedin theconstantpart,yieldingG symb H U0

Ψ0 I 0Θ0 I G ¥¦ T

skw 0 H l U1

l Ψ1 I The sectionpropertiescanbe computedby condensingthe warpingdegreesoffreedom,i.e.by left- andright-multiplyingtheinternalworkmatrixby thesolutionarrays,to obtainthestrainenergy in termsof compliancepropertiesof thesection,namely

∂∂ξ

∆δLi δϑT0C ξ ∆ϑ0

5.4. FINITE VOLUME BEAM FORMULATION 53

The dependenceof matrix C on thepositionwherethesolutionis evaluatedhasbeenhighlighted;however, asexpected,it canbeeasilyshown that suchdepen-denceis only formal: in fact the sectionpropertiesdo not dependon the axialabscissa,andmatrix C canbe evaluatedat ξ 0 without any lossin generality.Thestiffnessproperties9 D arefinally obtainedby invertingthecompliancematrixC.

5.3.6 Noteon the determination of the warping

The way the warping is defineddoesnot ensureit is decoupledfrom any rigiddisplacementof thesection,becausethewarpingshapefunctionsusuallycontainrigid displacementmodes. So, while the stiffnesspropertiesarecorrect,beingbasedon an appropriateevaluationof the strainenergy dueto both the warpingandthegeneraliseddisplacements,thegeneraliseddeformationsandthewarpingsareunderdetermined.A strain-energy basedcriterionfor thedecouplinghasbeenproposedby Borri and Merlini [14]; it is basedon the considerationthat bysettingthework of thewarpingagainstthesectionstresses,thelastaddenduminEquation5.11,equalto zero10,S

SδtTσS dS 0

thestrainenergy canbe written asfunctionof thegeneraliseddeformationsandof the internalforcesonly. Suchconstraintresolvesfor thesix underdetermineddisplacementunknowns. Thereasonsto preferana posterioridecouplingof theresultsareessentiallyrelatedto the lossin sparsityif suchconstraintis explici-tated,or to the increasein complexity if it is addedin a LagrangianMultipliersense.Thea posterioridecouplingrequiresto constrainthematrix prior to solu-tion dueto its singularity. By usinganarbitrarystaticallydeterminateconstraint,thewarpingin generalis not decoupled;at this point,by consideringadecoupledwarpingmadeof the original warpingplus a rigid displacement,the decouplingrigid displacementcanbedetermined,andtheinitially computedcoupledsolutioncanbecorrected.

5.4 Finite volumebeamformulation

The fundamentalideaof thefinite volumebeammodelconsistsin directly writ-ing theequilibriumequationof a finite portionof beam,consideringtheexternal

9Noticethat thesamesymbol,D, hasbeenusedfor boththematerialconstitutive law andthebeamsectionstiffnessmatrix; despiteit might bemisleading,it is intendedto underlinethey bothareconstitutivematrices.Thecorrectmeaningshouldbeclarifiedby thecontext.

10Or simply equalto aconstantvalue[12]


aswell as the internal forcesandcouples. The internal forcesandcouplesareexpressedasfunctionsof theconfigurationby meansof aconstitutive law.

5.4.1 Finite equilibrium

The forcesthat participatein the equilibrium arethe internalforcesϑ £ F M at thetwo extremitiesof theportion,ϑa andϑb, beingBb

a : £e a bf thedomainoftheportionanda, b theso-calledevaluationpointsfor theinternalforces,andtheexternalloadsappliedin thedomainBb

a. It resultsin§ pb x0 T ϑb § pa x0 T ϑa ¨ ba (5.12)

Matrix§

is definedas§ p5 G I p T

0 I H wherep is arelativeposition(adistance);thematrixsimplytransportstheinternalshearto thepolewheretheequilibriumof themomentsis computed11; x0 is thepositionof thepoleusedto computethemoments.Theexternalloadsaredefinedas ¨ b

a S b

a

§ p ξ x0 T τ dξ beingτ the distributedforcesandcouples,that werenot consideredin Eqs.5.6,5.7. Notice that thesameresultcanbeobtainedby writing theweakform of theequilibriumof thebeamin aweightedresidualsmanner, namelyby integratingthedifferentialequilibriumequations,Eqs.5.6,5.7,weightedbyaconstantpiecewise,or Heavyside,weightfunction,asshown in [69, 37].

5.4.2 Constitutive law

The internal forcesat the boundariesof the portion of beamcan be written intermsof the generaliseddeformationsby meansof an arbitraryconstitutive law.In caseof linearelasticity, or in generalwhena linearisedelasticconstitutive lawis required,the constitutive matrix computedin Section5.3 can be used. Theconstitutiveequationof a finite portionof beam,Eq.5.12,becomes§ pb x0 T bDb T

b ψb § pa x0 T aDa Ta ψa ¨ b

a (5.13)11Thetransposeof matrix © performswhathasbeentermedSouth-Westcrossproductby Borri

andhis co-workers,a generalisedcrossproductbetweentheconfigurationof a point, ª p ! R« , andforcesandmoments,ϑ ¬5ª F M « , at suchpoint, resultingin a changeof polefor themoments;arecentdescriptionof suchwork canbefoundin [15].


where® is a block-diagonalmatrix madeof two conventionalrotationmatrices,namely® ¯ ° R 0

0 R ±³²5.4.3 Linearisation

Thelinearisationof Eq.5.13involvesthelinearisationof matrices i , ® i , andofthegeneraliseddeformationsψi , i ¯ a µ b. It resultsin

∆ ¶R´· pi ¸ x j ¹ T ® iDiψi º ¯ ∆ ´· pi ¸ x j ¹ T ϑi» ´ · pi ¸ x j ¹ T ∆® iϑi» ´ · pi ¸ x j ¹ T ® iDi∆ψi µ (5.14)

wherei is anevaluationpoint,and j is areferencepoint,usuallyamultibodynode,with

∆ ´· pi ¸ x j ¹ ¯ ° 0 · ∆pi ¸ ∆x j ¹½¼ T

0 0 ± µ (5.15)

∆® i ¯ ° θ∆ ¼ Ri 00 θ∆ ¼ Ri ± µ

∆ψi ¯ ¾ RTi ¿ ∆l i»

l i ¼ θ∆ ÀRT

i ¿ ∆κi» κi ¼ θ∆ À¦Á ²

Theexpressionwith thelocal strainsψ hasbeenpreferredto thatwith theglobalonesbecauseits linearisationpointsoutthattheinitial axialslopeandcurvaturedonotdependon theconfigurationunknownsandthusdonotaffect thelinearisationof thedeformations.Theexternalloadsmayrequiresomelinearisation,at leastfor the role of matrix´ in computingthedistributedmomentdueto distributedtransverseforces.Theirlinearisationis notpresentedsinceit is straightforward,andit is notrequiredwhenonly concentratedloadsappliedat thepolex0 areconsidered.


Theinterpolationof positionandrotationparametersallows oneto directly writeEquation5.14 in termsof perturbationsof the nodal unknowns. A three-nodebeamelement,shown in Figure 5.1, hasbeenimplemented;as pointedout in


∗∗

ξ

Node

Reference point

Evaluation point

mI

mItII

tII

1

2

3f1

f2

f3

τ

I II

Figure5.1: Finitevolumethree-nodebeamelement.

[69, 37], this is the lowestorderelementthat gives the exact solution for end-appliedloads;theoptimalevaluationpointscorrespondto thetwo Gaussintegra-tion pointsthatexactly integratepolynomialsup to third degree[69, 37]. Eq.5.14mustbe written for threeportionsof beam,centeredon eachof the threenodes,with the boundaryportionsendingon the boundarynodes,so the equilibriumequationof theelementresultsinÂÃ ´ ¿ pI ¸ x1 À T 0¸ ´ ¿ pI ¸ x2 À T ´ ¿ pI I ¸ x2 À T

0 ¸ ´ ¿ pI I ¸ x3 À T ÄÅ ¾ ϑI

ϑI I Á ¯ ÆÇÈÊÉ I1É I IIÉ 3I I ËÌÍ

being1 µ 2 µ 3 the nodesof the beam,and I µ I I the evaluationpoints. An arbitrarypoint on the referenceline is p ¯ Nj p j , beingNj the j-th nodeparabolicshapefunction;anarbitraryrotationparametersetis g ¯ Njg j ; thesummationover theindex of the shapefunctionsis assumed.The position p relative to a nodeisobtainedfrom Eq. 5.1 as p j ¯ x j

»f j . The kinematicentities involved in the

equilibriumequationof thebeamare

p ¯ Nj · x j»

f j ¹ µg ¯ Njg j µl ¯ N Îj · x j

»f j ¹ µ

κ ¯ Gδ ¿ gÀ N Îjg j»

Rδ ¿ gÀ κ Ï 0Ð ;their linearisationyields

∆p Ñ¯ Nj · ∆x j» ∆g j ¼ f j ¹ µ

∆g ¯ Nj∆g j µ∆l Ñ¯ N Îj · ∆x j

» ∆g j ¼ f j ¹ µ∆κ Ñ¯ N Îj∆g j

» ∆g ¼ κ Ï 0Ð µ


wherethesimplificationsallowedby theupdated-updatedformulationhave beenexploited12.

5.4.5 Implementation notes

Thefinite volumebeamhasbeensuccessfullyusedin theanalysisof variousstruc-tural components.Its applicationto the modellingof rotor bladesrepresentsachallenge,andrequirestheawarenessof its behaviour in suchpeculiarapplication.This formulationis naturallyorientedtowardsaC0 descriptionof thegeneralisedstrains,andthusrequirestheability to determinethesix stiffnesstermsasshownin Section5.3. The useof a nonlinearformulationintrinsically accountsfor theprestresseffectsthat resultfrom thecentrifugalforceson a rotorcraftblade. Ontheotherhand,this requiressomecarein distributing theinertiaalongtheblade.A consistentformulationof the inertia forceshasbeendeveloped,but it hasnotbeenimplementedyet becauseit resultedin anunacceptableoverheaddueto theneedof numericallyintegratingtheinertiaforcesovereachportionof thebeam;infact,sincetheinertiaforcesdirectly dependon thedisplacementsinsteadof theirspatialderivatives,their contribution cannotbebroughtto a boundaryevaluationby integratingby parts.Theuseof afinite element-likeapproachto themodellingof rotorbladesmightsoundunusualto rotorcraftpeople,whoareusedto amodaldescriptionof theflexibility of theblades.Suchanapproachis dueto historicalreasonsandpossiblyresultedfrom theneedto have very small,efficient modelsfor highly time-consuminganalyseson computersvery limited both in memoryandCPUspeed.Moreover, amodalmodelallows theanalystto consideronly thedesiredcontributionsin termsof frequency spectrumandspatialcomponents,e.g.thedesiredmix of flapwisebendingandtorsion.It is notour intentionto criticizesuchapproach,which still representsa legitimatechoicefor certainanalyses,buta finite element-like approachsurelyallows the analysta higherflexibility andaccuracy in modelling structuralcomponents,and, in caseof a fully nonlinearmodel,astheoneherepresented,allows to overcomethefundamentallimitationof themodalapproach,theintrinsically limited flexible displacements.

12Any rotationparametrisationcanbeusedby substitutingg with p andby usingtheappropriatedefinitionsof matricesR andG.


5.5 Platemodel

The platemodel is definedin a way muchsimilar to the beam;in this case,thereferencedomainis a surface,and the sectionis representedby a line that, atleastin referenceconfiguration,is chosento be nearlynormalto thesurface. Insomesensetheplatemodelhereconsideredis dual to thebeammodelpresentedin theprevioussectionsof this chapter, in that thedomainin which it is definedis a surface,p, while thesection,which will be calledthe “fibre”, is a segment,i.e. a boundedone-dimensionaldomain. Most of the definitionsintroducedforthe beamare hereextendedto accountfor the changein dimensionalityof thedomainsof definitionfor thevariousentities.Thenomenclatureusedthroughoutthesectionsdedicatedto theplateanalysiscloselyresemblestheoneintroducedfor thebeam,to highlight theanalogies.Thereadershouldnoticehow mostof theexpressionsusedfor thebeammaintaintheir meaning,provided the appropriatedomainof definitionof theinvolvedentitiesis considered.

B ℜ2 domainof theplateS ℜ domainof theplatefibres B ¼ S ÒÓ V3 positionof apointp B ÒÓ V3 positionof thereferencesurfacet S ÒÓ V3 offsetof apoint from thereferencesurfaceR B ÒÓ LIN · V3 ¹ rotationmatrix (from local to globalframe)ei B ÒÓ V3 directionsof Rl B ÒÓ V3 ¼ V3 planetangentto thereferencesurface( ¯ ∇p)ρ ¼ B ÒÓ lin · V3 ¹ ¼ V3 curvatureof theplate( ¯ ∇RRT)ν B ÒÓ V3 ¼ V3 generalisedstrainsof thereferencesurfaceκ ¼ B ÒÓ lin · V3 ¹³¼ V3 generalisedelasticcurvatureof theplateψ B ÒÓ V6 ¼ V3 generaliseddeformationsF B ÒÓ V3 ¼ V3 internalforcefluxesM B ÒÓ V3 ¼ V3 internalmomentfluxesφ B ÒÓ V3 externalforcesperunit surfaceµ B ÒÓ V3 externalmomentsperunit surfaceϑ B ÒÓ V6 ¼ V3 generalisedinternalforcefluxesτ B ÒÓ V6 generalisedexternalforcesperunit surface

5.5.1 Definitions

A platemodel is definedby a referencesurfacep, whosepositionwith respectto theplateis arbitrarily chosen.Thesurfaceis definedon a local bidimensionaldomain,with local abscissæξ, η. A referenceframeof theplateat an arbitrarypoint is definedby arotationmatrixR. Directione3 is chosento benearlynormal

5.5. PLATE MODEL 59

to the referencesurfacep, while the othertwo directionsarearbitrarily selectedsuchthatin referencecondition,eT

1 lξ Ô 0 andeT2 lη Ô 0, beingl i thederivativesof

p with respectto theabscissæon thereferencesurface.An abscissaζ is takenindirectione3.

5.5.2 Kinematics of the plate

By mutuatingthesymbolsfrom thebeammodel,thepositionof anarbitrarypointcanbedescribedass ¯ p

»t, wheret isavectorin theglobalframethatdescribesa

relativepositionwith respectto thereferencesurfacep, obtainedby transforminga vector in the local frameby meansof R, namelyt ¯ Rt. The gradientof thedisplacementsallowsto characterisetheplatein termsof intrinsicstrainmeasures.By consideringthatboth p andRdependonthelocalcoordinatesof thereferenceplaneonly, andby assumingthat t doesnot dependon thepositionon theplate,or in otherwordsthatthereis nowarpingandtheplatehasconstantthicknessandproperties,thegradientof thedisplacementis

∇s ¯ ∇p» ∇Rt µ

or

∇s ¯ l» ρ ¼ t

wherel ¯Õ lξ µ lη µ 0 Ö is theslopeof thesurfacep (thegradientoperatorhasbeendefinedin Eq.5.8),andρ is thecurvatureof theplate,obtainedfrom thegradientof thereferenceframeas

ρ ¯ ax · ∇RRT ¹¯ Õ ax · R× ξRT ¹ µ ax · R× ηRT ¹ µ 0 Ö ²Theexpressionof ∇scanbeexplicitatedas∇s ¯ Õ lξ » ρξ ¼ t µ lη » ρη ¼ t µ 0 Ö ² Thedifferencebetweenthegradientof thepositionin deformedandin referencecon-figuration,both transformedbackto thematerialframe,determinesthemeasureof thestrainsof theplate

RT∇s ¸ RT0 ∇s0 ¯ · RT l ¸ RT

0 l0 ¹ » · RTρ ¼ t ¸ RT0 ρ0 ¼ t0 ¹¯ · l ¸ l0 ¹ » ¿ ρ ¼ t ¸ ρ0 ¼ t0 À

or, consideringthat t ¯ t0, dueto thenull warpingassumption,

RT∇s ¸ RT0 ∇s0 ¯ ν » κ ¼ t µ

wherethegeneraliseddeformationsof theplate,

ν ¯ l ¸ l0 µκ ¯ ρ ¸ ρ0 µ

havebeenintroduced.


5.5.3 Plateequilibrium

The differentialequilibrium of the plate resultsfrom consideringthat the workmadeby the internal forcesfor a rigid displacementis null. The virtual rigiddisplacementandrotationat point p ¿ ξ µ η À areδp ¿ ξ µ η À ¯ δp ¿ 0 µ 0À » δϕ ¿ 0 µ 0À ¼¿ p ¿ ξ µ η À ¸ p ¿ 0 µ 0ÀÀ andδϕ ¿ ξ µ η À ¯ δϕ ¿ 0 µ 0À , aspreviously shown for the beam.Thenull internalwork conditionresultsin

∇ ¿ δp ¿ ξ µ η À"Ø F ¿ ξ µ η À » δϕ ¿ ξ µ η À"Ø M ¿ ξ µ η ÀWÀ ¯ 0 µ (5.16)

whereF ¯ÙÕ Fξ µ Fη µ 0 Ö , M ¯Õ Mξ µ Mη µ 0 Ö arethefluxesof internalforceson thefibreof theplate.Theconditionin Eq.5.16resultsin

∇TF ¯ 0 (5.17)

∇TM» ¿ l ¼ À T F ¯ 0 (5.18)

where l ¯ ∇p hasbeenused. Notice that, by properlydefining the differentialoperators,andby exchangingthedimensionsof the referenceandof thesectiondomains,theplateintrinsic modelis perfectlydualto thatof thebeam.

5.5.4 Singularity and compatibility

Theequilibriumequations,Eqs.5.17,5.18containtheequilibriumaboutthedrilldegreeof freedom.By consideringthatusuallyaplateconstitutivelaw is notableto capturethestiffnessrelatedto suchdegreeof freedom,thedrill componentoftheequilibriumequationsimply statesthesymmetryof thein-planeshearfluxes,resultingin a singularconstitutivematrix for a finite plate.At thesametime, thedefinitionof thegeneraliseddeformationsof theplatethathavebeenintroducedinSection5.5.2do not imply thesymmetryof thestrains.By following theschemeproposedin [58], a generalway to enforcethesymmetryof thestrainsandof thestressesin theweakformulationof theequilibriumof theplatewill beintroducedin Section5.7,dedicatedto thefinite volumeformulationfor theplates.Thedeformationsmustsatisfyacompatibilityrelationthatis implicit in casetheyareexpressedasthegradientsof theconfiguration,but it mustbeenforcedwhena strain-basedformulationis considered,asin Section5.6. Theconditionresultsfrom theconsiderationthatthestrainsmustbenon-rotational,namely∇ ¼ ¿ ∇sÀ ¯0, whereoperator∇ ¼ ¿ Ø'À performstherotoroperation,yielding

∇ ¼ ν ¯ 0 µ (5.19)

∇ ¼ κ ¯ 0 ² (5.20)

This relationshipwill alsobeusedto obtaintheelasticpropertiesof theplateintermsof compatibleaswell asbalancedloadconditions.

5.6. PLATE FIBRECHARACTERISATION 61

5.6 Platefibr e characterisation

Theplatefibrecharacterisationfollows thesameapproachusedfor thebeamsec-tion. Thedomainof thefibre is muchsimplerthanthatof theplate,consistingin asegmentof line13 in thereferenceconfiguration.Theplatefibreanalysishasbeenimplementedonly in caseof linear, flat initial referencesurface. The extensionto thegeneralcaseof nonlinear(couplingbetweengeneraliseddeformationsandwarpingstrains)is straightforward,following theschemedevelopedfor thebeam;it is presentedanddiscussed.This part of the work is original andis presentedherefor thefirst time.Ú

, Û , Ü , Ý , ® , Þ stiffnessmatricesof thefibreΞ ¯ßMà¦µâáãµäå V6 æ 3 ¼ V3 straincollocationmatrixU0, U1 B ÒÓ V3 æ N ¼ V3 indicial fibrediscretewarpingP0, P1 B ÒÓ V3 æ N ¼ V3 indicial fibrenodalworkΨ0, Ψ1 B ÒÓ V6 ¼ V3 indicial generaliseddeformationsΘ0, Θ1 B ÒÓ V6 ¼ V3 indicial generalisedinternalforces

5.6.1 Kinematics of the fibr e

Thestrainsatanarbitrarypointonthefibreresultfrom thegradientof theposition,∇s, wherethegradientof thewarpingis consideredaswell, resultingin

∇s ¯ l» ρ ¼ t

»R∇t ²

The differencebetweenthe gradientat someconfigurationandthat in the refer-enceconfiguration,both rotatedbackto the materialframe,yields a measureofthestrainsin thefibre:

RT∇s ¸ RT0 ∇s0 ¯ ν » κ ¼ t

» ρ0 ¼ ¿ t ¸ t0 À » ∇ ¿ t ¸ t0 À ²where the previously definedstrain and curvatureof the referenceplanehavebeenconsidered.The conventionalmatrix notationfor the strains,namelyε ¯ß ε11 µ ε22 µ ε33 µ γ23 µ γ31 µ γ12 å , resultsfrom multiplying the differencebetweenthegradientsby asetof collocationmatrices,Ξ ¯nßMà¦µçáãµäèå , definedas

à ¯ÂééééééÃ 1 0 0

0 0 00 0 00 0 00 0 10 1 0

Ä'êêêêêêÅ µ á ¯ÂééééééÃ 0 0 0

0 1 00 0 00 0 10 0 01 0 0

Ä'êêêêêêÅ µ à ¯ÂééééééÃ 0 0 0

0 0 00 0 10 1 01 0 00 0 0

Ä'êêêêêêÅ µ13Thedomainis requiredto beregular, soit degeneratesin onesegmentonly. In caseof some

delaminationanalysis,multiple segmentsmaybeconsidered,i.e. a discontinuityin thedomainisallowed,providedthereis someconfiguration-dependent forcethatkeepsthesegmentstogether.


yielding

ε ¯ Ξ ¿ ν » κ ¼ t» ρ0 ¼ ¿ t ¸ t0 À » ∇ ¿ t ¸ t0 ÀÀ ²

Thelinearisationof thestrainsyields

∆ε ¯ Ξ ¿ ∆ν ¸ t ¼ ∆κ » ¿ ρ ¼ » ∇ ¿ Ø'ÀÀ ∆t À µwhile their virtual variationresultsin

δε ¯ Ξ ¿ δν ¸ t ¼ δκ » ¿ ρ ¼ » ∇ ¿ ØzÀWÀ δt À ²Thevirtual variationof thepositionsandof its gradientarerequiredfor thecom-putationof theexternalwork:

δs ¯ δp» δϕ ¼ t

»Rδt µ

and

∇δs ¯ ∇δp» ∇δϕ ¼ t

» δϕ ¼ ρ ¼ t» δϕ ¼ R∇t

» ρ ¼ Rδt»

R∇δt µwhich,by exchangingtheorderof application14 of operators∇ ¿ ØzÀ andδ ¿ Ø'À , resultsin

∇δs ¯ δl» δρ ¼ t

» ρ ¼ δϕ ¼ t» δϕ ¼ R∇t

» ρ ¼ Rδt»

Rδ∇t ²By taking a virtual variationof the referenceplanestrain ν, δl canbe replacedwith Rδν » δϕ ¼ l , while by consideringavirtual variationof theelasticcurvatureκ, δρ is replacedby Rδκ » δϕ ¼ ρ, finally yielding

∇δs ¯ Rδν » ¿ Rδκ À ¼ t» δϕ ¼ ¿ l » ρ ¼ t

»R∇t À » ρ ¼ Rδt

»Rδ∇t ²


The internalwork per unit volumeis δεTσ; its linearisationresultsin δ∆εTσ »δεT∂σ ë ∂ε∆ε ² By mutuatingthesymbolsfrom thebeamsectionanalysisformula-tion, theinternalwork perunit volumecanbewrittenas

δεT σ ¯ ÆììììÇ ììììÈδt × ξδt × ηδt

δψξδψη

Ë ììììÌììììÍT ÆìììììÇ ìììììÈ

Σ ítξΣ ítηΣ ítΣ íψξ

Σ íψη

Ë ìììììÌìììììÍ14It is permitted,providedtheconfigurationof theplateis sufficiently regular.


andits linearisationresultsin

∆ · δεT σ ¹ ¯ ÆììììÇ ììììÈδt × ξδt × ηδt

δψξδψη

Ë ììììÌììììÍTÂéééééÃ Ú íξξ

Ú íξη Û³íξ Üîíξξ ÜîíξηÚ íηη Û³íη Üîíηξ ÜîíηηÝïí ®¡íξ ®¡íηsym² Þ¢íξξ Þ¢íξηÞíηη

Ä êêêêêÅ ÆììììÇ ììììÈ∆t × ξ∆t × η∆t

∆ψξ∆ψη

Ë ììììÌììììÍ µthe star ¿ ØzÀ í indicatingthat the submatrices,definedin AppendixF, areper unitvolume. The generaliseddeformationsψ ¯ß ν µ κ å have beenusedfor easeofnotation.

5.6.3 External work

Theexternalwork is madeof two parts,asdescribedfor theplatesectionanalysis:theinternalforcestransmissionwork andthedeadloadswork,

∂2

∂ξ∂ηδLe ¯ ∂

∂ξ ð SδsTσSξ dS» ∂

∂η ð SδsTσSη dS» ð SδsTwV dS» ð c δsTwc Ø n dc

Due to the dimensionalityof the domainof the platefibre, their expressionsareverysimple

Transmissionwork

Thetransmissionwork basicallyconsistsin thesumof thegradientsin directionsξ andη of thework madeby avirtual variationof displacementagainsthestressesontherespectivesidesof thefibre,σSξ ¯ñà Tσ andσSη ¯má Tσ, or, in otherwords,by their in-planedivergence.Theintegrandsresultin· δsTσSξ ¹ × ξ ¯ δpT · ρξ ¼ σSξ

» σSξ × ξ ¹ » δνTξ σSξ × ξ» δϕT · lξ ¼ σSξ

»t × ξ ¼ σSξ

» ρξ ¼ t ¼ σSξ»

t ¼ σSξ × ξ ¹» δκTξ t ¼ σSξ» δtT σSξ × ξ » δtT× ξσSξ µ· δsTσSη ¹ × η ¯ δpT · ρη ¼ σSη

» σSη × η ¹ » δνTη σSη × η» δϕT · lη ¼ σSη

»t × η ¼ σSη

» ρη ¼ t ¼ σSη»

t ¼ σSη × η ¹» δκTη t ¼ σSη» δtT σSη × η » δtT× ησSη ²


By notingthatthe p, ϕ, l , ρ, ν, κ donotdependonζ, theintegrationvariable,andby defining

Fξ ¯ ð SσSξ dSµFη ¯ ð SσSη dSµMξ ¯ ð St ¼ σSξ dSµMη ¯ ð St ¼ σSη dSµ

thetransmissionwork becomesð S ¶c· δsTσSξ ¹ × ξ » · δsTσSη ¹ × η º dS ¯ δpT · Fξ × ξ » Fη × η ¹» δνTξ Fξ» δνT

ηFη» δϕT · lξ ¼ Fξ»

Mξ × ξ » lη ¼ Fη»

Mη × η ¹ » δκTξ Mξ» δκT

ηMη» ð S ¶ δtT σSξ × ξ » δtT× ξσSξ» δtTσSη × η » δtT× ησSη º dS²

Notice that, by independentlyperturbingδp, the differential force equilibriumequation∇F ¯ 0 is obtained;at thesametime, by independentlyperturbingδϕ,themomentequilibriumequation¿ l ¼ À F » ∇M ¯ 0 is obtained.

Deadloadswork

The deadloadswork doesnot requireany specialtreatment. It is importanttonoticethatthework of theforcesperunit volumeis simply representedby a lineintegralalongS, while theboundaryof S, c, is simplyconstitutedof thetwo pointsthatrepresenttheintersectionof Swith theupperandlowerfacesof theplate,andthustheintegraldegeneratesin two collocatedevaluations,ð c δsTwc Ø n dc ¯ sTwc òò u » sTwc òò lwheren is thenormalto thesurfaceof theplate,subscripts¿ Ø'À u and ¿ Ø'À l standingfor upperandlower. Becausethedeadloadsdo not participatein thecharacteri-sationof theplatefibre, they arenot discussedhere. They canbe formulatedinanalogywith thoseof thebeam,describedin AppendixE; however, they arenotdiscussedheresincetherehasbeennoplateapplicationyet in thepresentresearchwork.



The warping is discretisedby interpolatingthe nodalvaluesin a finite elementsense.Thenodalwarpingunknownsu areusedin conjunctionwith isoparametricshapefunctionsN to approximatethe warpingt ¯ N ¿ ζ À u ¿ ξ µ η À . The discretisedinternalwork canbe written in integral form, sincethe integrationis performedwith respectto ζ, andthenodalunknownsarecarriedoutof theintegraloperator,yielding

∂2

∂ξ∂ηδ∆Li ¯ ÆììììÇ ììììÈ

δu× ξδu× ηδu

δψξδψη

Ë ììììÌììììÍT ÂééééÃ Ú ξξ

Úξη Û ξ Ü ξξ Ü ξηÚηη Û η Ü ηξ Ü ηηÝ ® ξ ® η

sym² Þ ξξ Þ ξηÞ ηη

Ä êêêêÅ ÆììììÇ ììììÈ∆u× ξ∆u× η∆u

∆ψξ∆ψη

Ë ììììÌììììÍ ²The external work integrals relatedto the warping can be carriedout as well,yieldingð S ¶· δtT σSξ ¹ × ξ » · δtT σSη ¹ × η º dS ¯ · δuTPξ ¹ × ξ » · δuTPη ¹ × η µwherePξ, Pη arethenodalforcescorrespondingto theshapefunctionsusedin thediscretisation.

5.6.5 Solution

Theproblemresultsfrom equatinga perturbationof internalandexternalworkÂééééÃ Ú ξξÚ

ξη Û ξ Ü ξξ Ü ξηÚηη Û η Ü ηξ Ü ηηÝ ® ξ ® η

sym² Þ ξξ Þ ξηÞ ηη

Ä êêêêÅ ÆììììÇ ììììÈ∆u× ξ∆u× η∆u

∆ψξ∆ψη

Ë ììììÌììììÍ ¯ÆììììÇ ììììÈ

∆Pξ∆Pη

∆Pξ × ξ » ∆Pη × η∆ϑξ∆ϑη

Ë ììììÌììììÍ ²By assuminga linearapproximationof p in theform p Ñ¯ lξξ » lηη, andby con-sideringthefollowing indicial fluxesof internalforces

∆ϑξ ¯ ° I 0· lξξ ¹ ¼ I ± ¾ RTFξ0RTMξ0 Á µ

∆ϑη ¯ ° I 0· lηη ¹ ¼ I ± ¾ RTFη0

RTMη0 Á µ


asolutionis soughtof theform

∆u ¯ ¶ U0» · lξξ ¹ ¼ Uξ

1» · lηη ¹ ¼ Uη

1 º ϑ0 µ∆Pξ ¯ ¶ Pξ0

» · lξξ ¹ ¼ Pξξ1» · lηη ¹ ¼ Pη

ξ1 º ϑ0 µ∆Pη ¯ ¶ Pη0

» · lξξ ¹ ¼ Pξη1» · lηη ¹ ¼ Pη

η1 º ϑ0 µ∆ψξ ¯ ¶ Ψξ0

» · lξξ ¹ ¼ Ψξξ1» · lηη ¹ ¼ Ψη

ξ1 º ϑ0 µ∆ψη ¯ ¶ Ψη0

» · lξξ ¹ ¼ Ψξη1» · lηη ¹ ¼ Ψη

η1 º ϑ0 µwhereϑ0 ¯ Õ ϑξ0 µ ϑη0 Ö µ yielding

∆u× ξ ¯ lξ ¼ Uξ1 ϑ0 µ

∆u× η ¯ lη ¼ Uη1 ϑ0 µ

∆Pξ × ξ ¯ lξ ¼ Pξξ1ϑ0 µ

∆Pξ × η ¯ lη ¼ Pηξ1ϑ0 µ

∆Pη × ξ ¯ lξ ¼ Pξη1ϑ0 µ

∆Pη × η ¯ lη ¼ Pηη1ϑ0 ²

Actually, while thedrill degreeof freedomis consideredfor completenessof theformulas,it is undetermined,beingthedrill problemsingular;thematrix will beconstrainedprior to solution. The indicial fluxesof internalforcesmustbeaug-mentedby six self-balancedfluxes,threesetsof forcesandthreesetsof moments,that will be usedto restorethe compatibility of the deformations,asmentionedin Section5.5.4. By enforcingthe compatibility equations,Eqs.5.19,5.20, theindeterminationrelatedto thesix self-balancedsolutionsthatareallowedby thedifferentialequilibriumequations,Eqs.5.17,5.18,andthatmustbeaddedto thesetof 12 trial indicial load modes,will be reduced,resultingin a setof twelvecompatibleaswell asbalancedsolutionsto indicial loads.This topic will bedis-cussedin thenext section;from thesolutionstandpointit doesnotaddany furthercomplexity, only thesolutionis madeof 18 insteadof 12 loadmodes.Theproblemis solvedfirst in thelinearunknowns,yieldingÂÃ Ý ® ξ ® ηÞ ξξ Þ ξη

sym² Þ ηη ÄÅ ÆììÇ ììÈ lξ ¼ Uξ1

lξ ¼ Ψξξ1

lξ ¼ Ψξη1Ë ììÌììÍ ¯ ÆìÇ ìÈ

0

Θξξ1

Θξη1Ë ìÌìÍ


and ÂÃ Ý ® ξ ® ηÞ ξξ Þ ξηsym² Þ ηη ÄÅ ÆìÇ ìÈ lη ¼ Uη

1lη ¼ Ψη

ξ1

lη ¼ Ψηη1 Ë ìÌìÍ ¯ ÆÇÈ 0

Θηξ1

Θηη1 ËÌÍ ²

The linearsolutionsarecombinedto yield the right-handof theconstantpartoftheproblemÂÃ Ý ® ξ ® ηÞ ξξ Þ ξη

sym² Þ ηη ÄÅ ÆÇ È U0

Ψξ0Ψη0 ËÌÍ ¯ ÆÇ È 0

Θξ0Θη0 ËÌÍ» ÂÃ Û ξ ¸ Û T

ξ Ü ξξ Ü ξη0 0

skw² 0 ÄÅ ÆììÇ ììÈ lξ ¼ Uξ1

lξ ¼ Ψξξ1

lξ ¼ Ψξη1Ë ììÌììÍ» ÂÃ Û η ¸ Û T

η Ü ηξ Ü ηη0 0

skw² 0 ÄÅ ÆìÇ ìÈ lξ ¼ Uη1

lξ ¼ Ψηξ1

lξ ¼ Ψηη1 Ë ìÌìÍ ²

Thesolutionrequiresthematrix to bestaticallydeterminedby addingextra con-straints,becausethe warping functionscontainthe rigid displacementsalreadyheld by the generaliseddegreesof freedomof the fibre. The drill degree offreedommustbe constrainedeven in the generalisedunknowns becausea one-dimensionalmodelof thefibredoesnothaveany drill striffness.

5.6.6 Compatibility enforcement

Thecompatibilityconstraintsof Eqs.5.19,5.20appliedto a perturbationof gen-eraliseddeformationsstates

ρη ¼ ∆νξ»

lη ¼ ∆κξ ¸ ρξ ¼ ∆νη ¸ lξ ¼ ∆κη» ∆νξ × η ¸ ∆νη × ξ ¯ 0 µ

∆κξ × η ¸ ∆κη × ξ ¯ 0 ²Applied to thepreviouslydeterminedsetof solutions,thecompatibilityconditionyields ° ρη ¼ lη ¼

0 0 ± Ψξ0 ¸ ° ρξ ¼ lξ ¼0 0 ± Ψη0

» Ψηξ1 ¸ Ψξ

η1 ¯ 0 ²


-h/2

0

h/2

-1 0 1 2 3 4 5 6

thic

knes

s

stress

stress yz

unit transverse shearself-bal. x momentself-bal. y moment

self-bal. x forceself-bal. y force

compatible unittransverse shear

Figure5.2: Transverseshearstressdistribution— compatiblevs.non-compatibleandself-balancedsolutions.

MatricesΨ represent18 balancedsolutions,including 6 self-balancedsolutionsthatcanbelinearlycombinedto theothersto obtain12compatiblesolutions.Ac-tually only four solutionsareused,theremainingtwo referringto thedrill degreeof freedomandthusbeingconstrainedto avoid singularities.As anexample,thetransverseshearstressdistribution throughthe thicknessof a 0/90/0/90laminateresultingfrom thecompatibilityenforcementprocedure,comparedto thoseduetothe non-compatiblesolutionandto the self-balancedsolutions,is shown in Fig-ure 5.2. The self-balancedmodesandthe compatibility enforcementprocedurearedescribedin detail in [33].

5.6.7 Characterisation of the fibr e

After thesolutionis madecompatibile,it canbeusedto computethefibreproper-ties. They resultfrom substitutingthe indicial solutionsinto thedefinitionof thediscretisedinternalwork, thusyielding thecompliancematrixof thefibre

∂2

∂ξ∂η∆δLi ¯ δϑT

0C∆ϑ0 ²Theusualstiffnessconstitutive matrix D canbeobtainedfrom C by inverting it.Thedirectinversionis notpossiblesincematrixC is singularbecauseit showsnullcompliancefor thedrill degreeof freedom;this solutionis meaningless,because

5.7. FINITE VOLUME PLATE 69

the drill degreeof freedomwasarbitrarily constrainedto allow the solution ofthe problem. The compliancematrix canbe pseudo-inverted,or the generaliseddeformationscanbe condensedto determinethe minimum setof 8 independentdeformationsof theplate: the threemembranestrains,the two transverseshearsandthethreecurvatures15.

5.7 Finite VolumePlate

The finite volumeplateandshell formulationresultsfrom the direct writing ofthe equilibrium of a finite portion of plate. By dividing a plate in subportions,theequilibriumof eachportioncanbewrittenby integratingthefluxesof internalforcesalongtheboundaryof theportionandby integratingthedistributedloadson thesurfaceandin thevolumeof theplate.By expressingtheinternalfluxesintermsof nodaldisplacementsandrotationsin afinite elementway, theusualstiff-nessmatrix results.Themainadvantagesof thefinite volumeformulationarethereducedlocking effect if comparedto correspondingfinite elementformulations,andthereducedintegrationorder, namelya line insteadof asurfaceintegration.

5.7.1 Finite Equilibrium

Theweakequilibriumof the finite plateis written by integratingthedifferentialequilibriumequationsweightedby apiecewiseconstantweightfunction,asmen-tionedfor the finite volumebeam. A portion of a generalplate,representedbydomainBc, supportsaweightfunctionw, thatassumesunit valuein Bc andis nulloutsideBc; asa result,its gradientis null everywhereexceptat theborderof Bc,whereit is representedby a Diracδ thatmultipliesa vectornormalto thebound-ary c anddirectedinside.Theequilibriumequationsof theplate,Eqs.5.17,5.18,multiplied by w andintegratedoverBc yieldð Bc

w · ∇TF» φ ¹ dS ¯ 0 µ (5.21)ð Bc

w ¶ ∇TM» ¿ l ¼ À T F

»µº dS ¯ 0 µ (5.22)

wherethedistributedforcesandcouplesφ, µhavebeenadded.Theproductw∇TFcanbetransformedin

w∇TF ¯ ∇T ¿ wF À ¸ ¿ ∇wÀ T F µ15Two arethebendingcurvatures,andoneis the twist curvature;the latter is symmetric:it is

thesamein thetwo directions,asthemembraneshearstrainis.


where∇w ¯ ¸ δ ¿ p ¸ cÀ ¿ ∇cÀ ¼ . Theintegrationby partsof w∇TF yields

ð Bc

w∇TF dS ¯ 1ë 2 ð c ¿ ∇cÀ ¼ F dc ¸ ð Bc¿ ∇wÀ T F dSµ

whereStokestheoremhasbeenappliedto thedivergenceterm. It is easyto provethatthetwo integralsat right-handsimply resultin

ð Bc

w∇TF dS ¯ ð c ¿ ∇cÀ ¼ F dc

The integrationof the momentequilibrium equationis a bit moreawkward be-causeof theforcemomentterm,but, by notingthat

w∇TM ¯ ∇T ¿ wM À ¸ ¿ ∇wÀ T M µw ¿ l ¼ À T F ¯ ∇T ¿ wp ¼ F À ¸ ¿ ∇wÀ T p ¼ F ¸ wp ¼ ∇TF µ

wherel ¯ ∇p hasbeenused,and,by using∇F» φ ¯ 0, theintegrandin Eq.5.22

resultsin

w ¶ ∇TM» ¿ l ¼ À T F

»µº ¯ ∇T ¿ wM

»wp ¼ F À¸ ¿ ∇wÀ T ¿ M » p ¼ F À » wp ¼ φ ²

Thesameconsiderationsmadefor theforceequilibriumintegrationby parthold;in detail,thesurfaceintegrationreducesto anintegrationontheboundaryc of Bc.Thetwo equilibriumequations,Eqs5.21,5.22resultinð c ¿ ∇cÀ ¼ F dc ¯ ¸ ð Bc

φ dSµ (5.23)ð c ¿ ∇cÀ ¼ ¿ M » p ¼ F À dc ¯ ¸ ð Bc¿ µ » p ¼ φ À dS² (5.24)

The two equationscanbe condensedin oneby extendingto the plate the armsmatrix ´ andthegeneralisedinternalforcefluxesϑ, resultingin

ð c ¿ ∇cÀ ¼ ´ ¿ p ¸ x0 À ϑ dc ¯ ¸ ð Bc

´ ¿ p ¸ x0 À τ dSµwhereτ arethegeneraliseddistributedloads,andthepolex0 hasbeenintroducedin themomentarm;it is arbitrary, becausep resultsfrom theintegrationof l .

5.7. FINITE VOLUME PLATE 71

5.7.2 Compatibility

As mentionedearlier, thegeneralisedstrains,asdefinedin Section5.5.2,do notdescribea puredeformation,sincethey containa rigid rotation. Their symme-try must be enforced,by addingthe non-rotationalitycondition to the equilib-rium equations.A penaltyfunction approachis considered.Considerthe non-rotationalityconditionof Eqs.5.19,5.20,weightedby thesamefunctionusedforthe equilibrium equationsandintegratedover Bc; in particular, consider, for theweightingfunction,thestructure16 W ¯nß 0 µ 0 µ Iw å . It yieldsð Bc

WT∇ ¼ ν dS ¯ 0 µð Bc

WT∇ ¼ κ dS ¯ 0 ²ConsiderWT∇ ¼ ν first; the sameconsiderationshold for κ. By applying theproductdifferentiationrule, it becomes

WT∇ ¼ ν ¯ ¸ ∇T ¿W ¼ ν À » ¿ ∇ ¼ W À T ν ²By integratingby parts,andby applyingStokestheoremto thedivergenceterm,ityieldsð Bc

WT∇ ¼ ν dS ¯ ¸ 1ë 2 ð c ¿ ∇cÀ T ν dc» ð Bc¿ ∇ ¼ W À T ν dS²

It canbeeasilyshown thatthetwo integralsat right-handareequal,thusyieldingð Bc

WT∇ ¼ ν dS ¯ ¸ ð c ¿ ∇cÀ T ν dc ²Thustwo compatibilityequationsresult:ð c ¿ ∇cÀ T ν dc ¯ 0 µð c ¿ ∇cÀ T κ dc ¯ 0;

they can be addedto the correspondingequilibrium equationsby meansof apenaltyfunction γ that dimensionallyis a stiffness(a force per unit length forν anda momentfor κ), or in form of algebraicconstraints,in a Lagrangianmul-tiplierssense.

16A moregeneralstructurewould beW ó¦ª Iwô Iwô Iw « , but it canbe easilyshown that, sincew, aswell asν andκ, doesnot dependon the abscissain direction3, only the third componentof W is requiredto obtainthethird component,theonly non-nullone,of theresultof operationsinvolving secondorderentities,like∇ õ ν.


5.7.3 Implementation notes

Thefinite volumeplateor shellhasnotbeenimplementedyet; for this reason,nodiscretisationnor linearisationis presented.Theformulationhasbeendevelopedandinvestigated,but atafirst glancesuchaflexible elementhasnotbeenretainedsignificantfor rotorcraftanalysis,at leastat thelevel of detail requiredby a con-ventionalmultibodyanalysis.It might beusefulfor theanalysisof complicated,hingelessor bearinglessrotor hubs,which sometimesusevery flat componentsto implementstiff-in-plane,torsionallyweakbladeroots. Basicallyit resultedasa spin-off of the beamformulation,asit is apparentthat mostof the resultscanbeobtainedfrom thebeamformulationby simply exchangingthesupportof thestructuralcomponentandthatof thesection,andby appropriatelyredefiningtheintegralanddifferentialoperators.Itsdevelopmentwill beprosecutedin thefuturefor moregeneralapplications.

5.8 Modal flexibility

A generalwayto introduceflexible behaviour in amultibodymodelis throughtheuseof modes.Historically this hasbeenthefirst approachfollowed in commer-cial codes[81]. Advantagesof themodalapproachare:1) theability to describethe behaviour of complex structuralcomponents,not reconductibleto beamsorplates,with a comparatively low numberof degreesof freedom;2) theability tofocuson thedesiredrangeof frequencies,thusallowing theuseof explicit inte-grationschemeswith stepsizecontrolto ensurethestabilityof theintegration;3)theeasein writing problemsin relative coordinates,which makestheanalysisofopen-loop,or tree-like,mechanismsverycompactandefficient.

The drawbacksmainly are: 1) the small flexible displacementimplicit assump-tion, to allow the linear combinationof modes;2) the restrictionto linear elas-tic structuralcomponents;3) the poor descriptionof the effect of concentratedloads/massesif dynamicmodesareconsideredonly, without enrichingthemodalbasewith appropriatestaticshapes.

In theoutlinedrotorcraftanalysisthemodalflexibility canbeusefulif theflexi-bility of themainbody of thehelicopter, or thewing-fuselageof a tiltrotor is tobeconsidered,sincea finite element-like discretisationis unreasonablewhenthedynamicsof therotorarethemainfocusof theanalysis.As oftenaddressedin theliterature,theflexibility of therotor supportandof the fuselagearefundamentalto correctlypredict the vibratory level in the cockpit [26]. No modalflexibilityhasbeenusedin theanalysespresentedin this work; however it is beingimple-

5.9. AERODYNAMIC FORCES 73

mentedin thecodethathasbeendeveloped,andits usewill beinvestigatedin thenearfuture. Thereis no practicalinterestin resortingto a modalmodelfor rotorblades,becausethe finite elementapproachis expectedto allow a moregeneralmodellingwith theability to accountfor geometricnonlinearitiesin a morereal-istic way thanmodesare. As a consequence,the modesareseenasa possible,reasonablechoicewhena trade-off betweenthe level of detailandtheefficiencyof the computationmust be reached. Anyway the questionis still open,sincemayresearchersandmany researchrotorcraftcodesinsist in usingmodalmodelsfor theblades,seefor instanceChopra’sUMARC [57] andJohnson’sCAMRAD[61, 62], asopposedto thefinite elementapproachof Bauchau’sDYMORE [7].

5.9 Aerodynamic forces

Thetaskof modellingtheaerodynamicforcesthatacton a rotorcraft,especiallyduring unsteadyadvancingmotion, is formidable. Whenattentionneedsbe fo-cusedon very specificphenomena,like Blade-Vortex Interaction(BVI), noisegeneration,interactionof the main rotor wake with the fuselageor the tail ro-tor, a good model of the unsteadywake is mandatory. Unfortunately, the taskof modellingtheunsteadyaerodynamicsof anadvancingrotorcraftis still imma-ture, asshown for instanceby the only partially satisfactoryresultsobtainedinRef. [16], wheresomestate-of-the-artrotorcraftanalysiscodeshavebeenusedtoinvestigatethebladeloadsof a Puma helicopterwith increasinglysophisticatedaerodynamicmodels,includingComputationalFluid Dynamics (CFD) determi-nationof the pressuredistribution on a sweptbladetip, to matchexperimentalmeasures.

5.9.1 Strip-theory, quasi-steadyaerodynamic forces

A brief descriptionof the aerodynamicmodelusedin this work is presented17.The lifting devices,suchasrotor bladesandaircraft wings andcontrol surfacesare modelledby the strip-theory, with steadyor quasi-steadyaerodynamicco-efficients,asdescribedin Ref. [54]. The coefficients accountfor Mach effectby meansof Glauert’s correction,with furthercorrection,i.e. reducedlift andin-creaseddrag,for veryhighlysubsonicMachnumbers.They alsoaccountfor static

17It might seemsurprisingthat a couplepagesonly are spentaboutaerodynamicswhen anentiredissertationon themodellingof rotorcraftis presented.Therearemany reasons,which canbe summarisedin a few words: the dissertationpresentswhat is novel, or hasbeendevelopedduring thecurrentresearchactivity, or at leastis key to thecomprehensionof what is presented.Nothingnew is beingpresentedin themodellingof theaerodynamicsof rotorcraft,soonly abriefsummaryof themodelsthathavebeenusedis included.


post-stallcorrectionandreverseflow, andfor dynamicstallcorrectionin anempir-ical manner;thedragresultingfrom radialflow is alsoconsidered.Theboundaryconditionsfor theaerodynamicforces,i.e. theconfigurationof theblade/wingsta-tionsatwhich theforcesarecomputed,resultfrom thecombinationof themotionof the referencepoint at the desiredstation,with the additionof a referenceve-locity thatrepresentstheasymptoticairstreamspeedandof theeffectsof spatiallyresolvedgusts

v ¯ vb»

V∞»

vG µω ¯ ωb µ

wheresubscriptsb, ∞ andG respectively referto body, asymptoticandgustquan-tities. A rigid-surface,lumpedaerodynamicelement,anda beamaerodynamicelementhavebeenformulated.They only differ in themannertheboundarycon-ditionsarecomputedandtheloadsareappliedto thestructure,sothesimplecaseof lumpedaerodynamicforceswill be considered;the othercasecanbe easilyobtainedby substitutingtherigid-bodyvelocitieswith thoseobtainedby interpo-lating the velocitiesof a pool of nodes. The velocity of a point at a prescribedstationis

vb ¯ x» ω ¼ f µ

wherex andω arethevelocity andtheangularvelocity of thenode,and f is theoffset from thenodeto thestation. A referenceframeon theairfoil is relatedtothe nodeframe, identifiedby matrix R, by a rotationmatrix Ra, that is usedtobring thesectionvelocitiesin theairfoil frame.Thelocal velocitiesarethus

vb ¯ RTa RTvb

ω ¯ RTa RTω

An aerodynamicoperator Þ ¿ v µ ω À is usedto computethe forces and couplesÕ F µ M Ö in the aerodynamicreferenceframe,which aretransformedbackto theglobalframeandappliedto thenode,yielding

F ¯ RRaF

M ¯ f ¼ RRaF»

RRaM

This aerodynamicmodelis not sophisticated,andshouldbeconsideredonly asameansto introducethefundamentalaerodynamicloads,irrespectiveof any mem-ory effects,suchasthe wake, the BVI, andso on. Sucha model is consideredsufficient for instantaneousstabilityanalysis,i.e. local linearisationandlocal sta-bility assessment,while, for a global, or periodicstability analysisof the rotor,a moredetailedandsophisticatedmodel is mandatory. It is not the aim of thiswork to developsucha model,so thevery first approximationgivenby thestriptheoryis acceptedbecauseit is consideredsufficient for theanalysisof theflightconditionsweareinterestedin.

5.9. AERODYNAMIC FORCES 75

5.9.2 Induced velocity

A simplecorrectionto thesteady, collocatedevaluationof theaerodynamicforcesis given by the modellingof the inflow causedby the rotor. A dynamicinflowmodelhasbeenconsidered,basedon the work of Peters [76]. A dynamicsys-tem,madeof threeunknownsthat representthenon-dimensionalinducedveloc-ity and its first-ordermomentsin pitch and roll directions,is written, with therotor thrustandthepitchandroll momentsasforcingterms.Theinducedvelocityat eachstationthusdependson the radial andazimuthalposition,andpreservessomememoryof theglobal forcesactingon the rotor. Ideally it is addedto theasymptoticsteamvelocity whenforming the input for theaerodynamicoperator,so the presenceof an inflow model is transparentto the aerodynamicoperator.Theglobal thrustandmomentsarecomputedby addingthecontribution of eachaerodynamicelement,referredto the frameof the mast. More sophisticatedin-flow modelshave beenformulated,with higherordermomentaandbetterspatialresolutionof theinducedvelocity, but, dueto theintrinsic approximationof suchmethod,theincreasein accuracy is notconsideredto besignificantto theproblemunderinvestigation.


Part II

Control

77

Chapter 6

Rotorcraft control

Thecontrolof arotorcraftmayassumeratherdifferentmeanings.Therearemanyinterestingaspectsof theflight envelopeof a rotorcraftsomeform of controlcanbeappliedto, rangingfrom thecontrolof theflight path,usuallyaccomplishedbythepilot but with increasingautomatedassistance,to thereductionof vibrationsinthecockpitandin critical components,e.g.avionics,to loadsalleviation,noisere-duction,fluttersuppression,andmore.All thesegoalshavedifferentrequirementsandpossiblyrequiredifferentarrangementsto bemade.Dif ferentsolutionshavebeenproposed,andarebeingproposedatpresent;therequirementthatrepresentsa sortof commondenominatoris theneedto beeffective in a wide spectrumofflight conditionsthatresultin ahighly varyingbehaviour of thecontrolledsystem.

6.1 Intr oduction

This work is focusedon load alleviation andvibration reduction. It is a funda-mentaltask,sincethehelicopteris a restlesssourceof vibrations,dueto thewaythrustis generated.Theblademotion is periodic,so,whenin forwardflight, thecombinationof the bladerotation and of the helicopteradvancespeedsresultsin a periodic flow field on the blades. This on turn producesload unbalance,whoseoneperrev. harmonicis cancelledby therotor flappingmotion. However,higher-harmonicsof therotationspeedaretransmittedthroughthemastto therestof the rotorcraft. In caseof ideal blades,only the Nb/rev. andhigherharmonicsof the thrustaretransmitted,beingNb the numberof blades,while small unbal-ancesin theinertialor aerodynamicpropertiesof thebladescancauseappreciablevibratorylevel evenat 1/rev. Thevibrationsresultin reducedlife of critical com-ponents,significantlyavionics, in reducedfatiguelife of structuralcomponents,reducedoperabilityof the crew andreducedcomfort for the passengersandthepayloadin general.Thecancellationof thevibrationsis fundamentalfor a better

79

80 CHAPTER6. ROTORCRAFTCONTROL

exploitationof rotorcraftfor civil andmilitary use.Theproblemhasgainedatten-tion in the‘70s, whenthefirst studieson active vibrationreduction,significantlyby meansof Higher-HarmonicControl (HHC), began[78, 66,93]. Thecurrentlyacceptedmeaningof HHC is thatof asimultaneouscontrolof thebladesby meansof theconventionalbladepitch controlmechanism,theswashplate.Recently, at-tentionmovedtowardsmoreversatileandefficient controltechniques,which canbegenericallyidentifiedasIndividual BladeControl (IBC), see[53, 52, 26, 45]amongtheothers,consistingin independentlyactuatingeachblade;nonetheless,HHC is still considered[72, 79, 74]. Perhapssomeconfusionexists in thetermi-nology, becauseusuallyHHC is associatedto periodiccontrol,sincethesimulta-neousactuationof thebladesresultsin superimposingperiodicpitchingmotionsthatcanbedifferentfrom bladeto bladeby properlychoosingthefrequency andthe phaseof eachinput, while IBC, usually associatedto completelyunrelatedcontrol for eachblade,canbeimplementedto obtaina periodic,harmonicactua-tion of theblades.Moreover, while HHC is commonlyrelatedto theswashplateasactuator, thereis no suchclearidentificationof the actuationtechniqueto beusedbestfor IBC. Very differentdeviceshave beenproposedandarecurrentlyinvestigated,and in somecasesimplemented. Thereare threebasicsolutions,shown in Figure6.1,consistingin

1. deflectingatrailingedgetrim tabin theouterportionof thebladethatcausestheflexible bladeto twist, thuschangingits angleof attackin a distributedmanner;

2. applyinga pitch changeat theroot of theblade,e.g.by varyingthe lengthof thepitch link;

3. directly twisting thebladeby meansof distributedinducedstrainactuators.

Theseactuationsolutionscanbeobtainedin differentways.

6.2 Trim tab

The trim tab solutionis very attractive andis undergoing intensive investigationby many differentresearchteams.Many wind tunnelrotor modelsexploiting theactive trim tabhave beenbuilt andtestedin recentyears[87, 70,71]; a review ofthestateof theartcanbefoundin [26, 45,47]. Dif ferentmeansof controllingthetabhavebeenexplored;themostpromisingconsistingin theuseof smartmateri-alsto controlaconventionaltrailing edge,hingedtab. While lessthan ö 2 degreesof pitchareenoughto achieveareasonablecontrolauthoritywhencontrollingthepitch of thewholeblade,at least ö 15 degreesof excursionareneededto obtain

6.2. TRIM TAB 81

Trailing edge flap

Blade pitch

Blade twist

Figure6.1: Rotorbladeactuationtechniques.


effective controlby a trim-tab[45]. In this caseonly little power is required,butit mustbeappliedto a really largestroke, sovariousstroke amplificationmech-anismsarebeing investigated,sincesmartmaterialssuchaspiezoelectrics,andelectro-andmagneto-strictive materials,the bestcandidatesfor the actuationofthe tab, show a very limited maximumstroke. While most of the stroke am-plification schemesrely on mechanicaldevices,suchasbimorphpiezoelectrics,piezo-stackdrivenscissors,andsoon,averyinterestingideais representedby thepiezoelectricpump[46]. A piezoelectricdevice is usedto pumphydraulicfluidin a largesectioncylinder by a smallstroke piston,to drivea smallsection,largestrokeactuator. A torsionalactuatorhasbeenobtainedby axially loadinganopensectioncylinderwith apiezo-stackat oneendto obtaina rotationat theotherend[47]. The ideaof controllingthecamberof thebladeby strain-inducedactuatorsis facingthearena,but sinceit provedto bea challengeevenfor thefixed-wing[92], it maybestill too immaturefor rotorcraftapplication.

6.3 Bladepitch control

For the direct control of the bladepitch, both high-frequency hydraulicand in-ducedstrainactuatorsare investigated.The comparatively high frequenciesre-quiredto control the higher-harmonicloadscanbeobtainedby hydraulicactua-torswhenonly limited stroke is required[45]; hydraulicactuatorscanreachup to50 ÷ 60Hz,whichmeans10 ÷ 12/rev. in actualhelicopters,for limitedstroke. Thisis thecaseof thesmallpitch amplitudesthatneedbesuperimposedto theflight-control relatedpitch angles,i.e. thecollective andthecyclic pitch. Piezoelectricstackshavebeenproposedaswell, sincethey haveaverybroadbandof operationanddonotrequireany hydraulicpowerto bebroughtin therotatingsystem.Thereis no airborneIBC systemyet; oneform of IBC that is enteringproductionis theTotal Vibration Reduction(TVR) by Kawasaki[4], which relieson actuatorsatthe rearmountsof thegearbox for the reductionof higher-harmonicvibrations,while thelengthof thepitch links canbeadjustedby meansof a jackscrew, elec-trically actuatedatvery low speed,to cancelthe1/rev. vibrationscausedby bladeunbalance.Themajorunknown of anactivepitchcontroldevice is thereliability;in fact a failure of the systemcould result in the completelossof pitch control;thedeviceshouldlock thebladepitch in caseof failure,andshouldallow at leastthepitch controlrelatedto flight control,i.e. collectiveandcyclic.

6.4. INDUCED TWIST ACTUATION 83

6.4 Induced twist actuation

Themostchallengingwayof controllingthepitchof thebladeis by directly twist-ing thewholebladeby inducedstrainactuators.Thestrain-inducedtwisting of aslenderbody, like a rotor blade,requirestheability to induceshearstrainsin thebeamsection,that resultin a global twist of thestructure.A very promisingma-terial for distributedandembeddedinducedstrainactuationhasbeenfoundin thepiezoelectrics.Thepiezoelectriceffect is representedby the capabilityof a ma-terial to producean electricfield when loaded,and,on the contrary, to deformwhensubjectto anelectricfield. In usualnotation[3], thelinearisedpiezoelectricconstitutive law is:¾ S

D Á ¯° sÏ E Ð dT

d ε Ï T Ð ± ¾ TE Á (6.1)

whereS6 æ 1 andT6 æ 1 arethestrainandstressarrays,andE3 æ 1 andD3 æ 1 aretheelectric field and the electricdisplacementarrays. Piezoeletricdevices for dis-tributedinducedstrainapplicationsaremanufacturedin two-dimensionallaminæ,to beappliedon,or embeddedinto, thepassive,or host,structure.Theusualim-plementationconsistsin thin, thickness-wise(direction3) polarisedlaminæ,thatshow anisotropicbehaviour in theirplane,thusbeingunableto inducethedesiredtwisting. In fact,theshearstrainin theplaneof thepiezoelectriclaminais:

γxs ¯ 2 cos¿ α À sin ¿ α À ¿ S1 ¸ S2 À (6.2)

wheredirections lies in the planeyz of the beamsection,andx is the axis ofthebeam,while S1 andS2 aretheprincipal strainsin theplaneof thepiezoelec-tric in thematerialframeandα is therelativeanglebetweenthematerialandthebeamreferenceframes,resultingin no shearwhen the two inducedstrains,S1

andS2, areequal,ashappensfor planeisotropicpiezoelectricmaterials.An in-planeanisotropicpiezoelectricmaterialis requiredto obtaindifferentelectricallyinducedstrainsin directions1 and2. A materialwith suchpropertieshasbeenobtainedby applyingthefibre compositetechnologyto theceramicpiezoelectricmaterial,resultingin the ActiveFibre Composites(AFC) [8]. Previous researchresultedin thedevelopmentof theInter-DigitatedElectrodes(IDE) principle[48]asa usefulmeansto induceanisotropicin-planeactuation(Figure6.2). Theelec-trodesarealignedin pairsof conductive stripson both the upperandthe lowersurfaceof thepiezoelectric.Thepairsof stripsarealternatelychargedplanewise,so that an alternate,in-planeelectric field normal to the strips (direction 1) isgenerated.The piezoeletricmaterialis initially polarisedby meansof the stripsthemselvesduring the manufacturingprocess.The resultingdevice exploits themainpiezoelectriccouplingcoefficient (d11), which is usuallylargerthanthesec-ondaryones(d12ø 13) andoppositein sign (d12ø 13 ¯ ¸ kd11, with k ù 0 ² 2 ÷ 0 ² 5);


Epoxy Matrix

Piezoceramic Fibers

Inter Digitated Electrodes

1 2

3

Figure6.2: ActiveFibreCompositeswith Inter-DigitatedElectrodes.

boththefibresandthepolarisationarein direction1. Whenusedin conjunctionwith AFC, theinterdigitatedelectrodescanapplythecontroltensionexactlyin thedirectionof thefibre, thuscompletelydecouplingtheinducedstrainin thedirec-tion on thefibre from thestrainsin theotherdirections.Theelectricallyinducedstrainis appliedin thedirectionof thepiezoelectricfibresonly, resultingin atrulyanisotropicactuationin theplaneof theactiveply. Equation6.2shows thatwhenthefibresareoriented45o apartfrom thebeamaxis,andthe two normalstrainsdiffer, themaximalcoupledaxial-shearactuationof theply is obtained;theaxialactuationloadscanbe cancelledby stackingthe active plies oriented90o apartfrom eachother.Theinducedtwist actuationis beinginvestigated1 andwill bepresentedin Chap-ter 13. The characterisationof the materialandof the beamsection,followingthe beamsectioncharacterisationformulationdescribedin Section5.3, will bepresentedin Section9.1. With currentlyavailablepiezoelectricmaterials,theau-thority of a reasonable,embeddableinducedstrainactuationfor a rotor bladeisvery limited. Comparedto a minimal requirementof about ö 2 degreesof twistfrom root to tip [45, 80], about ö 0 ² 35 degreeshave beenobtainedby Chopra in1993with a minimumgoalof ö 1 deg. [18]2, usinga ratherdifferentmaterialbut

1Somepreliminaryresultshavebeenpresentedin [36]2Reference[18] hasnot beendirectly consulted,informationwereobtainedfrom Ref. [45].

6.4. INDUCED TWIST ACTUATION 85

exploiting the sameprinciple of embeddingan inplaneanisotropicpiezoelectricmaterialoriented45o apartfrom thebeamaxisto inducetwist; Rodgers andHa-good obtainedabout ö 1 ² 5 degreeswith AFC andIDE [80] andaboutthesameis expectedby Wilkie et al. [89, 90, 91], but at thecostof dramaticallyreducingthe torsionalstiffnessof the blade. This is necessarynot only to allow moder-atelylow authorityactuatorsto staticallytwist theblade,but alsoto movethefirsttorsionalresonancefrequency of thebladedown to the frequency bandat whichthecontrol is required. In this way theactuatoroperatescloseto resonance,ex-ploiting large twist angleswith limited controleffort but with appreciablephasedelay. This designgoalrequiresa very carefulinvestigationof theinfluenceof astiffnessreductionon theaeroelasticstability of therotor. Thelatter resultshavebeensubstantiallyconfirmedin [36] with acompletelyunrelatedanalysis.


Chapter 7

DiscreteControl

Thischapterdescribesthefundamentaltheoryof discretepredictivecontrol,whichhasbeenimplementedfor rotorcraftactive control. Predictive control hasbeenhistorically formulatedin discretetime form dueto its usualimplementationindigital control systems;the theory hasbeenformulatedfor continuoustime aswell [1]. Thegeneral,Multi-Input Multi-Output (MIMO) discretetime formula-tion proposedby Juang [63, 23, 64] will be usedthroughoutthe chapter;exten-sionswill be madeto colourednoise(moving average),measureddisturbances(feedforward),andarbitraryweightingof bothpredictionerrorandcontroleffortin the error function. The conventionaldiscretetime formulationhasbeenpre-servedin view of digital controlapplications[40].

87

88 CHAPTER7. DISCRETECONTROL

y ¿ k À measureat timeku ¿ k À input at timek (eitherimposedor from control)e¿ k À errorat timekf ¿ k À measurederrorat timekai i-th ordermeasureregressioncoefficientbi i-th orderinput regressioncoefficientci i-th ordererrorregressioncoefficientdi i-th ordermeasuredregressioncoefficientm numberof measures(andof errors)n numberof inputso numberof measurableerrorsp orderof themodelq advancingcontrolhorizonr recedingpredictionhorizons advancingpredictionhorizonΘ collectionof modelparametersϕ ¿ k À collectionof measuresat timeky ¿ k À predictedoutputat timekµ forgettingfactorλ control inputweightfunction

7.1 DiscreteTime Equation

A discretetime, Auto-Regressive, Moving Average,with eXogenousinput (AR-MAX) equationhastheform:

y ¿ k À ¯ a1y ¿ k ¸ 1À » ²W²W² » apy ¿ k ¸ pÀ»b0u ¿ k À » ²W²W² » bpu ¿ k ¸ pÀ»e¿ k À » c1e¿ k ¸ 1À » ²W²W² » cpe¿ k ¸ pÀ (7.1)»d0 f ¿ k À » ²²W² » dp f ¿ k ¸ pÀ µ

wherey ¿ t À , u ¿ t À arethe outputandinput arraysat time t, e¿ t À is the error arrayat time t, f ¿ t À is a measurablebut uncontrollableinput at time t; a j , j ¯ 1 µ ²²W² µ p,b j , j ¯ 0 µ ²W²W² µ p, c j , j ¯ 1 µ ²W²W² µ p andd j , j ¯ 0 µ ²W²W² µ p arethematricesof a p-order,time-independent,lineardiscretesystem.Thenumberof equationsis representedby the numberof outputsm; matricesai arem ¼ m, asmatricesci are;matricesbi arem ¼ n, beingn the numberof inputs;finally, matricesdi arem ¼ o, beingo thenumberof measurableerrors. Thedifferencebetweenthe inputsu andthemeasurederrors f is purely formal, beingthe u reachablein principle,andthususableto controlthesystem,while the f areonly observableandthuscanbeusedonly asmeasuresfor a feedforward control. Usually the matricesof the system

7.2. SYSTEMIDENTIFICATION (ID) 89

areunknown, only measuresof inputsandoutputsbeingavailable;theerrore isunmeasurableby definition,while f is assumedto beperfectlymeasurable.Theorderof the system,p, may differ for eachsignal; the caseof a Finite ImpulseResponse(FIR)

y ¿ k À ¯ b0u ¿ k À » ²W²W² » bpu ¿ k ¸ pÀis the limit for a systemwith order0 for y andvery high order p for the u, assuccessfullyimplementedin [24]. It hasbeenimplementedandinvestigated,butthevery complex dynamicsof rotorcraftseemto requiretoo long a time for theresponseto decay. In the following the sameorder p for all the signalsis as-sumedthroughoutthe chapter, theextensionto differentregressionordersbeingstraightforward.

7.2 SystemIdentification (ID)

Theyet unknown systemmatricescanbestackedin a matrix Θ, while theobser-vationscanbestackedin anarrayϕ ¿ k À , asfollows:

Θ ¯ ú a1 µ ²W²W² µ ap µ b0 µ ²W²W² µ bp µ c1 µ ²W²W² µ cp µ d0 µ ²W²W² µ dp û µϕ ¯ ü y ¿ k ¸ 1À T µ ²W²W² µ y ¿ k ¸ pÀ T µ

u ¿ k À T µ ²W²W² µ u ¿ k ¸ pÀ T µe¿ k ¸ 1À T µ ²W²W² µ e¿ k ¸ pÀ T µf ¿ k À T µ ²W²W² µ f ¿ k ¸ pÀ T ý T ²

Thepredictedoutputis

y ¿ k À ¯ Θϕ ¿ k À µ (7.2)

and the differencebetweenthe currentand the predictedoutput representstheerrorat thecurrenttime step,which is unknown by definition. Matrix Θ dependson k asfar asit is estimatedfrom a finite setof measures;it approachestheexactvalueprovidedthetruesystemhastheform of theassumedmodel.Equation7.2givesa meansto estimatetheerrorat every time stepin a recursive manner. Theerror may be dueto unmeasureddisturbances,errorsin measures,anderrorsintheparametersof themodel(type,order, andsoon):

e¿ k À ¯ y ¿ k À ¸ y ¿ k À ²The observationsat time stepsrangingfrom i to j canbe stacked by columns:y ¯ y ¿ i : j À , ϕ ¯ ϕ ¿ i : j À , e ¯ e¿ i : j À , resultingin

e ¯ y ¸ Θϕ µ (7.3)


wheretheexpectedoutputthat resultsfrom theyet to be identifiedsystem,ye ¯Θϕ, is used. If theerror is unbiased,Equation7.3 doesnot dependon theerroritself (theerrordoesnotparticipatein arrayϕ) andthusΘ canbesolvedfor afinitesetof measuresto determinetheoptimalvalueof theunknownparameters.In caseof biasederror, instead,it canbe determinedby recursively addingcolumnstoEquation7.3,andusingeachparameterestimateto computethecurrentestimateof theerror. A globalmeasureof theerroris:

J ¯ 12

eeT ²Theminimisationof J with respectto Θ givesa leastsquaresfit of thesystem:

Θ ¯ yϕT · ϕϕT ¹ † µwherethe † denotesthe pseudo-inversion,that is requiredin casethe systemisonly semi-definite.In this case,the excitation is not persistent,or the systemisnotcompletelycontrollable.

7.2.1 Recursive Implementation

Therecursiveexpressionsof matricesϕϕT andyϕT are:· ϕϕT ¹ j þ 1 ¯ · ϕϕT ¹ j » ϕ ¿ k » j À ϕ ¿ k » j À Tand: · yϕT ¹ j þ 1 ¯ · yϕT ¹ j » y ¿ k » j À ϕ ¿ k » j À T ²The inverseof matrix ϕϕT canbedirectly updatedinsteadof factorisingtheup-datedmatrix,by usingtheLDLT factorisation,sincethematrix is symmetricandpositivedefiniteor semidefinitein theworstcase;thepositivedefinitenesscanbeartificially enforced.In this way, thenumericallossof accuracy canbereducedwhile improving theefficiency of thecomputation.Therecursivealgorithmis:

Φ ¿ k À † ¯ µΦ ¿ k ¸ 1À † » ϕ ¿ k À ϕ ¿ k À T µ (7.4)

ψ ¿ k À ¯ µψ ¿ k ¸ 1À » y ¿ k À ϕ ¿ k À T µ (7.5)

Θ ¿ k À ¯ ψ ¿ k À Φ ¿ k À µ (7.6)

e¿ k À ¯ y ¿ k À ¸ Θ ¿ k À ϕ ¿ k À µ (7.7)

Equations7.4,7.5areusedto updatethematrices

Φ ¿ k À ¯ ÿ ∑j 1 ø kϕ ¿ j À ϕ ¿ j À T † µ

ψ ¿ k À ¯ ∑j 1 ø ky ¿ j À ϕ ¿ j À T µ

7.2. SYSTEMIDENTIFICATION (ID) 91

whereaforgettingfactorµ hasbeenusedto identify acomparatively slowly time-varyingsystem.Equation7.6 is usedto updatetheestimateof thesystemparam-eters;finally, Equation7.7 is usedto estimatethe error at the currentstep. Asshown in thefollowing section,artificial stabilisationof themoving averagepartof thesystemis required,sinceunstableerrordynamics,thatcanoccurduringtheidentification,haveno physicalmeaning[2].

7.2.2 Stabilisation of the Parameter Estimates

The matricesai can representeithera stableor an unstablesystem,dependingon the natureof the systemto be identified. Usually matricesbi, aswell asdi ,shoulddescribea stablesystemunlessa non-minimumphasesystem(a systemwith zeroesoutsidetheunit circle) is considered.But matricesci , whena biasederrorsystemis considered,shoulddescribeastablesystem,sincethedynamicsoftheerror cannotbeunstable1. During the identification,andsignificantlyduringthe initial phaseof the recursionprocedure,the systemdescribedby matricesci can temporarilybecomeunstable,leadingto a loss of physicalmeaningfortheerrormodel. A stabilisationalgorithmis needed,thatpreservesthedynamicpropetiesof the error, namelythe phaseof the eigenvaluesandthe shapeof theeigenvectors.A statespacerealisationof suchsystemis

ÆìììÇ ìììÈe¿ k À

e¿ k ¸ 1À...

e¿ k ¸ p»

1À Ë ìììÌìììÍ ¯ÂéééÃ ¸ c1 ²W²² ¸ cp 1 ¸ cp

I ²W²² 0 0...

......

...0 ØWØØ I 0

Ä êêêÅ ÆìììÇ ìììÈe¿ k ¸ 1Àe¿ k ¸ 2À

...e¿ k ¸ pÀ Ë ìììÌìììÍ µ

or

E ¿ k À ¯ ME ¿ k ¸ 1À ²If ρc are the eigenvaluesof matrix M, andmax¿ ρc À Ô 1, the dynamicsof theerrorbecomesunstable.It canbeeasilystabilisedby contractingthepolesinsidetheunit circle, in otherwordsby scalingthedynamicsof theerrorby acoefficientw suchthat,beingk a stability treshold,e.g.k ¯ 0 ² 85 ÷ 0 ² 90, ci ¯ wci , with w ¯k ë max¿ρc À , becomethenew errordynamicscoefficientswhenw Ô 1. Theerrorsat timesk ¸ 1 µ k ¸ p on turnmustbedividedby w.

1The error is by definition the resultof an ergodic process;the error regressioncoefficientssimply representtheresidualisationof thedynamicinformationthatcannotfit in thesystem.If thedynamicsof theerrorcontainany significantinformation,suchassomeunstabledynamics,thesemustfit in themodel.


7.2.3 Adaptive forgetting factor

Theforgettingfactorµ, introducedin Equations7.4,7.5,determinesthememoryof the identification. It is very important,basicallyfor two reasons,onebasedon numericalconsiderationsandthe otheron requirementsfor adaptive control.Thefirst reasonis relatedto thestartingof theidentification.Whentheextended,recursive leastmeansquaresare implemented,the estimatesof the parametersare requiredright from the very first steps,to allow to estimatethe error; butthe problemcannotbe solved until the cross-covariancematrix is positive def-inite. To overcomethe problem,the matrix is initialised as an identity matrixthat is quickly cancelledby theadditionof thecovarianceterms,andin thelimitbecomesnegligible. The useof a forgettingfactorallows to quickly canceltheinitial, roughestimatesof theparameters,whichdecayquadratically, since,beingµ 1, the contribution to the covariancematrix relatedto a time stepk stepsbehindis weightedin theparameterestimateby µk. Thesecondreasonis relatedto thefactthat,whenidentifyingasystemthatslowly varieswith time,theidenti-ficationmustbeableto follow ascloseaspossiblethenew system.An alternativeto theuseof a forgettingfactoris to resortto amoving window, asusedby Juang[64] in a ratherdifferentway to computetheDeadbeat2 controlmatricesdirectlyfrom theraw data.This choiceis consideredinteresting,but to allow anefficientimplementationit requirestoaddthecontributionthatisenteringthemeasurewin-dow to thecovariancematrix andsimultaneouslyto subtractthecontribution thatis falling out of thewindow; if thefactorisedmatrix is directly updated,suchop-erationcanbecomenumericallyinconsistentandleadto adestabilisingbehaviour[2]. By forgetting what hasbeenidentified in the past,only the recentsystembehaviour affectstheidentificationandappreciablechangesin thesystemcanbequickly reflectedby the identification.On theotherhand,if thesystemdoesnotchangemuchfor long time, a short-memoryidentificationrequiresa high persis-tent excitation to be able to continuouslykeeptrack of the system. This mightbe a drawbackbecauseit requiresthe systemto be continuouslyexcited, some-timeswith detrimentaleffectsonpowerconsumption,fatiguelife of components,comfort of the crew and of the payload. A good compromisehasbeenfoundby implementinganadaptive forgettingfactor, basedon a globalmeasureof thevarianceof theerror. Theforgettingfactoris integratedby a differenceequation,µ ¿ k À ¯ ρµ ¿ k ¸ 1À » ¿ 1 ¸ ρ À µmax, to reachanasymptoticvalueof µmax 1, corre-spondingto nearlypermanentmemory;whenthevarianceof theerrorovercomesapredefinedtreshold,implying thatappreciablechangesin thesystemtookplace,the forgettingfactoris pulled backto a comparatively small valueµmin, thusal-lowing theidentificationto refreshtheestimatesof theparametersin ashorttime.

2Discussedin Section7.3.1.

7.3. PREDICTIVECONTROL 93

7.3 PredictiveControl

As soonasanestimateof thesystemto becontrolledis available,eitherby para-metricmodellingor by blackbox identification,thehorizonof thepredictioncanbe easilyextended.By evaluatingEq. 7.1 at time t ¯ k

»1, the predictedvalue

resultsin

y ¿ k » 1À ¯ a1y ¿ k À » ²W²W² » apy ¿ k ¸ p»

1À»b0u ¿ k » 1À » ²W²W² » bpu ¿ k ¸ p

»1À»

c2e¿ k ¸ 1À » ²W²W² » cpe¿ k ¸ p»

1À (7.8)»d2 f ¿ k ¸ 1À » ²W²² » dp f ¿ k ¸ p

»1À µ

thedifferencebetweenthepredictedandtheactualvaluesbeingtheerror. Noticethatthemeasurederrorat timesk

»1 andk, aswell astheerrorat time k, arenot

consideredbecausethey areunknown, andby definitionthey arenot predictable.By substitutingthepredictedvalueof theoutputat time t ¯ k, Eq.7.8becomes

y ¿ k » 1À ¯ a11y ¿ k ¸ 1À » ²W²W² » a1

py ¿ k ¸ pÀ»b0u ¿ k » 1À » b1

0u ¿ k À » ²W²W² » b1pu ¿ k ¸ pÀ»

c11e¿ k ¸ 1À » ²W²W² » c1

pe¿ k ¸ pÀ»d1

1 f ¿ k ¸ 1À » ²W²² » d1p f ¿ k ¸ pÀ µ

wherethenew systemmatricesarerecursively definedas:

a0i ¯ ai µ

b0i ¯ bi µ

c0i ¯ ci µ

d0i ¯ di µ

a ji ¯ a j 1

1 a0i»

a j 1i þ 1 µ

b ji ¯ a j 1

1 b0i»

b j 1i þ 1 µ

c ji ¯ a j 1

1 c0i»

c j 1i þ 1 µ

d ji ¯ a j 1

1 d0i»

d j 1i þ 1 µ

apþ 1 ¯ 0 µbpþ 1 ¯ 0 µcpþ 1 ¯ 0 µdpþ 1 ¯ 0 ²

Thepredictederrorat stepk andbeyond is assumedto benull sincetheerror isassumedto be uncorrelatedwith the outputs,the inputs, the pasterrorsandthepastmeasurederrors,while theestimatesof theoutputaresupposedto beexact.Thepredictedvalueat time t ¯ k

»j becomes:

y ¿ k » j À ¯ a j1y ¿ k ¸ 1À » ²W²W² » a j

py ¿ k ¸ pÀ»b j

1u ¿ k ¸ 1À » ²W²W² » b jpu ¿ k ¸ pÀ»

c j1e¿ k ¸ 1À » ²W²W² » c j

pe¿ k ¸ pÀ»d j

1 f ¿ k ¸ 1À » ²W²W² » d jp f ¿ k ¸ pÀ»

b00u ¿ k » j À » ²²W² » b j

0u ¿ k À ²


Let s bethenumberof stepsaheadof theprediction.Thepredictedoutputsfromtime t ¯ k to time t ¯ k

»s ¸ 1 become:¾ y k s 1

.

.

.y k Á ¯ as 1

1 as 1p

.

.

.. . .

.

.

.a0

1 a0p ¾ y k 1

.

.

.y k p Á» bs 1

1 bs 1p

.

.

.. . .

.

.

.b0

1 b0p ¾ u k 1

.

.

.u k p Á » cs 1

1 cs 1p

.

.

.. . .

.

.

.c01 c0

p ¾ e k 1...

e k p Á» ds 11 ds 1

p

.

.

.. . .

.

.

.d0

1 d0p ¾ f k 1

.

.

.f k p Á » b0

0 bs 10

.

.

.. . .

.

.

.0 b0

0 ¾ u k s 1 ...

u k Á µor:

Ys ¯ AYp»

BUp»

CEp»

DFp»

PUs ² (7.9)

Thearraysandthematricesin Equation7.9areobtainedby stackingtheequationsof the output at the above mentionedtime steps,i.e. Ys containsthe predictedoutput at s future time stepsfrom the currentone; Yp, Up, Ep and Fp containthe(measured)outputs,inputs,theunmeasurableandthemeasurederrorsat theprevious p time steps,and thus are known; Us containsthe control inputs thatmustbedeterminedto obtainthedesiredbehaviour. Thepredictedoutputshouldbeequalto adesiredsequenceof values,namelyYs ¯ Yd, resultingin:

Yd ¯ AYp»

BUp»

CEp»

DFp»

PUs ² (7.10)

7.3.1 GeneralisedPredictive Control

ThesocalledMinimumVarianceControl [1] descendsfrom Equation7.10withs ¯ 1, by directly imposingthe desiredoutputat stepk andsolving for the re-quiredcontrolinput. Undertheassumptionthatthesystemhavea full rankdirecttransmissionterm (namely, matrix b0 be invertible), andprovided that the sys-tem is minimum phase,a single-steppredictionhorizonis sufficient. Moreover,theresponsein onestepfollows thedesiredbehaviour regardlessof the requiredcontroleffort, exceptfor the(unpredictable,becauseuncorrelatedby assumption)errorse¿ k À and f ¿ k À . As a consequence,thevarianceof theerroris minimal. TheGeneralisedPredictiveControl, on the otherhand,representsan extensionanda generalisationof this behaviour. The control still tries to force the systemtofollow thedesiredoutputstartingfrom thecurrentstep,but thedesiredbehaviouris imposedoverahighernumberof stepsahead.A predictionhorizonhigherthanthecontrolonecanbeused;in thiscasethedesiredresponseis imposedin a least


squaresense.Moreover, thecontroleffort is accountedfor by weightingthecon-trol outputagainstthepredictionerror, to avoid saturationof theactuatorsandtoorougha behaviour. An interestingform of predictive control is calledDeadbeatControl. It hasnot beenconsideredin this work sinceit can be obtainedas aspecialcaseof a moregeneralformulationof the GPC(the sameappliesto theMinimum Variance),andbecauseit resultedlessefficient andlessversatilethantheGPCin theinvestigatedcases.Thecontroloutputresultsfrom theminimisa-tion of thefunctional3

J ¯ 12¶ ¿ Yd ¸ Ys À T ¿ Yd ¸ Ys À » λUT

s Usº (7.11)

with respectto thecontrolinputUs, yielding:

Us ¯ · PTP» λI ¹ †PT ¿ Yd ¸ AYp ¸ BUp ¸ CEp ¸ DFp À µ

whereλ is thecontrolweightfunction.Thecontrolinputat time t ¯ k is givenby:

u ¿ k À ¯ αcYp» βcUp

» γcEp» δcFp

» εcYd µwhereεc is the last block-row of matrix Q ¯ · PTP

» λI ¹ †PT , andthe feedbackmatricesareαc ¯ ¸ εcA, βc ¯ ¸ εcB, γc ¯ ¸ εcC andδc ¯ ¸ εcD.

7.3.2 Inter pretation of the Predictive Control

The minimum variancecontrol clearly representsa form of zero-polecancella-tion. The control cancelsthe systempolesand zeroesby inverting the systemA 1B. This operationis permittedonly if the systemis stableand minimumphase;the resultingclose-loopsystemstatically respondsto the current,unpre-dictableinput only. TheGPCattenuatesthis abruptbehaviour by simply shiftingthe polesandzeroesof the systemtowardshigher frequencies.This operation,aswell astheminimumvariancedoes,mayleadto unstablebehaviour whenthecontrolis appliedto non-minimumphasesystems.By appropriatelychoosingtheweight function λ, both non-minimumphaseandunstablesystemscanbe con-trolled, with limited lossin performances.Thechoiceof themodelorderandofthepredictionandcontrolhorizonsarekey to theeffectivenessof thecontrol.Theorderp mustbehighenoughto accountfor all themeaningfulpolesof thesystem(aruleof thumbsaysthatp ¼ mshouldbeequalto or slightly higherthanthenum-berof physicalpoles).But toohighanordercouldresultin apoor, noisyandtimeconsumingidentification,that tries to follow a bunchof numerical,continuously

3By consideringthat thesystemidentificationandthecontroldesignrequireto minimisetwofunctionalsvery similar in form andnature,Juang [64] integratedthe two phases,obtainingtheDeadbeatcontrolmatricesdirectly from themeasures.


moving poles.Thepredictionhorizons shouldbeashigh asp to ensurethat thecompletedynamicsof the systemis accountedfor, including any non-minimumphasebehaviour of the system,at leastup to the modelorder;higherpredictionhorizonsdo not add further information to the prediction,but, togetherwith asmallercontrol horizon, result in an overcollocatedenforcementof the desiredbehaviour, thusoverconstraining,andimplicitly reducing,thecontroleffort.

7.3.3 Temporal weighting

Theerrormeasureof Equation7.11canbegeneralisedby introducingtwoweight-ing matrices,W andR, thatgive differentmomentumto thepredictionerrorandto thecontroleffort dependingon thedistancefrom thecurrenttimestep,namely

J ¯ 12¶ ¿ Yd ¸ YsÀ T W ¿ Yd ¸ Ys À » λUT

s RUsº µ (7.12)

yielding

Us ¯ · PTWP» λR¹ †PTW ¿ Yd ¸ AYp ¸ BUp ¸ CEp ¸ DFp À ²

TheGPCresultsfrom setting4 W ¯ R ¯ I with appropriateλ, while thedeadbeatcontrolcanbeobtainedby setting

W ¯ Q ¯ ° I 00 0 ±

regardless5 of λ, with s ¯ 2p, beingsubmatricesI and0of order ¿ p ¼ mÀ ¼ ¿ p ¼ mÀfor W and ¿ p ¼ nÀ ¼ ¿ p ¼ nÀ for R respectively, yielding· PTWP

» λR¹ †PTW ¯ ° 0 0P†

db 0 ± µmatrixPdb ¯ P ¿ 1 : p ¼ mµ p ¼ n

»1 : 2p ¼ nÀ beingtheQ of thedeadbeatcontrol;

in otherwordsit resultson onesidein forcing thecontrol effort to be null afterp steps,while on theothersidein weightingonly thecontroleffort after p steps.The pseudo-inversionis requiredsincematrix Pdb in generalis rectangularandcouldberank-deficient.Becausethesystemis well-posed,thecontrolis null afterp steps,andthereis no constrainton the responseduring the first p steps.Anyintermediatecombinationof weightingmatricescanbeusedto taylor thecontrolbehaviour, from scalingtheway the responseat differenttime stepsis weighted

4RememberthatmatricesW, R in generalhavedifferentdimensionsunlessthesystemhasthesamenumberof inputsandoutputs.

5Providedλ ó 0 to ensurethecontroleffort is null afterexactly p steps.


in J, to weightingthecontroleffort alsoin thefirst stepsto avoid control rough-ness.This formulationsomehow unifiesthedifferentformsof predictive controlheredescribed,andextendsthe family of the generalisedpredictive controllerstowardsa completetayloringof theerrormeasure.Evenmorecomplicated,non-quadraticerror functions,accountingfor actuatorsaturation,outputconstraints,andvirtually any otherobjective function,in thespirit of thesuboptimalcontrol,have beenproposed[83, 51], with the major drawbackof requiringa nonlinearsolutionto computethecontrolsignals.In thementionedworks,theidentificationwasperformedby aneuralnetwork, soaNewton-Raphsonprocedurewasalreadyrequiredfor the identification.This aspectof thepredictive controlhasnot beenthoroughlyinvestigatedin rotorcraftapplications,but shows aninterestingversa-tility , even if the additionof two new setsof control parameters,the weightsWandR, makesit morecomplicateto tunethecontroller, in theabsenceof asimple,reliabledesignalgorithm. An adaptive, self-adjustingalgorithmis sought,basedon someperformancemeasure,thatcomputestheweightingfactorsin matrix R,or at leastthevalueof theweightλ.


Chapter 8

Multidisciplinary problems

This chapterlooks like a todo list, indeed. A lot of work is still underdevel-opmentin the field of multidisciplinarymodelling. In this dissertationonly thepartdirectly relatedto rotorcraftcontrol is illustrated,but efforts towardsthe in-tegratedsolutionof multidisciplinaryproblemsinvolving aeroelastic,electricandhydraulicmodelsareunderway.

8.1 Control

The analysisof active control-relatedproblemsrequirethe modellingof severalcomponentswhenahighdetail level is desired.They canbegroupedasfollows:

1. Actuators

2. Sensors

3. Networking

4. Generalpurpose

Thephysicalactuationandsensingprinciplesdeterminehow thebehaviour of themechanicalsysteminteractswith thecontrolandmeasuresignals.Regardlessoftheir nature,thereis somerelationshipof thekind

s ¯ s ¿ xÀ µf ¯ f ¿ aÀ µ

wherex generallyrepresentsa kinematicunknown, regardlessof thedifferentia-tion orderthat is actuallyexploitedby thesensingprinciple;s anda respectivelyarethe measureandthe actuationsignals,while f is the force generatedby theactuator, regardlessof its nature.

99

100 CHAPTER8. MULTIDISCIPLINARY PROBLEMS

8.2 Actuators

Asspecialisedactuatorsfor rotorcraftcontrol,thefollowinghasbeenimplemented:

1. Swashplate

2. Piezoelectricbeam

3. Other

8.2.1 Swashplate

Theswashplateresultsfrom theassemblyof basicjoint elements,usually: two in-planejoints thatforcethenon-rotatingplateto slidealongthemast; threevariable-distancepin joints that enforcethedistanceandthe attitudeof thefixedplatewith respectto therotorcraftframe; one in-planejoint that preventsthe axial rotation of the fixed plate withrespectto the fuselage,emulatingthe scissors;asan alternative, the scis-sorscanbeentirelymodelledby two planehingesandonesphericalhinge,requiringtwo additionalnodes; oneplanerotationjoint thatconstrainsthetwo plates; onein-planejoint thatpreventstheaxial rotationof the rotatingplatewithrespectto thehub,emulatingthescissors.

An alternativesolutionconsistsin: oneprismaticjoint that forcesa nodeto slide alongthemast,allowing norotationwith respectto therotorcraftframe; oneuniversaljoint that constrainsthe fixed plateon the sliding node,re-strainingtheaxial rotationonly; threevariable-distancepin joints that enforcethedistanceandthe attitudeof thefixedplatewith respectto therotorcraftframe; oneplanerotationjoint thatconstrainsthetwo plates; onein-planejoint thatpreventstheaxial rotationof the rotatingplatewithrespectto thehub,emulatingthescissors.

8.2. ACTUATORS 101

Theswashplatetransmitsthepitch to thebladeby meansof a pitch link, usuallymodelledas a rigid or flexible distancepin joint. The elongationof the non-rotatinglinks canbe independentlyassignedasfunction of the time, or of somecontrolsignal. To obtaina smootherbehaviour, the impositionof theelongationtime raterepresentsa moreviablechoice,that is morerespective of theprinciplethatis usedtoapplythecontrol,namelybymeansof controlforcesthatresultfroma controllerthat follows somedesiredactuatorposition; in this casea changeinelongationratecorrespondsto an impulsive changein control force. To easetheapplicationof the desiredcontrols,a distributor elementhasbeenimplemented,which determinestheelongationsof thelinks basedon thedesiredcollectiveandcyclic pitch.

8.2.2 Piezoelectricbeam

The Active Twist Rotor requiresthe impelementationof a beammodel for thebladesthat allows to exploit the piezoelectriceffect at the constitutive level, toaccoutfor any effectsrelatedto the geometricnonlinearityof the blademotion.Thepiezoelectricbeammodel,from abeamcharacterisationandastructuralanal-ysisstandpoint,is describedin Chapter9; hereits useis briefly anticipatedfroma control standpoint.Thefinite volumebeamelement,presentedin Section5.4,describesathree-nodebeamelementwhoseconstitutivepropertiesareconsideredat two intermediatesections.Thepiezoelectricgeneralisationintroduceselectricfield-dependentinternal forcesat thesesections,that participatein the determi-nationof the beamequilibrium. Due to the natureof the piezoelectricpatchesandto the negligible low frequency electrodynamicsof the systemsunderanal-ysis, it is reasonableto assumethat theelectricvoltageis appliedto thepatchessimultaneouslyateverypoint,boththroughoutthesectionandalongtheaxis.So,evenif a longblade,with refinedmeshdiscretisationis considered,thenumberofelectricunknowns is limited to the numberof independentpiezoelectricpatchesthataredistributedalongtheblade;thecontrolsignalis applieddirectly to thoseunknowns.

8.2.3 Other actuation means

Thegeneralityof theimplementationof exogenousforcesallowsahighdegreeoffreedomin determiningtheiramplificationfactor, asdescribedin AppendixH.1.1.As a result,any exogenousforce,mechanicalaswell asgeneralised,candependon the valueof an arbitrarycombinationof unknowns, thusexploiting a simplebut effective form of indirect feedback.This dependenceis not exploited in thecomputationof the Jacobianmatrix, of course,thus introducingsomeexplicit


Control+

+

External

Node

Length

PitchLink

Figure8.1: Pitchcontrolscheme

behaviour in thesolutionalgorithm,but this is usuallyacceptablein termsof con-vergenceratewhensuchbehaviour is confinedin few degreesof freedomwithcomparatively smallgains.As anexample,anodecanbegroundedby aspringbyintroducinga forcewhoseamplitudeis functionof thepositionof thenodein onedirection. This trick hasbeenexploited,for example,to superimposethecontrolsignalto someexternalpitch control signal. Considerfor instancethecollectivepitch. Its amplitudeis containedin a variableθactual, andthe lengthof thepitchlinks is readfrom suchvariableatrun-timeby theimposedlengthjoint. Thevalueof θactual resultsfrom a simpleequationof the form θactual θexternal θcontrol .The two inputsareappliedby meansof generalisedforce elements,the former,θexternal, representinga truly exogenouscontribution, e.g. the desiredreferencepitch,andthelatter, θcontrol , resultingfrom thecontroller, asshown in Figure8.1.Eventually, suchequationwill beextendedby including thedynamicsof thehy-draulic actuatorsheredescribedby the variablelength distancejoints, and theresultingdesiredelongationwill beconvertedin someelectrovalvecontrolsignal.Thesameapproachcanbeused,for instance,to stabilisetherotationspeedof therotor, insteadof directly prescribingit with a joint. A couple,proportionalto thedifferencebetweenthe desiredandthe actualspeed,andto the speedtime rate,canbe a goodfirst approximationof a speedcontrol scheme.The gainsof theproportionalandderivativepartsmustbedesignedwith careto avoid any possibleinstability.

8.3 Sensors

Thefollowing sensorshavebeenimplemented:

1. Linearandrotationalaccelerometers

8.3. SENSORS 103

2. Straingagesattachedto beamelements

3. Piezoelectricbeamelements

4. Direct measureof any unknown, andof many derivedentities.

8.3.1 Accelerometers

Accelerometers,attachedto thenodes,have beenimplemented.They returnthemeasureof theaccelerationin a prescribeddirection,which is fixed in thenodereferenceframe. Suchmeasurecan be filtered througha transferfunction, asdescribedin Section8.5, thataccountsfor thedynamicsof thedevice,consistingin a band-passfilter with a zeroat theorigin followedby a pole,a flat responsepart terminatingwith a resonancepeak,andwith two coincidentpolesgoverningthehigh frequency decay. Thebuilt-in transferfunctionmaybeoverriddenby auser-definedone.

8.3.2 Strain gages

The beamelementsallow to easily introducesomestrain measure.The directmeasureof thegeneralisedstrainsis a very simpletask,althoughit is importantto rememberthat thestrainsresultingfrom thedirect interpolationof thederiva-tivesof thenodaldisplacementsandrotationsarehighly inaccurate,a reasonablemeasureof the strainsresultingonly from an internal force andcouplebalancereconstructiona posteriori. To allow somemorerealisticmeasure,a straingageelementhasbeenimplemented,which computesthestrainsat somepoint of thebeamsectionby combiningthestrainsresultingfrom axialextensionandflexuralbending.

8.3.3 Piezoelectricbeams

Thesensingequationof thepiezoelectricbeamallows to introducesomeinterest-ing measureof the averagestrainover a portion of beam. Although very inter-estingin casea collocatedcontrol is addressed,this kind of measurementhasnotbeeninvestigatedyet in active twist rotorcraftapplications.

8.3.4 Dir ectunknown measure

The last type of measure,althoughvery simple and at a first glanceunrelatedfrom any physicalmeasureprinciple,canbeviewedasthesimplificationandtheidealisationof an arbitrarymeasureprinciple; asan example,onecanconsider


thedirectmeasureof a reativerotationanglein aplanerotationhinge:thereis noneedto detailhow themeasureis obtained,whetherby apotentiometeror bysomenon-intrusive electromagneticdevice or anything else,asfar asthe dynamicsofthetransferfunctionfrom theangleto theelectricsignalin thefrequency rangeofinterestcanbeneglected.Moreover, themeasuredsignalcanbefiltered throughan arbitrary transferfunction elementto emulatethe dynamicsof someactualmeasurementdevice,asdescribedin Section8.5.

8.4 Networking

Themodellingof control systemsup to somedetail level may requirethe directinclusionof someelectriccomponentbehaviour. Basicnetwork devicessuchasresistors,capacitors,inductors,diods,tensionaswell ascurrentgeneratorshavebeenimplemented.They acton scalardegreesof freedomthatrepresenttheelec-tric tensionsat the nodesof the electricnetwork. To parallel the structuralfor-mulation,theelectrictensionsarethenodaldisplacements,andthenodalcurrentbalanceequationsmimic theforcebalanceequationsof thestructuralcase.As anexample,afterdefiningtheoutputy andtheinput u asthecurrentandthetensionaffectinga node,andx asaninternaldegreeof freedomwhenrequired,considera linearresistorin theform:

y1

y2 k 1 1 1 1 u1

u2 wherek 1 R is theconductance,which parallelsa spring;a linearcapacitorintheform

y1

y2 k 1 1 1 1 u1

u2 wherek C is thecapacitance,which parallelsadamper;a linearinductorin theform

y1

y2 11 x

x k 1 1 u1

u2 wherek 1 L is the inverseof the inductance.Thoseelementsaregeneral,andcanbe usedto describeideal componentscommonto many typesof problems.An accurateprogrammingallows to usethemin conjunctionwith any degreeoffreedomin asafeandintuitivemanner.

8.5. GENERAL PURPOSE 105

In somecases,“algebraic”constraintequationsbetweentensionsneedbedefined,for instancein caseof voltagegenerators;in suchcases,the resultingreactionunknownsareelectriccurrentsthatparticipatein thenodecurrentbalance,i.e.

y x

u u0

To addgeneralityto the formulation, most of suchcomponentshave beenim-plementedreferring to arbitraryequationsinvolving arbitrary unknowns. Suchelementshavebeencalled“generalpurposeelements”,or GENELs. For instance,asingledegreeof freedomvoltagegeneratorcanbeusedto imposethetensionofa nodeof thenetwork, or to imposethedisplacementaswell asthevelocity of adegreeof freedomof astructuralnode,or thepressureof somehydraulicnetworknode,andsoon. Strictly speaking,GENEL elementsarethosedescribedin Sec-tion 8.5, the networking onesratherrepresentingsome“bulk” elementtype,butthedifferenceis verysubtle.

8.5 Generalpurpose

The strict definition of “generalpurposeelements”(GENEL) refersto elementsthat areuser-defined1, andwrite somearbitraryrelationshipbetweenunknownsthatdonotneedto haveany precisephysicalmeaning.Examplesof implementedGENELs are: State-spacesystems,of theform

x Ax Buy Cx Du

wherethex areinternalstatesandu, y aretheinputsandtheoutputsrespec-tively. This elementallows to modelanarbitrarytransferfunctionbetweensignalsandthusis very usefulin addingthedynamicsof actualdevicestobulk measureor controlsignals,or in implementinganalogfilters. GeneralisedPredictive Control elements,asdetailedin Chapter7; in suchcase,adiscretesignalof theform

uc t ! ∑i

aciy t i ! ∑i

bciu t i ! #"$"%"1The implementationof generalpurposeanduser-definedelementshasbeensimplified and

enhancedby allowing theuseof run-timelinkedelements.


is computedfrom arbitrarymeasuredsignalsto estimatethe next controlinput. At thesametime, thecontrolmatricesareredesigned,if required,toadaptthecontrolaccordingto thedynamiccontentof themeasures. DynamicInflow element,exploiting theformulationproposedin Ref. [76],wheretheinflow velocityatsomepointon therotordisk is written in termsof a trigonometricseriesexpansion,whosecoefficientsresultfrom thetimeintegrationof normalisedvelocityparametersthataccountfor thedynamicsof theaerodynamicsystem.

The differencebetweennetworking and generalpurposeelementsis apparentnow: networking elementsareimplementedin a generalform becauseanalogiesbetweendifferentdisciplineshavebeenexploited,while generalpurposeelementshave thesamefunction regardlessof thedisciplinethey areappliedto, asin thecaseof controlelementsansstatespacesystems,wherethenatureof inputsandoutputsareirrelevant,or areconfinedto onedisciplineonly, becausethey performaveryspecialisedtask.SonetworkingelementsareimplementedasGENELsonlyto exploit codereusabilityandto addflexibility to theprogram,insteadof dupli-catingtheimplementationof basicelementsfor everydiscipline.

8.6 Other problems

Themostcommonelementsrelatedto themodellingof any network baseddisci-plinehavebeenpresentedin Section8.4; they provideabasisfor definingsimpli-fied,essentialscalarvaluednetwork problems.But any discipline,whenpracticalproblemsmustbe addressed,may requirespecialisedelementsto considerspe-cific problems;for instance,if an incompressiblelaminarfluid in a pipe is to bemodelled,ageneralpurposeresistormaysuffice,but whenaturbulentflow condi-tion needsbeanalysed,anequivalentnonlinearresistorshouldbeused;moreover,whentransitionfrom laminarto turbulentflow andviceversais required,a dedi-catedelementneedsbeimplemented.Moreover, a fundamentealcareis requiredto interfacedifferentnetworks;prosecutingin thehydraulicparallel,thehydraulicactuatorthat interfacesthehydraulicplantwith themechanicalsystemrequiresadedicatedmodellingif any detail is desired,e.g.inertiaandcompressibilityof thefluid, pressurelossesat orifices,friction at interfacesbetweenmoving parts,andsoon. For thesereasons,themodellingof hydraulicplantsis beingpursued,andthemodellingof electricnetworksis beingimproved,basedondemandsfrom thetaskof multidisciplinaryanalysisof controlledrotorcrafts.

Chapter 9

Piezoelectricbeamanalysis

Thebeamanalysisformulatedin Chapter5 is extendedto theanalysisof beamsin-cludingpiezoelectricmaterials.Thesectionis characterisedby extractingelectricdegreesof freedomrelatedto theconductorsappliedto thepiezoelectricpatchesincorporatedinto the structuralcomponent,and the finite volume beammodel[69,37] is extendedby includingpiezoelectriccouplingterms[35] andthechargebalanceequationresultingfrom theelectrostaticsimplificationof Maxwell laws.

9.1 Beamsectioncharacterisation

Thebeamsectionanalysisprocedurepresentedin Section5.3 hasbeenextendedto the analysisof beamsectionsembeddingpiezoelectricdevices, to allow thecharacterisationof embeddedactuators[32, 34], andsignificantly of the activetwist bladediscussedin Section6.4 [68, 36].

9.1.1 Electric field

A piezoelectricdeviceembeddedin astructuralcomponent,undertheassumptionthat the frequenciesit operatesat arerelatively small, behaves like a capacitor,namelyits electromagneticbehaviour is negligible. As a consequence,only theelectric field inside the device needsbe modelled. The formulation presentedhereinallowsthemodellingof theelectricfield evenanywhereoutsidethedevice,undertheassumptionit canberegardedasstatic.Thisfeaturecanbeusefulin casethe lossof electricfield dueto the electricpermittivity of the surroundingenvi-ronmentneedsbeaccountedfor. In amoregeneralsense,asdescribedin [30], thepiezoelectriceffect mayberegardedasa specialisationof a moregeneralmulti-field characterisationof materialsthataccountsfor mechanical,electric,magneticandthermalinteractions.The linearpiezoelectricconstitutive law of Eq. 6.1 re-

107

108 CHAPTER9. PIEZOELECTRICBEAM ANALYSIS

sultsfrom the linearisationof thestressesσ andof the electricdisplacementDe

underthe assumptionof small frequencies,resultingin negligible influenceofmagneticandelectromagneticeffects,andof very slow changesin temperatureT, accountedfor by evaluatingthematerialpropertiesat theactualreferencetem-perature. Considera sectionof the beam,in the materialframe1. The electricpotentialscalarfield V is definedanywherein theelectricdomain, which canbelimited to the piezoelectricdevice or extendedto the restof thestructureandtothe air or vacuumoutsidethe beam. The oppositeof its gradientrepresentstheelectricfield E ∇V, which is naturallyexpressedin the materialframe. Inanalogywith theelasticformulation,it canbedecomposedin thederivativealongtheaxisof thebeam,andin thegradinetin theplaneof thesection,namely

E 1 0 0 T V& ξ (' e V !)operator' e +* ! being

' e +* ! ,- 0 +* ! & η .* ! & ζ /0 "Notice that, in analogywith the structuralanalysis,a generalisedmeanelectricfield orientedalongthe axis of the beamcould have beendefined,parallelingasortof “rigid”, or global,axialstrain,i.e.uniformamongthesection;thispossibil-ity hasnotbeenexploitedsincetheusualimplementationof piezoelectricpatchesinvolves the transversepolarisationof the piezoelectricmaterials,and thus themeanelectricfield in axial directionis null. This is not thecasewheninterdigi-tatedelectrodesareconsidered,becausethey canapplyastrongaxialelectricfieldcomponent,but in any case,sinceit usuallyis very closelyspatiallyalternated,adifferentprocedurehasbeenconsideredto mimic theIDE behaviour, asdescribedin Chapter13.


The internalwork in thebeamsectioncanbeextendedfrom theelasticform in-troducedin the conventionalbeamsectionanalysisto the piezoelectriccasebyconsideringtheelectricdomainof thesection.Thevirtual variationof dielectricenergy perunit volumeis δDe

TE; to obtaintheenergy perunit lengthof thebeam,it mustbeintegratedovertheelectricdomain.Thereis noneedto explicitly define

1Becausethe electricpotentialis scalar, it is independentof the referenceframechoice;thereferenceframe is significantwhen its gradientis considered.Anyway the material referenceframeis usedto allow aneasydescriptionof theintegrationdomainandof thedirectionsthatareconsidered.

9.1. PIEZOELECTRICBEAM SECTION 109

DomDom

e

p

s

Lamina

Dom

Structural Domain

Electric Domain

Piezoelectric Domain

Figure9.1: Piezoelectricbeamsection.

any piezoelectricdomain;only, wherein the intersectionbetweenthe structuralandthe electricdomainsthe materialconstitutive law is truly piezoelectric,thestressesσ andthe electricdisplacementDe dependon both the strainsε andontheelectricfield E. A sketchof a typicalpiezoelectricbeamsectionis reportedinFigure9.1,wherethedifferentdomainsarehighlighted.Theinternalwork perunit lengthis

∂∂ξ

δLi 21Ss

δεT σ dS 31Se

δETDe dS;

its perturbationis

∂∂ξ

∆δLi 1Ss 4 δεT 4 ∂σ

∂ε∆ε ∂σ

∂E∆E 5 ∆δε ! T σ 5 dS

31Se 4 δET 4 ∂De

∂ε∆ε ∂De

∂E∆E 565 dS

wherethe partial derivativesof stressesandelectricdisplacementarethecoeffi-cientsof thelinearpiezoelectricconstitutivelaw

∂σ∂ε D 7 E 8 ∂σ∂E eT

∂De

∂ε e∂De

∂E ε 7 ε 8

while theperturbationof electricfield is

∆E ,- 100 /0 ∆V& ξ ' ∆V !9


andits virtual variationis

δE ,- 100 /0 δV& ξ ' δV ! "

Thematrix representationof the internalwork is a trivial extensionof thestruc-turalcase,in whichthedielectricandtheelectroelasticcouplingtermsappear;it isdetailedin AppendixG. Theinternalwork perunit volumeδL :i δεT σ δETDe

is

δL :i ;<<<<= <<<<>δt & ξδV& ξδtδVδψ

?<<<<@<<<<A

T

;<<<<<= <<<<<>Σ :tξΣ :Vξ

Σ :tΣ :VΣ :ψ

?<<<<<@<<<<<A

while its perturbationresultsin

∆δL :i ;<<<<= <<<<>δt & ξδV& ξδtδVδψ

?<<<<@<<<<A

T ,BBBB-C :ss

C :se D :ss D :se E :sC :ee D :es D :ee E :eF :ssF :se G :s

sym" F :ee G :eH /JIIII0 ;<<<<= <<<<>∆t & ξ∆V& ξ∆t∆V∆ψ

?<<<<@<<<<A

the matricesbeingdefinedin AppendixG. Subscripts +* ! s and +* ! e respectivelystandfor structuralandelectric.Noticethattheproblemshouldnotbesymmetric,dueto the differentsignsthat matrix e takesin the “acuation” andin the “sens-ing” equationsof thematerial;symmetryis restoredby changingthesignof the“sensing”equations,at thepriceof losing thepositivedefinitenessof thematrix.Thechangeof signis not merelya trick; it is allowedin view of thearbitrarinessof thevirtual variationof potential.

9.1.3 External work

In strict analogywith the structuralcase,the external dielectric work is madeof a transmissionwork, the axial transmissionof the electricdisplacement,plusan externalwork madeby charging the conductorsthat boundthe piezoelectricpatches.In this casethecharge work is fundamentalsinceit representsthewaypiezoelectricsarecontrolled.Thesectionis topologicallydividedin piezoelectricregionsof influence,that representportionsof the planethe sectionlies in thatareself-containedfrom an electricstandpoint.This is a simplification,becausethe electric region shouldextend to the universe,but, sincethe electric field is

9.1. PIEZOELECTRICBEAM SECTION 111

usuallyconcentratedinto thepiezosdueto their high dielectricpermittivity com-paredto that of the air or vacuum,andbeingthe conductingpartssubstantiallyequipotential,it might be convenientnot to modelany electricfield outsidethepiezelectricmaterials,thus logically dividing the sectionin asmany regionsasthe patchesare. In eachactive region oneof the conductors,if any2, needsbegroundedto avoid thesingularityof theproblem.Theelectricloadis appliedbyindependentlyapplyingunit chargeperunit lengthto eachof theremainingcon-ductors.The conductorsareequipotentialby definition, thussuchconstraint,inthe discretisedmodel,canbe imposedby numberingeachnodeon a conductorwith the samelabel. Thenthe assemblyprocedurewill automaticallytake careof imposingtheconstraint.Theconductornodesarepreserved in thecondensa-tion of thewarpingunknowns,sincethey holdthetensionappliedthroughapiezoandthusrepresentglobal degreesof freedomof thesmartsection.Theexternalelectricwork resultsin

∂∂ξ

δLe ∂∂ξ 1 Se

δVTDe dS 31ce

δVTq dc "The last work term is relatedto the free charge densityrequiredto charge theequivalent capacitor, wherece representsthe boundaryof the piezoelectricre-gions,including the lines of discontinuitythat ideally describethe internalcon-ductors;sincethepotentialV is constanton theconductors,suchwork contribu-tion simply resultsin thesumof theproductsof thevirtual variationsof tensionby thetotal chargesperunit lengthon theconductors

∂∂ξ

δLe 1Se K δVTDeL & ξ dS ∑

ce

δVcqc;

the sectioncharacterisationwill requirethe solutionof the problemfor indicialchargedensities.

9.1.4 Solution

Thesolutiondoesnot basicallydiffer from thatof thepurelystructuralproblem,so it is not detailedhere. One importantremark,first introducedin [34] to theauthors’knowledge,concernstheequipotentialityconstraintalongtheaxisof thebeam. A conductoris equipotentialby definition; if the conventionalprocedurefor thestructuralcharacterisationis followed,theelectrictensiondegreesof free-domwill have linearform alongtheaxisof thebeam,at leastin caseof transverseshearloads.While this is correctfor internalelectricnodes,it violatestheequipo-tentiality constrainton the conductors,which on the contrarymustobey to the

2Onemight be interestedin analysingthe passive behaviour of a structureembeddingpiezo-electricpatches,with no electricwork termsdefined,andthuswith noactiveelectrodes.


non-holonomicconstraintof null axial derivativeof thetension,thetensionitselfbeingunknown. This requiresanextra constrainton thepotentialof theconduc-torsduringthefirst, linearstepof thesolution,thatmustberemovedfor thelast,constantsolutionstep. As shown in [68], the reactionchargesthat result fromthisextra,overdeterminedelectricalconstraintatfirst step,linearlydependon theaxialpositionanddo notaffect thesectioncharacterisation.

9.1.5 Remarks

Therearemany possibleextensionsto thebeamsectioncharacterisationpresentedin this work, rangingfrom the generalisationto the nonlinearelectro-magneto-thermoelasticproblem,consideringaspectssuchasthe modellingof thecurrentinto the conductorsthat areusedto polarisethe piezoelectrics,with the thermaleffect it hason the surroundingmaterial,to the propagationof the charge alongthe beamaxis, at very high speed,to the modellingof porousbeamsincludingelectro-and magneto-rheologicfluids, and more. One importantaspectis theability to modelverycomplex compositelaminatesectionswith high accuracy indetailssuchasboundaryandextremity effects[42], andto allow theanalysisofcritical conditionssuchas fracturemechanics[43], with possibleextensionstodelaminationof composites,aswill beinvestigatedin forthcomingworks. Whiletheresearchintereston suchtopicsis very high, its immediateapplicationis be-yond the currentneedsof practicalengineeringin smartmaterialsapplications,mainly dueto intrinsic technologicallimitations of currentmaterialsandto thelack of information aboutmaterialcharacterisationother than the basic, linearelasticpropertiesof verycommonpiezoelectricmaterials.

9.2 Finite volumepiezoelectricbeam

Thepiezoelectricformulationof thefinite volumebeammodelis straightforwardwith regardto the beamequilibrium equations;in fact, the electriccontributiononly affectstheconstitutivelaw of thebeam,introducingsomedependenceof theinternalforcesandcoupleson thevoltageappliedto thepiezoelectricpatches.

9.2.1 Equilibrium

Theinternalforcesat theevaluationpointsof thefinite volumebeamelement,incaseof smartbeamsectionsdependon the electricdegreesof freedomaswellason the generalisedstrains. The actuatingpart of the linearisedbeamsectionconstitutive law is

ϑ Dψ θTV

9.2. FINITE VOLUME PIEZOELECTRICBEAM 113

thenovel term,θTV, representsthepiezoelectriccontributionto theinternalforces,θ beingthegeneralisedpiezoelectricmatrix resultingfrom thebeamsectionchar-acterisationprocedure,andV beingtheelectricvoltagesappliedto theconductorsembeddedin thesection.The additionof suchcontribution to thefinite volumebeamformulationis straightforward.

9.2.2 Chargebalance

Theelectricchargebalanceequationneedsbeformulated.It is obtainedby con-sideringthe charge balanceper unit length,containedin the piezoelectricbeamsectioncharacterisation,i.e.

q θψ εV (9.1)

being θ andε the piezoelectriccouplingand the dielectricmatricesof the sec-tion, andψ andV thegeneraliseddeformationsof thebeamin thelocal frameandthetensionsappliedto thepiezoelectricpatches.Theweightedintegral of Equa-tion 9.1yieldsthediscreteform of thechargeaccumulatedon theconductorsin afinite portionof beam

Q M ba 1 b

aw θψ εV ! dξ (9.2)

whereQ M ba is the resultingpolarisationcharge andw is thefinite volumeweightfunction, i.e. a stepwiseunit function. This equationcanbe written “as is”, andthechargecanbe transformedinto a currentby meansof a derivatorGENEL, orEq.9.2canbewritten in differentialformN M ba 1 b

aw K θ ˙ψ εV L dξ

directlyyielding acurrentcontribution thatcanbeaddedto anelectricnode.

9.2.3 Discreteform

It is interestingto noticethat a very simplediscreteform of the charge balanceequationcan be easily obtained. Considerthe discretisationproposedin Sec-tion 5.4 for thethree-nodefinite volumebeamelement.Thediscretechargebal-anceequationneedsnot be integratedover thesubdomainsusedfor themechan-ical formulation; on the contrarythe whole beamelementcanbe consideredatonce,sincetheelectricpotentialis constantall alongthe beam. In this case,byconsideringanumericalintegrationby Gausspointcollocation,andrecallingthattheoptimalevaluationpointsfor thesegmentationof thebeamcoincidewith the


two Gausspointsthatexactly integratepolynomialsup to third degree,with unitweights,thediscretechargeequationcanbecomputedas

Q M 31 PO l0 O θψ !QM I O l0 O θψ !RM I I S O l0 O ε !TM I UO l0 O ε !QM I I ! Vbeing1 and3 thetwo end-nodesof thebeamelement,andI andI I thetwo evalua-tion points.Thechargerelatedto thebeamelementcanbeconnectedto asensingnetwork.

Part III

Applications

115

Chapter 10

Preliminary studies

The proposedmethodologiesfor the analysisof multidisciplinarysystemshavebeeninvestigatedandassessesfirst by consideringsimpleexamplesandbench-marks,usuallytakenfrom theliterature.

10.1 Rigid body mechanisms

10.1.1 Pendulum

A simplependulumis considered,madeof a concentratedunit massconstrainedto be at unit distancefrom the origin of the global referenceframe,underunitgravity accelerationin the negative directionof y axis. This planeproblemhasbeentakenfrom Ref. [17], Chapter6, pp.150–157.Theproblemis definedas

x u 0

y v 0

u x lλ 0

v y lλ g

x2 y2 l2

In thementionedreference,theproblemhasbeenstabilisedby addingthederiva-tiveof theconstraintequation;on thecontrary, in thepresentimplementation,themodelhasbeenbuilt from simpleentitiesin amultibodyframework. Onesecondof simulationis considered;the mentionedreferenceintegratesthe problembytwo-step,second-orderbackwarddifferences.A consistentsetof initial conditionsx 0! 1, y 0! 0, x 0! 0, y 0! 1, λ 0! 1 is considered.On thecontrary,in thiswork theinitial valuesof theconstraintreactionandof thederivativesof themomentaareinconsistent.Nonetheless,thedesiredsecond-orderaccuracy is ob-tained,andmoreover, by usingthesecond-orderformulaproposedin Section4.2

117

118 CHAPTER10. PRELIMINARY STUDIES

Table10.1:Pendulumaccuracy (ρ∞ 0 " 0).Variable Solution h V" 01 h W" 005 h W" 0025

x 0.1349949261 3.1e-3 7.8e-4 2.0e-4y 0.9908462897 5.7e-5 1.4e-5 3.6e-6x -1.710951582 3.1e-4 7.6e-5 1.9e-5y 0.2331035448 3.0e-3 7.4e-4 1.9e-4λ 3.972538869 2.4e-4 5.9e-5 1.4e-5

Table10.2:Pendulumaccuracy (ρ∞ 0 " 6).Variable Solution h V" 01 h W" 005 h W" 0025

x 0.1349949261 9.5e-4 2.4e-4 6.0e-5y 0.9908462897 1.8e-5 4.4e-6 1.1e-6x -1.710951582 9.2e-5 2.3e-5 5.6e-6y 0.2331035448 9.1e-4 2.2e-4 5.7e-5λ 3.972538869 7.1e-5 1.8e-5 4.3e-6

with an asymptoticradiusof 0 " 6, a higheraccuracy hasbeenobtainedthanthatresultingfrom Ref. [17]. Tables10.1,10.2reporttheerror in thependulummo-tion afteronesecondfor differenttimesteps;thefirst is apuretwo-stepbackwarddifference,while thesecondexploits theproposedsecond-ordertwo-stepintegra-tor. They shouldbecomparedto Table6.2.1in Ref.[17]. In particular, Table10.1showsaconstantincreaseof about20%in theerrorwith respectto thereference,which is ameanableto thenonconsistentinitial conditionsof thepresentmethod1,while Table10.2showsareductionof about60X 80%in theerrordueto thehigheraccuracy of theproposedmethodwhenpartialalgorithmicdissipationis used.

10.1.2 Spinning top

A verysimplespinningtop is considered,madeof a rigid body, whosepropertiesareindicatedin Table10.3; threecasesareanalysed,following Ref. [25]2. Thethreecasesdiffer in the initial linearandangularvelocity. Thespin top hasits xaxisorientedin thedirectionof theglobalone,while its zaxisis rotatedawayfromtheglobaloneby 10 degreesaboutthex axis. Thecenterof gravity is at distanced 0 " 20 units from the pin point in the local z direction. All the propertiesaredimensionless.Gravity orientationis oppositeto theglobalz axis.

1SeeSection4.3for adiscussionon thestart-upof thesimulation.2First reportedin F. Mello’sPh.D.dissertationin 1989atGeorgia Tech., notdirectlyconsulted.

10.1. RIGID BODY MECHANISMS 119

Table10.3:Spintopproperties(Ref. [25])d 0.20

mass 1.00Jx 0.75Jy 0.75Jz 0.40

grav. -3.00

Table10.4:Spintop initial conditions(Ref. [25])Case1 Case2 Case30.19776 0.04181 0.

v 0. 0. 0.0. 0. 0.0. 0. 0.

ω 0.9888 0.20905 0.7.5167 6.2964 6.3794

The initial conditionsin the threecasesarereportedin Table10.4; eachof thethreecaseshasbeenintegratedwith a time stepof 0.001s for a durationof 10 s,to obtaina baselinecomputation.The last casehasbeenintegratedwith longertime stepsto obtaininformationon thesensitivity of themethodto thetime stepfor a problemthat, dueto the presenceof two overlappingmotionsat differentspeeds,onerelatedto thebasicspinningof thetopandtheotherto theprecessioncausedby thegravity, is somewhatstiff.Figures10.1,10.2,10.3reporttheplot of thex andy positionsof thespin top inthethreecasesconsidered.Theplotsperfectlymatchthoseof Ref. [25].Figure10.4 shows the differencesin Case3 whendifferent time stepsarecon-sidered.The differencesareappreciableonly for ∆t 0 " 05, to which a rotationincrementof about0.3rad.for eachstep,about1 10 π, corresponds.

10.1.3 Bipendulum

A rigid bipendulumis considered,againfrom Ref. [25]. The propertiesarere-portedin Table10.5. Thetwo rigid links areconnectedby a sphericalhinge;thefirst link is groundedat the otherextremity to the origin of the global referencesystemby a pin joint. The two links arealignedalongthenegative half of theyaxis,andarerotatingaboutthez axisat a speedω 4 " 0 rad/s.Dueto thegrav-


-0.1

-0.05

0

0.05

0.1

-0.1 -0.05 0 0.05 0.1

yY

x

Case 1

Figure10.1:Spintop— case1

-0.05

-0.025

0

0.025

0.05

-0.05 -0.025 0 0.025 0.05

yY

x

Case 2


10.1. RIGID BODY MECHANISMS 121

-0.06

-0.03

0

0.03

0.06

-0.06 -0.03 0 0.03 0.06

yY

x

Case 3


-0.06

-0.03

0

0.03

0.06

-0.06 -0.03 0 0.03 0.06

yY

x

Case 3, dt=1e-3Case 3, dt=5e-3Case 3, dt=5e-2

Figure10.4:Spintop— convergence


Table10.5:Bipendulumproperties(Ref. [25])L 1

mass 0.4565JtransZ 0.0308Jaxial 0.0001grav. -32.175

-2

-1

0

1

2

-2 -1 0 1 2

yY

x

Bar 1Bar 2

Figure10.5:Bipendulum— xCG vs.yCG

ity, opposedto axisz, themotionbecomesthree-dimensional.Figures10.5,10.6,10.7show thepathof thecentersof gravity of thetwo links; Figure10.8showsthetimehistoryof thetwo reactionsin directionz. Theplotspracticallyoverlapthoseof Ref. [25]. An extra plot is shown in Figure10.9,containingthetimehistoryofthetotal reactionbetweenthetwo joints.

10.2 Flexible elements

10.2.1 Flexible pendulum

Two problemsareconsidered.The first, the so-calledBathependulum[5], is aparticularlyill-posedproblem,consistingin averystiff pendulumsubjectto large

10.2. FLEXIBLE ELEMENTS 123

-2

-1

0

1

2

-2 -1 0 1 2

z[

x

Bar 1Bar 2

Figure10.6:Bipendulum— xCG vs.zCG

-2

-1

0

1

2

-2 -1 0 1 2

z[

y

Bar 1Bar 2

Figure10.7:Bipendulum— yCG vs.zCG


-20

0

20

40

60

80

0 0.2 0.4 0.6 0.8 1 1.2

Z-r

eact

ion\

Time (s)

Joint 1Joint 2

Figure10.8:Bipendulum— z reaction

0

20

40

60

80

100

0 0.2 0.4 0.6 0.8 1 1.2

Tot

al r

eact

ion]

Time (s)

Joint 1Joint 2

Figure10.9:Bipendulum— total reaction


amplitudeoscillations.Thesecondis a moreconventionalproblem,integratedasanODE,showing how algorithmicdissipationmaybeusedto obtainthedesiredbehaviour in termsof convergenceto a forcedsolution.

Bathependulum

Thepropertiesof Bathe’spendulumaresummarisedin Table10.6.Thependulumstartsin horizontalpositionandis droppedwith null initial velocity. Theperiodofthe ^ 90degreesoscillationsis about4 s,while thatof thesupportis about2.0e-4.Newmarkintegrationis shown by someauthorsto fail whentoo long a time stepis used( _ 0 " 025), while energy preservingandenerydissipatingalgorithmscanuselarger time steps(of theorderof 0.5, oneorderof magnitudelarger)even ifwith alargephaseerror. Thepresentalgorithm,whentheproblemis implementedusingtherotationasunknown, i.e.

ϑ ϕ u wϕ 2ϑw g sin ϑ !%!` l w lϕ2 k m! u g cos ϑ !9

being ϑ the angularpositionof the pendulumandu the elongationof the rod,obtainsverygoodresults,while theglobalcoordinateimplementation,namely

mx qx 0 my qy 0

qx k x l0 l ! 0 qy k y l0 l ! g

aswell astheindex 1 DAE implementation,

mx qx 0 my qy 0

qx x lλ 0 qy y lλ g

λ k l l0 1! 0 wherel ba x2 y2, in themultibodyframework3 behavesmoreliketheNewmarkscheme,i.e. it requiresacomparatively shorttimestep( _ 0 " 04 X 0 " 10, dependingon thealgorithmicdissipationandon the typeof joint at theclampedendof the

3Actually, therewerealgebraicconstraintsaswell, becausein thecurrentimplementationtherodelementrequirestwo connectionnodes,sothepin pointneedsbegroundedby a clampjoint.


Table10.6:Bathependulumproperties(Ref. [5]).L 3.0443 m

mass 10.0 kggrav. 9.80 m/s2

EA 1.e10 N

-90

-60

-30

0

30

60

90

0 2 4 6 8 10

Ang

le (

deg)c

Time (s)

cubic, dt=0.01multistep, dt=0.01

cubic, dt=0.50multistep, dt=0.50

multistep (global coord.), dt=0.05

Figure10.10:Bathependulum— angularposition.

rod) not to fail, while still yielding accurateresults,becauseof theextremestiff-nessof the problem. Figure10.10shows how the angularpositionis integratedby someintegratorsof theproposedfamily. Notice that, in casethehigher-orderschemeis considered,a time stepof 0.5 yields a solutionthat is still very closeto the accurateone,with limited phaseerror; on the contrary, whenthe secondorderschemeis used,it yieldsanappreciablephaseerrorbut no divergencefromthesolutionor excessive lossin total energy dueto algorithmicdissipation.Bothcasesconsideranasymptoticradiusρ∞ 0 " 6. This is a typical caseof ill-posedproblem;by usinga rigid link insteadof anextremelystiff rod, thestiffnesscanbe eliminatedwithout appreciabledifferencesin the solutionfor slow dynamicssimulations.In fact, if accuracy is thegoal,very shorttime stepsarerequiredtoexploit the flexibility of the rod; on the contrary, whena quick integrationwithlong timestepsis sought,theflexible rod is definitelynot required.


Table10.7:Deformablependulumproperties.L 1 m

mass 1 kggrav. 9.81 m/s2

EA 9.81e4 N

Deformable pendulum

Consideradeformablependulum,whosepropertiesaresummarisedin Table10.7.Thelink frequency is 100timesthatof thependulumfor smalloscillations.Thependulumstartsat the stablerestpositionwith initial speedv 6 " 3 m/s,whichis sufficient to reachtheunstablerestpositionwith someresidualkinetic energy.The initial conditionsareconsistentandbalanced,but having therod null initialelongation,theinertiaforcedueto thecentrifugalaccelerationstrainsit duringtheveryfirst stepsandcausesoscillationsthatareproblem-related,andnotinducedbythealgorithm.Thesystemis integratedwith a time stepof 0.02s, correspondingto aboutahundredthof theperiodof thependulum,but verycloseto theperiodoftherod. As a result,themotionof therod cannotbecapturedby the integration,but it leavesnumericaloscillationsin thesolution.Theproblemis integratedwithboth the multistepand the cubic formulaswith algorithmicdampingρ∞ 0 " 6,comparedto abaselinesolutionthatcorrectlyintegratestheoscillationsof therod,asshown in Figure10.11. Figure10.12shows that the effect of the algorithmicdampingis to drive thesolutioncloseto the onewith no oscillations,wheretheinitial elongationof therod hasbeenaccountedfor. In fact,in suchcase,thereisno sourceof spuriousoscillations.

10.2.2 Buckling of axially compressedbeam

Thebuckling critical loaddueto a conservativeaxial forcethatcompressesa baris a classicalbenchmarkfor nonlinearbeamformulations.Fromlinearelasticitytheory, thevalueof thecritical loadis:

P ed π2 f 2 EJ

l2

for a clamped-freebeam. This static stability result is correctwhen the axialdeadload is conservative [9]. Thefinite volumelinearisedpre-stressmatrix of astraight,untwistedbeamfollows from the linearisationof themomentarmsma-trix, Eq. 5.15,abouta previously calculatedpre-stresscondition;by discretisingthe displacementsat the evaluationpointsby meansof the nodaldisplacements


-0.0006

-0.0004

-0.0002

0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0 0.2 0.4 0.6 0.8 1

Elo

ngat

ion,

m

Time, s

ReferenceMultistep, h=2.e-2

Cubic, h=2.e-2

Figure10.11:Deformablependulum— elongation.

-0.0006

-0.0004

-0.0002

0

0.0002

0.0004

0.0006

0.0008

0.001

0 0.2 0.4 0.6 0.8 1

Elo

ngat

ion,

m

Time, s

h=2.e-2, rho=1.h=2.e-2, rho=1., initial equilibrium

h=2.e-2, rho=.6

Figure10.12:Deformablependulum— correctinitial elongation.


asdetailedin Section5.4.4, the following expressionfor the pre-stressstiffnessmatrixof thethree-nodebeamelementis obtained:

Kp ,BBBBBBBB-0 0

tI0 gh NI i δi1I ! tI0 g NI i fi g0 04 tI0 gh NI i δi2I ! tI I0 gh NI I i δi2I ! 5 4 tI0 g NI i fi g tI I0 g NI I i fi g 50 0

tI I0 gh NI I i δi3I ! tI I0 g NI I i fi g/ IIIIIIII0

xi

ϕi The critical loadsand the buckling shapeshave beencalculatedfor a straight,uniform beammodelledwith a single element,with equally spacednodesandloadedby an axial compressionforce. Two buckling modeshave beenfound ineachtransversedirection,whosecritical valuesare:

Pf v 6 d 5 ^ji 21f EJl2 (10.1)

The lower valuegivesan error of about1 " 48%. Quasi-staticsimulationsof thebuckling conditionhave beenperformedusinga slowly growing axial compres-sion loadanda small transverseload. An abruptincreaseof the lateraldeviationtakesplacewhentheaxialloadapproachesthelowestcritical value.Subsequently,the transverseload is removed, and if the critical valuehasbeenexceeded,thebeamremainsbent.Thenumericalsimulationscanonly determinelowerandup-per boundsfor the critical values,becausewhenthe exact valueis approached,theJacobianmatrixof thesystembecomessingular. Thelowerandupperboundsobtainedby meansof a singleelementmodelexactly boundthe analyticvaluegivenby Eq. (10.1). Theboundsobtainedusinga 4 elementmodelarecorrecttowithin ^ 0 " 04%with regardto theanalyticalvalue,showing theeffectivenessof arelatively coarsemodel.Thebeamcanresista loadthat is largerthanthecriticalloadwhenthefull nonlinearbehaviour is takeninto account,but a largecurvaturedevelops,asshown in Fig. 10.13.In thisfigureafour-elementbeamcarriesa loadup to twice thebuckling critical value. Figure10.14shows the internalforcesattheevaluationpointsdueto the2 Pcr load.

10.2.3 Rotor blademodal analysis

Theresultsof themodalanalysisof a rotor bladearepresentedto assesstheva-lidity of thefinite volumebeammodel.A linearisedimplementationof thefinitevolumebeam,with both lumpedandconsistentinertia matrices,hasbeenusedto performa standardeigenvalueanalysisof a previously validatedrotor blademodel[27]. Theseresultshave beenpublishedin Ref. [37], wherethemultibody


-0.2

0

0.2

0.4

0.6

0.8

0 0.2 0.4 0.6 0.8 1

y (a

dim

.)k

x (adim.)

1/2 Critical Load1 " "

3/2 " "2 " "

Figure10.13:Deformedshapesof a 4 three-nodeelementbeamunder1/2,1, 3/2and2 timesthecritical buckling load

-60

-40

-20

0

20

40

60

0 0.2 0.4 0.6 0.8 1

Inte

rnal

For

ces

Abscissa (adim.)

Axial Force, NShear Force

Bending Moment

Figure10.14:Internalforcesdueto twice thecritical buckling load,at theevalu-ationpointsof thefour beamelementmodel


Table10.8:Frequenciesof ahelicopterrotor bladeMode ModeType Exper. Ref. [27] Lumped Consist.

1 1st Beam 7.42 7.49 7.49 7.492 2nd Beam 21.90 21.93 21.91 21.893 3rd Beam 42.55 43.17 42.72 42.884 1st Chord 43.36 44.45 44.51 44.475 1st Torsional 65.70 66.35 68.32 66.226 4th Beam 70.81 73.34 72.64 72.627 5th Beam 105.78 110.20 107.51 107.858 2nd Chord 121.97 125.34 125.69 125.57

implementationof the finite volumebeamhasbeenpresentedfirst. The modelrepresentstherotor bladeof a commercialhelicopter. Thebladeis madeof a ‘C’shapedaluminummainspar, closedby arearspar. Thewholestructureis coveredby an aluminumsheetandrepresentsaboutonethird of the chord. The trailingpartof theairfoil is honeycombedandcoveredby analuminumskin. Thetrailingedgeis madeof a‘V’ shapedaluminumrib. A steelantiabrasivestrip is putontheleadingedge.Non-structuralmasseshavebeentakeninto account,aswell asglueandpaint weight. Fourteencrosssectionsof the bladehave beenindependentlyanalysedby meansof the beamsectionanalysispresentedin Section5.3 [27].The sparsandthe honeycombhave beenmodelledby meansof brick elements,while two dimensionallaminæhave beenusedfor the skin andthe antiabrasivestrip. A threedimensionalmodelwasthengenerated,madeof fifteenthree-nodebeamelements(Figure10.15). The Frequency ResponseFunction(FRF) of thebladehasbeencalculatedby analysingthePower SpectralDensity(PSD)of thebladeexcitedwith arandomforceatoneend.Theresponsehasbeencomputedbymeansof themulti-bodycode,usingthesamemodelof the linearanalysis.Theresultsof thespectralanalysesarereportedin Table10.8.Themodalfrequenciesobtainedby meansof thePSDof theresponsecomputedwith themulti-bodycodearein agoodagreementwith theexperimentandwith previousanalyses.

10.2.4 Flexible leverage

The analysisof a flexible leverageis presented. This example is taken fromRef. [84]4. It consistsin two rigid links groundedat oneend,andconnectedto abeamto theotherend,asshown in Figure10.16.Thebeamcarriesalumpedmass

4Thereferencewasnot directly available;datahasbeenaccessedthanksto LorenzoTrainelli[85].


Model Discretisation

Analysed Cross-sections

Influence Length

End nodes Mid-point nodes

Figure10.15:Sketchof thediscretisedhelicopterblade

at midpoint; themassis largecomparedto theinertiaof thebeam.An impulsiveforceis appliedto themassalongtheaxisof thebeam(x direction);theforcehasbeenactuallymodelledasa triangleimpulse,with a total durationof 0.256s andamaximumvalueof 2 N. Thepropertiesof thesystemarereportedin Table10.9.This problemis very significantbecause,dueto thegeometryof theleverage,af-ter about0.5secondsthemechanismreachesa singularpoint, in which thebeamandthefirst link arealigned;asaconsequence,theoppositeendof thebeamstopsabruptlyandinvertsits motion. This causesimpulsive loadsto propagatenearlyaxially alongthe beam,asshown in Figures10.17and10.18,andan impulsivereactionforceon the left hinge,seeFigure10.21. Themovementof theflexibleleverage,afterthesingularpoint, is dominatedby bendingoscillationsdueto thehigh-frequency, internaldynamicsof thebeam,asshown in Figures10.22,10.24,10.25,while themid node,attachedto thebig mass,behavessmoothly, seeFig-ure 10.23. Figures10.19and10.20show the transverseshearandthe bendingmomentat the left endof the beam. The oscillatorybehaviour is apparent;butsincethediscretisationis very different,dueto the impulsive natureof the load,theresponsehasa very differentspectralcontent.Thebeamhasbeendiscretisedwith 2 and4 beamelements,correspondingto 5 and9 nodesrespectively. A rigidcasehasalsobeenconsidered,asa baseline.Figure10.26shows a portionof thefrequency contentof thebendingmomentcloseto the leftmostendof thebeam;it clearlyshows that the two discretisationsbehave quitedifferently. Notice thatnoneof the frequenciesdepictedis in the rangeof the axial frequenciesof thebeam,the lowestbeingof theorderof 3.5 kHz. Actually, a time stepof 1.5e-5sis requiredto correctlycapturetheaxial frequencies( _ 20 stepsper cycle). Theaxial frequenciesof thefreebeamaregivenby ω λ a EA m, whereλ is:

1. 0 for therigid bodymode;

2. for thesymmetricmodes,in which themassstandsstill andthebeamoscil-latessymmetrically, λ π l 2kπ l , with theintegerk l 0; thefirst modehasω _ 3 " 6 kHz;

10.3. GENERALISEDPREDICTIVECONTROL 133

3. for theskew symmetricmodes,wherethelumpedmassoscillatesaboutthemidpoint andthe two partsof thebeamoscillatein phaseopposition,λ istherootof

tan λl 2!m M ml ! λl 2 0;

thefirst modehasω _ 3 " 5 kHz. NoticethatwhenM 2m! is very large,thesolutionat the limit approachesthat of the symmetricmodesfrom below,asin the presentcase,becausethe lumpedmassactslike a clamp;on thecontrary, whenM 2m! is verysmall,thesystemdegeneratesinto asimpleuniformbeam,approachingλ 2kπ l from above.

The coarsermodel hasthe peaksrelatedto the first modesat little higher fre-quencies,with no frequency contentabove 60 Hz, while the model with finerdiscretisationshows the samepeaksof the previous one shifted towardslowerfrequencies,with the addition of higher modesup to 250 Hz. In this specificproblem,theadditionof degreesof freedomaddsfrequency contentto themodelandmakesthingsworsebecausethe excitation activatesall the modes. So, notonly to achieve accuracy, but also to be able to find a solution, the integrationrequiressmallerandsmallertime stepsto capturethe propagationof the effectsof thesingularpoint throughoutthedeformablepartof themodel. In fact,whilethefour-beammodelhasnot beenableto passthesingularpoint with a timesteplongerthan4.e-4s, thetwo-beammodelcouldusea time stepup to 1.e-3s,withvery little differencesbetweenthetwo cases.Whena variablestephasbeenusedwith the four-beammodel,thefirst partof the integrationwascarriedout with atimestepof 5.e-3s; thetime stepdroppedto 2.e-4sduringthetransitionthroughthe singularpoint; graduallyit wasrestoredto about8.e-4X 9.e-4for the restofthe simulation. Comparedto the maximumfixed time stepfeasiblesimulation,only about40%of thestepswereperformed,with about60%of theiterations.Theplotsin Figures10.22–10.25arecomparableto thosefoundin theliterature;no exact solution to this problemis available due to its intrinsic stiffness: thehigherthemeshrefinement,thehigherthe numberof modesthatareexcitedbytheimpulsiveload,resultingin achaotic,discretisationdependentresponse.

10.3 GeneralisedPredictiveControl

10.3.1 Thr eemassessystem

A verysimplesystem,shown in Figure10.27,is studied.It is takenfromRef.[63].The systemis madeof threemassesin serieswith threespringsand dampers,groundedatoneend;anexcitationforceis appliedat thefreeend,andthecontrol


Table10.9:Flexible leverageproperties.beamlength 0.72 mlink 1 length 0.36 mlink 2 length 0.36i 2 mlumpedmass 0.5 kgbeammass 9.76e-3 kg

Axial stiffness 3.65e5 NBendingstiffness 3.04e-2 Nm2

-0.6

-0.4

-0.2

0

0.2

-0.2 0 0.2 0.4 0.6 0.8 1 1.2

ground hinge 1

hinge 1 hinge 2

ground hinge 2

link 1link 2

beam

singularpoint

Figure10.16:Flexible Leverage— scheme.


-60

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-20

0

20

40

60

80

100

120

140

0 0.2 0.4 0.6 0.8 1

Axi

al fo

rce

(N)n

Time (s)

2 beam model4 beam model

Figure10.17:Flexible Leverage— axial forcecloseto theleft endof thebeam.

-60

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-20

0

20

40

60

80

100

120

140

0.6 0.61 0.62 0.63 0.64 0.65 0.66 0.67 0.68 0.69 0.7

Axi

al fo

rce

(N)n

Time (s)


Figure10.18:Flexible Leverage— zoomof Figure10.17.


-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

0 0.2 0.4 0.6 0.8 1

She

ar fo

rce

(N)n

Time (s)


Figure10.19:Flexible Leverage— transverseshearforcecloseto theleft endofthebeam.

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0 0.2 0.4 0.6 0.8 1

Ben

ding

mom

ent (

Nm

)

n

Time (s)


Figure10.20: Flexible Leverage— bendingmomentcloseto the left endof thebeam.


-120

-100

-80

-60

-40

-20

0

20

40

60

0 0.2 0.4 0.6 0.8 1

Join

t Rea

ctio

n (N

)o

Time (s)


Figure10.21:Flexible Leverage— left link x reaction.

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0.2 0.25 0.3 0.35 0.4 0.45 0.5

yp

x


Figure10.22:Flexible Leverage— intermediatenode1 path.


-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

0.35 0.4 0.45 0.5 0.55 0.6 0.65

yp

x

Rigid model2 beam model4 beam model

Figure10.23:Flexible Leverage— mid-nodepath.

-0.5

-0.45

-0.4

-0.35

-0.3

-0.25

-0.2

0.5 0.55 0.6 0.65 0.7 0.75 0.8

yp

x


Figure10.24:Flexible Leverage— intermediatenode2 path.


-0.5

-0.3

-0.1

0.1

0 0.2 0.4 0.6 0.8 1

yq

x

left hingeintermediate node 1

midnode

intermediate node 2

righthinge

Figure10.25:Flexible Leverage— nodepathsof the4 beammodel.

0

2

4

6

8

10

12

0 50 100 150 200 250

Ben

ding

mom

ent (

Nm

)

n

Frequency (Hz)


Figure 10.26: Flexible Leverage— frequency contentof the bendingmomentcloseto theleft endof thebeam.


Regulator

DesignerControl

Disturbance

a 2 a 1

m1m2m 3

Figure10.27:Threemasses— scheme

measuresaretheaccelerationsattheothertwo points,thusimplementingasystemwith no directtransmissionterm. Thepropertiesare:m1 r m2 r m3 r 1, k1 r 1,k2 r 2, k3 r 3; the dampingis assumed5 proportionalto the squareroot of thestiffness,i.e. ci r 0 s 05t ki . Thesystemhas6 poles.

GeneralisedPredictive Control

Differentvaluesfor theorderof theidentifiedsystemp aswell asfor thecontrolweight λ have beentested.The predictionandcontrol advancinghorizonshavebeenchosenequalto p. The integrationtime stepis 0.01s; thesamplingfor thediscretecontroller is taken every 10 time steps.The systemis excited by a unitamplitudeharmonicforce at 0.4 Hz; a white noisewith 0.001amplitudeis ap-plied andmeasuredto identify thesystem.Thecontrol is activatedafter40 s ofsimulation. An order p r 6 hasbeenused,with λ r 10u 3. The control weightis graduallyreducedto thenominalvaluein abouttensecondsto avoid anabruptinterventionof thecontrol.Figures10.28,10.29show thetwo measuredaccelera-tionsandthecontrolforce,andthedisplacementsat thethreenodes,respectively.

Stabilisation

Whenthefirst dampingcoefficient is setto a negativevaluec1 rwv 0 s 15t k1, theresponse,Figure10.30,shows the effectivenessof this form of adaptive controlfor unstablesystems.It is interestingto noticethat the control is not collocated

5Thereseemsto beanambiguityin Ref. [63] in thedefinitionof thedamping.


-1.5-1

-0.50

0.51

1.5

0 20 40 60 80 100t

Node 1 accel.Node 2 accel.

-1.5-1

-0.50

0.51

1.5

0 20 40 60 80 100t

Control force

Figure10.28:Threemasses— controlsignals

-1

-0.5

0

0.5

1

0 20 40 60 80 100t

Node 1Node 2Node 3

Figure10.29:Threemasses—displacements


-1

-0.5

0

0.5

1

0 20 40 60 80 100t

Node 1Node 2Node 3

Figure10.30:Threemasses— stabiliseddisplacements

andtheunstablesectionof thesystemis closeto oneof themeasures,but it is notdirectly controllable.Theeffect of thecontrolleris to drive mass3 in phasewithmass2, andthento move thesetwo massesin phaseoppositionwith mass1, theunstableone,to cancelits oscillations.Thecontrolweightλ canbereducedevenmore,with performanceimprovementsexpecially in the unstablecase,but withexcessively roughbehaviour of thecontrolforce.

Chapter 11

Tiltr otor modelanalysis

Intr oduction

The 1/5-scaleaeroelasticmodelof the V-22 (Figure 11.1) wasbuilt and testedin the periodfrom 1983to 1988,to supportthe preliminarydesignandthe fullscaledevelopmentof thetiltrotor aircraft laterknown astheJVX. Thewind tun-nel testsbeganat the TransonicDynamicTunnel (TDT) of the NASA LangleyResearch Center(LaRC) on a semispanmodel,andwereglobally performedinthreedifferentfacilities,includingtheBoeingHelicopterVSTOL tunnelfor boththesemispanandthefull spanmodelconfigurations.Thesemispanmodelis cur-rently referredto astheWing RotorAeroelasticTestingSystem(WRATS),andislocatedat theLaRC.A tiltrotor aircraft is a complex systemthathasthebehaviour of both a conven-tionalairplaneandof arotorcraft,with peculiarmaneuvres,e.g.theconversion.Itsaerodynamicsarecharacterisedby thehigh influenceof therotoron theairstreamthat affects the wing, in both helicopterandairplaneconfiguration. In fact theWRATS projecthasbeenmainly focusedon the reductionof thevibration levelinducedby theseinteractionsin theairplanemodeby meansof anactively con-trolled swashplatewith theHigherHarmonicControl (HHC) technique,coupledto anactiveflap[74]. Thebladesof therotor representacompromisebetweenhe-licopterandpropellerblades.Sincethey areoptimisedfor thehighaxialairstreamspeedtypical of the airplanemode,they arehighly twisted, thusshowing highelasticcouplingsbetweentwist andin- andout-of-planebending[73]. Thecon-versionmaneuvre,whenperformedat typical ratesfor aircraftcontrol,introducesgyroscopiceffectsthatareunusualin conventionalhelicopters.Theflexibility ofthe wing canmagnify the effectsof groundandair resonances,the latter beingtypical of automaticallycontrolledrotorcrafts.Many of theseproblemsarewellknown, but haveneverbeenfacedto thisextent,while othersarecompletelynew.

143

144 CHAPTER11. TILTROTORMODEL ANALYSIS

Figure11.1:WRATSModelat Langley’sTransonicDynamicsTunnel

For this reason,a particularly intensive experimentalcampaignpreceededandaccompaniedthe whole developmentof the JVX [65], supportedby numericalanalysesof therelatedsubproblems.Eigenvalueanalysesof the rotor dynamics,the determinationof flutter margins of the rotor, of the wing andof the ensem-ble, by meansof analyticalmodelsbasedon comprehensive rotorcraft analysiscodes,anddynamicsimulationsof therotormechanismsby meansof earlymulti-body codeshave beenperformedat eachstepof the wind tunnel investigations[77], [82]. A pictorial historyof proprotorinvestigationat Langley is depictedinFigure11.2,wherein recentyearsthe proposedmultibody analisisformulation,implementedin a codecalledMBDyn, is mentioned. The figure is taken fromtheviewgraphsaccompanying a talk thatDr. Kvaternikrecentlygave at Langley.Figure11.3refersto testcampaign529,performedin August‘98. Theresultsofthepresentwork havebeenpresentedin [38], andsubsequentlypublishedin [39]andpresentedin [41].

Mostof theanalysesof theWRATSmodelwereperformedby meansof compre-hensive rotorcraftcodes,with properlysimplified models. In this work the useof a single,state-of-the-art,multipurposemultibodycode,capableof performingmostof therequiredsimulationsstartingfrom asinglebulk modelthatcanbespe-cialisedfor eachanalysis,is consideredasa possiblemeansto accomplishmostof thetaskspreviouslyundertakenby specialisedanalysistools.

Eachsubpartof the tiltrotor is modelledandanalysedin its basickinematicand

145

Figure11.2:Pictorialhistoryof proprotor/tiltrotorinvestigationat Langley.


Figure11.3: Fromthe left, Pierangelo Masarati andDr. Mark W. Nixon in theTDT duringAugust‘98 testcampaign529.

dynamicfeatures;subsequentlythepartsareassembledtogether, andthesystemis analysedasa whole. By usingthesamecodeandthesamemodellingfor thesinglepartsandtheassembly, andby usinga rathergeneralapproachin thekine-maticandmechanicaldescriptionof thesingleparts,any undueapproximationisavoided.

11.1 Tiltr otor submodels

Thetiltrotor hasbeensplit in thefollowing subsystems:

1. Theblade,madeupof aflexbeamandapitchhingemechanism,connectingthebladeto thehub,andapitch link transmittingthepitchcontrolfrom theswashplate.Eitherrigid or flexible bladeshavebeenconsidered.

2. Thegimbal,a constantvelocity joint madeof all of its mechanicalcompo-nentsin orderto giveanaccuratekinematicdescriptionof thejoint.

3. Theswashplate,madeof thetwo plates,thetwo scissorsthatconstraintheaxial rotationof theplateswith respectto thepylon andto thehub,andthethreenon-rotatinglinks thatcontrolthecollectiveandthecyclic positionoftheplates.

11.1. TILTROTORSUBMODELS 147

Rigid Blade

BladeDeformable

Aerodynamic Fairings

Gimbal

Flexbeam

Cuff, Bearing

Swash Plate

Conversion Mechanism

Semispan Wing

Fixed Control Link

Rotating Control Link

and Pitch Link

Figure11.4:AnalyticalModel

4. Thehalf-wingmodel,madeof thedeformablewing, thepylon, modelledasa rigid body, theconversionhinge,thedownstopspringandthemast.

Thecompletemodelis sketchedin Figure11.4.

11.1.1 Blade

The singleblademodelhasbeenusedto analysethe dynamicpropertiesof theisolatedblade,suchasfrequenciesandaerodynamicproperties.Threedifferentmodelshave beenconsidered,with increasingdiscretisationrefinement.All themodelssharethedescriptionof theflexbeam,thatusesathree-nodebeamelement,andof thecontrols.Thebladeis joinedto theflexbeamby asphericalhingeat theouterend,andby a spanwiseorientedin-line joint, 2.2 in outwardsof the rotoraxis.Thebendingof theflexbeamaccountsfor flexible flapandlagmotion,whilethepitchrotationis allowedby thebearings.A distancejoint betweentherotatingswashplateandanoffsetpointaft of eachbladecuff modelsthecontrollink. It canbebothrigid or flexible, to accountfor theflexibility of thecontrolsystem.A rigidbladehasbeenconsideredfirst, whichprovedto bepoorbecausetheyoke is verystiff, soeventheveryfirst in-planebendingmodeimpliesappreciabledeformationof thebladeitself. On theotherhand,therigid blademayrepresentanacceptable


Table11.1:Cantileveredbladefrequencies,HzMode Exp UMARC NASTRAN MBDyn

4 elem. 2 elem.1 Beam 12.29 12.3 11.5 11.3 11.71 Chord 34.11 34.1 33.4 33.1 32.72 Beam 52.44 53.0 56.7 55.8 55.01 Tors. 113.35 111.4 127.0 119.0 122.0

0

20

40

60

80

100

120

140

1 Bea

m

1 Chord

2 Bea

m

1 Twis

t

GVTUMARCNASTRANMBDyn 2 El.MBDyn 4 El.

Figure11.5:Cantileveredbladefrequencies,Hz

tradeoff whenonly theperformancesof therotor areaddressed.A flexible bladehasbeensubsequentlymodelled,with two andfour beamelementsrespectively.Thefirst frequenciesof thecantileveredbladeobtainedby thepresentanalysisarereportedin Table11.1andin Figure11.5, comparedto GroundVibration Tests(GVT), andto UMARC andNASTRAN FiniteElementAnalysiscodesresults.

11.1.2 Gimbal

Thegimbalmodelhasbeenusedto determinethekinematicandgyroscopicprop-ertiesof therotor. It consistsin a constantvelocity joint, madeby two universaljoints, linked to the mastandto the hub respectively at onearm of eachcross.Theotherarmof thecrossesis connectedto a linkage,that transmitsthe torque

11.1. TILTROTORSUBMODELS 149

betweenthe mastandthe hub andkeepsconstantthe distancebetweenthe twoU-joints. Thehubis alsolinked,by meansof an in-line joint, to a sphericaljointon themastthatallows thegimbalmotion. Thegimbalallows therigid flappingof the whole rotor and,sincethe directionof the angularvelocity tilts togetherwith thehub,no Coriolis forcesdueto this motion resultin thebladeswhentheflappingis steady. At thesametime,the1 perrev. flappingmotionhasnostiffnessdueto centrifugaleffects,but only thatprovidedby asetof springs.

11.1.3 Swashplate

The swashplatemodel hasbeenusedto analysethe kinematicsof the controlsystemand,togetherwith thegimbalandtherigid blade,to evaluatethepitch-flap-lag couplingsfor thewholecollectivepitch range.It hasbeenusedalsoto applythedesiredcontrolsto therotor duringrealisticmaneuvresimulations.It consistsin the two plates,modelledas rigid bodiesjoined by a planehinge. The fixedplateis linkedto thepylon by meansof an in-line joint that forcesit to translatealongthe mast. A fixed anda rotatingscissorconstrainthe rotationof the twoplatesabouttherotor axis,with respectto thepylon andthemast.Threevariabledistancejoints areusedto controlthetranslationalongthemast(collectivepitch)andattitude(cyclic pitch) of the non-rotatingplate. The elongationof the fixedcontrol links is imposedby meansof a dedicatedgeneralpurposeelementthatsplits the threefundamentalcontrol inputs, namelycollective, and fore/aft andlateralcyclic pitch angles,into the elongationsof the links. Figures11.6–11.7respectively representthe kinematicpitch-flap coupling due to the gimbal andtheflexible flapping,andthecontrol stiffnessasfunctionof thecollective pitch,comparedto dataobtainedfrom Bell Helicopterandfrom modelcalibrations.

11.1.4 Wing-Pylon

Thehalf wing modelhasbeenusedfor aeroelasticclearanceof theisolatedwing.Both dynamicandaeroelasticpropertiesof thewing in forwardflight have beenanalysed.It consistsin two beamelementsfor thewing, andin thepylon, mod-elledwith a rigid body. Thepylon is connectedto thewing by meansof aflexiblespindle,modelledwith abeamelement,andadownstopspring.Thespindlemod-els the conversionbearing. Its bendingallows the pylon to rotatewith respectto the wing aboutthe roll and yaw axes; it mimics the flexibility of the trans-missionshaftat the conversionbearing.The conversionactuatorconstrainsthisrotation, andcontrolsthe conversionangle. In the wind tunnelmodel, springswith differentpropertiesareusedto simulatethe behavior of the conversionac-tuator in helicopterandairplaneconfiguration,both on- andoff-downstop. The


-26

-24

-22

-20

-18

-16

-14

-20 -10 0 10 20 30 40 50 60 70 80

delta

3, d

egx

Collective Pitch at 75% of the Blade, deg

Bell Helicopter ResultsMBDyn

Figure11.6:Pitch-flapcouplingasfunctionof thecollectivepitch θ75%

5000

6000

7000

8000

9000

10000

11000

12000

13000

14000

0 10 20 30 40 50 60 70

Con

trol

Stif

fnes

s, in

-lb/r

ad


ExperimentMBDyn

MBDyn - no flexbeam

Figure11.7:Controlstiffnessasfunctionof thecollectivepitch θ75%

11.2. PRELIMINARY CONSIDERATIONS 151

Table11.2:Wing frequencies,HzMode GVT NASTRAN MBDyn

downstoponBeam 6.00 6.16 5.9Chord 8.45 9.33 9.1Twist 12.5 12.6 12.5PylonYaw 16.5 18.9 17.2

downstopoffBeam 5.51 5.45 5.4Chord 8.45 8.74 8.8Twist 10.6 10.6 11.0PylonYaw 16.7 16.7 16.6

mainfrequenciesof thewing-pylon arereportedin Table11.2andin Figure11.8,comparedto bothGVT andNASTRAN results.

11.2 Preliminary considerations

Somepreliminaryconsiderationson theflappingmotion: therotationaxisfor theflappingdueto thegimbal,namely1 perrev. flapping,is locatedon therotor ro-tation axis, 2 in above the rotor plane,while the onefor the flappingdueto theflexbeamdeformation,namelythe coneand y 1 per rev. flapping,lies about1.5in outboardalong the bladeaxis. The pitch control is linked to the blade75o

aft of thebladeitself andthusintroducesa pitch-flapcouplingδ3 zr v 15o that isnegative (flap up causespitch up) for the1 per rev. motion,andslightly positivefor theflexible flapping. It is known that, for a stiff-in-planerotor, a positive δ3

cangiveraiseto apitch-flapaeroelasticinstabilitywhenthefirst out-of-planeandin-planefrequenciesnearlymeet[60]. The occurrenceof this instability in thesimulationsrequireda deeperanalysisof theflexibility of theyoke. In detail,theflexibility of the inner part of the yoke, from the hub to the inner pitch bearing,proved to be fundamentalin describingthe correctcouplingbetweenthe bladepitch andthe flexible, symmetricflappingmotion. After this part wasproperlymodelled,the slight, symmetricinstability in the analyticalmodeldisappeared.Figure11.9shows thechangein pitch-flapcouplingfor theconeflappingmotionboth for rigid andflexible root of the flexbeam. The experimentaldataseemtoindicatethatthecorrectflexible pitch-flapcouplingbesomwhatbetweentherigidandthedeformableflexbeamcase,biasedtowardstheflexible caseat lower col-lectiveangles,andtowardstherigid caseathighercollective.A three-dimensionalmodellingof theyoke is possiblyrequiredto determinetheexactcouplingeffect.


0

4

8

12

16

20

Beam

ChordTwis

t

P. Yaw

GVTNASTRANMBDyn

0

4

8

12

16

20

Beam

ChordTwis

t

P. Yaw

GVTNASTRANMBDyn

Figure11.8:Wing frequencies,Hz — downstopoff (top) andon (bottom).

11.3. ROTOR MODELS 153

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

-20 -10 0 10 20 30 40 50 60 70 80

rad/

rad


Pitch-Cone Coupling

Bell Helicopter ResultsMBDyn - Flexible Root

MBDyn - Rigid Root

Figure11.9:Pitch-conecouplingasfunctionof thecollectivepitch θ75%

11.3 Rotor models

The model,consistingin the rotor with rigid/deformableblades,thegimbal andtheswashplate,hasbeenusedto investigatethestabilityof therotor, with particu-lar regardto thepitch-flap-lagcoupling,andto evaluatetheaerodynamicresponseto the controls. The rigid blademodelmatchedthe first out-of-planefrequency,but gave poor resultseven for the first in-planefrequency, so it wasof little usein otherthanperformanceanalyses.Theflexible blademodelsagreedvery wellwith availabledatafor the low frequenciesof the rotor, both in the rotatingandnon-rotatingcasesin vacuo. Both thesinglebladeandthecomplete,threebladerotormodelshavebeenanalysed.For thispurpose,UMARC hasbeenmodifiedtoallow themodellingof multiple bladerotorsin thefinite elementsanalysismod-ule, with multiple load pathsto accountfor the control links. Tables11.3 and11.4 show that the presenceof the gimballedhub modifiesthe naturalfrequen-ciesof thesystemby breakingthesymmetry. In fact,non-symmetricmodesarefound,asshown bothby theanalysisandtheexperiment.Resultsfrom Bell Heli-copterswereavailablefor thelockedgimbalcase,sincethey wereobtainedfor asingleblademodel.They referto anold configurationof thehub,with calibratedspringsat thebladeroot to simulatethestiffnessof thecontrols.Theseresultsarenot completelyrepresentativeof thecurrentconfiguration.TheGVT resultswithlockedgimbalarealsonot completelysignificant,sincethegimbal couldnot beperfectlylocked.As anexample,in Tables11.5and11.6,therotatingfrequencies


Table11.3:Singlebladewith flexbeam(lockedgimbal),non-rotating,HzMode BELL GVT UMARC MBDyn

10 deg. 50deg. 10deg. 10deg. 10deg. 50 deg.Cone 6.6 7.2 6.6 6.3 6.8 7.82nd Flap 26.6 35.1 25.2 30.8 28.5 39.03rd Flap 68.8 77.3 69.3 77.9 73.5 82.01st Lag 19.3 12.5 20.6 19.3 19.5 12.61st Twist 114.5 109.9 112.6 110.0 109.0 107.0

Table11.4:Full rotor (freegimbal),non-rotating,HzMode GVT MBDyn

10deg. 10deg. 50deg.Gimbal 2.0 1.8 1.5Cone 6.8 7.0 7.82nd Flap 25.0 26.5 36.52nd Flapasym. 64.2 57.1 55.03rd Flap 76.2 78.0 82.51st Lag 19.7 19.0 12.72nd Lag 91.3 98.0 92.01st Twist 112.1 109.0 107.5

of thecomplete,threebladerotor arereportedat two typical rotatingspeedsandcollectivepitches,relatedto hoverandcruiseflight conditions,respectively.

11.4 Wing model

Thewing modelshows goodcorrelationfor thelowestmodes,asreportedin Ta-ble 11.2. The resultsare comparedto experimentalmeasurementsand to nu-mericalresultsbasedon NASTRAN code[75]. Thebeam1 andtwist modesarestronglycoupledandareinfluencedby thepropertiesof thedownstop,thespringthat is usedto lock theconversionactuatorin airplanemode.At presentthereisno conversionactuatoron the wind tunnelmodel,so it is simulatedby a setofspringsthatmodelits stiffnessin differentconfigurations.

1As theflapwise,or out-of-planebendingmodesareconventionallycalled.

11.4. WING MODEL 155

Table11.5:Rotatingfrequencies,888rpm,θ75% rWv 3 deg.,HzMode Myklestad UMARC MBDynGimbal - 14.8 14.8Cone 17.2 17.3 17.51st Lag 22.4 20.8 24.0“Coll. Lag” 42.0 44.0 36.02nd Flap 37.3 49.6 41.02nd Flapasym. - 70.2 65.03rd Flap 75.3 90.3 73.0Flap/Twist 89.3 92.7 90.0Lag/Twist - 113.4 104.0Twist - 116.0 110.0

Table11.6:Rotatingfrequencies,742rpm,θ75% r 55 deg.,HzMode Myklestad UMARC MBDynGimbal - 12.4 12.6Cone 14.7 14.9 15.11st Lag 15.3 15.8 16.52nd Flapasym. - 42.3 44.2“Coll. Lag” 32.7 45.9 46.92nd Flap 45.3 45.6 49.13rd Flapasym. - 46.9 60.33rd Flap 66.0 60.1 65.2Flap/Twist 89.3 90.6 97.8Lag/Twist 90.0 90.8 89.73rd Lag - 92.0 92.9Twist - 116.0 108.5


11.5 Wing-rotor models

Thepreviously mentionedmodelshavebeenmergedby mountingtherotor mod-elson theflexible wing. In therigid bladeversion,thecompletemodelhasbeenusedto evaluatethe performancesof the aircraft during maneuvres,an exampleof which is the conversion. The deformableblademodelhasbeenusedto testthe stability of the elasticallymountedrotor and to assessthe feasibility of themulti-body model for the simulationof the whole, detaileddeformablesystem.Moreover, theeffectsof theflexibility of thebladeson thedynamicsof thesys-tem, in termsof transmissionof the higherharmonicsof the rotor systemto thebodyof theaircraft,havebeeninvestigated.Thefirst wing modesarenotdirectlyaffectedby themodellingof theflexibility of the rotor. The torsionmodeof thewing is very close,andat someairstreamspeedscoincident,to the rotor speed;this givesraiseto resonancethatcanbeseenin thefrequency analysesof the in-ternal forcesof the wing. Four wing modesaremainly considered:the beam,chordandtorsionmodesof thewing itself, andthesocalled“pylon yaw” mode,alow frequency yaw oscillationof thepylon dueto theflexibility of theconversionactuator. Whenconsideredin thefixedframe,theretreatingrotor modesinteractwith thewing modes.Thiscanbeclearlyappreciatedfrom afrequency analysisofthewing responsewhentherotormodesareexcited.Mostof thesemodescannotbe easily identifiedwhenthe aerodynamicsaremodelled,sincethey arehighlydamped. For this reason,a comprehensive analysisof the structuralpropertiesof the modelhasbeenperformedby simulating in vacuooperations,while theaeroelasticpropertieshave beenestimatedin differentways.Thedampingof thewing modesin forward flight hasbeenestimatedby systemidentificationof the(damped)responseto a giveninput, asis usuallydoneduringactualwind tunneltests,while the aeroelasticpitch-flapcouplinghasbeenestimatedby measuringthephaseshift betweenanharmoniccontrolinput andtheflappingresponse.

11.6 Testcases

11.6.1 Responseto controls

Figures11.10,11.11referto a collectivepitch maneuvre.They show theinternalmomentsat the wing root andthe geometricpitch of blade#1 asthe collectivecontrolis raisedfrom 0 to 10degreesin onesecond.Thesimulationis performedin helicoptermode;thenominalhoverrotationspeedof 888rpmis reachedin onesecondto obtaina trimmedcondition(not shown). Thereis no airstreamspeed.Thedifferencebetweenthegivencontrolandtheactualpitch of thebladeis dueto thedeformationof theflexbeamandof theflexible link.

11.6. TESTCASES 157

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(ft-

lb)|

t, s

Wing Torsion

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t, s

Wing Out-of-plane Bending

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0

10

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

(ft-

lb)

t, s

Wing in-plane Bending

Figure11.10:Internalmomentsat thewing root duringa 10 deg. collectivepitchmaneuvre,flexible blademodel


0

2

4

6

8

10

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

pitc

h, d

eg

t, sec

blade #1 pitchcontrol input

Figure11.11:Blade#1 pitch duringa 10 deg. collectivepitch maneuvre,flexibleblademodel

Figures11.12,11.13refer to a 5 degreesfore/aft cyclic pitch maneuvre.As therotor tilts forwards,the high frequency in-planemodesof the wing areexcited,asshown by the plot of the internalmoments.The oscillationsin the pitch linkare 1/rev., partially due to 1/rev. flexbeamflapping that is superimposedto thegimbal flapping (which implies no appreciablepitch link loads),that is neededto counteractthe gimbal springs. It shouldbe notedthat while the frequencycontentof the pitch link loadsis acceptablycorrect,the amplitudeof the loadshasnot beenvalidatedand it is likely to be even an order magnitudedifferentfrom measurements,dependingon thefrequenciesthatareconsidered.In facttheloadsin thelinks heavily dependon theaerodynamicsof theblade,whosemodelis relatively poor, especiallywith regardto thedynamicsof theaerodynamics;soonly theconstantandthe1/rev. loadsmaybeconsideredreasonable.

11.6.2 Conversion maneuvre

Figure11.14refersto theconversionmaneuvreperformedby adeformableblademodel.It showstheinternalforcesat thewing root. Theconversionis performedat a10 deg/sconstantangularspeed.Oscillationsof theinternalforcesdueto theuntrimmedinitial conditionsareappreciablydampedasthe maneuvreproceedsto theend,at 9 s. The following abruptraiseof the internalforcesis dueto thetransientcausedby theendof themaneuvre.

11.6. TESTCASES 159

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Wing Torsion

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t, s

Wing Out-of-plane Bending

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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

(ft-

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t, s

Wing In-plane Bending

Figure11.12: Internalmomentsat the wing root during a 5 deg. fore/aft cyclicpitchmaneuvre,flexible blademodel


10.6

10.8

11

11.2

11.4

11.6

11.8

12

12.2

12.4

12.6

12.8

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

link

axia

l for

ce, l

b

t, sec

blade #1 link

Figure11.13:Blade#1controllink axial forceduringa5 deg. fore/aftcyclic pitchmaneuvre,flexible blademodel

11.6.3 Gust response

Theflexible blademodelhasbeenusedto simulatetheresponseto acosinusoidalverticalgustin airplanemode,of 10ft/s amplitude.Both thestabilityandthesen-sitivity of the tiltrotor have beenaddressed.Figures11.15,11.16show thewingout-of-planebendingmomentdue to the gustat differentairstreamspeeds,forbothoff- andon-downstopconfigurations.In figure11.15theoff-downstopcon-figurationis clearlylessdampedthantheotherone,in factthestability boundaryin air is about137 Kts, comparedto 173 Kts of the on-downstopconfiguration.Whentherotatingspeedis increased,thestabilityboundarymovestowardslowerspeeds,asshown by previousanalysesandexperiments[77].

11.7 Computational notes

The completeflexible modelhasnearly600 degreesof freedom. The physicalflexible elementshave beenusedthroughoutthe analyses,without any modalcondensation.A typical modelof thetiltrotor is madeof 45 nodes,39 rigid bod-ies, 35 joints of differentkind, 18 beamelements,14 aerodynamicelements,6control-relatednodesand 4 control-relatedelements. Most of the simulationshave beenperformedwith off-the-shelfPCs. The time stepinitially requiredforthe rigid blademodelwas∆t r 0 s 0005s, while the deformableblademodelre-

11.7. COMPUTATIONAL NOTES 161

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0 2 4 6 8 10

(lb ft

)

t (s)

"Wing Torsion"

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0

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0 2 4 6 8 10

(lb ft

)

t (s)

Wing Out-of-Plane Bending

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0 2 4 6 8 10

(lb ft

)

t (s)

Wing In-Plane Bending

Figure11.14:Internalmomentsat thewing rootduringtheconversionmaneuvre,flexible blademodel


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0

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0 0.5 1 1.5 2 2.5 3 3.5 4

Ben

ding

(lb

-ft)

Time (s)

120 Kts130 Kts134 Kts138 Kts

Figure11.15:Gust—off-downstop:wing bending

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0

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0 0.5 1 1.5 2 2.5 3 3.5 4

Ben

ding

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-ft)

Time (s)

160 Kts170 Kts172 Kts174 Kts

Figure11.16:Gust—on-downstop:wing bending

11.7. COMPUTATIONAL NOTES 163

quired∆t r 0 s 00025s to startcorrectly. Whena variabletime stepwasused,therigid blademodelsimulationsquickly reacheda valueof 0 s 003 ~ 0 s 0035s,whilethedeformableblademodelonesreachedabout0 s 001~ 0 s 0012s. Theconversionsimulationrequiredabout4.5 hourson a PentiumPRO 200 for a total of 40000fixedsizetimesteps(10sat∆t r 0 s 00025s). Whenperformedwith variablestepsize,it requiredaboutonehour. After modelrefinement,andwith a soft start,theflexible blademodelis ableto startwith ∆t r 0 s 001s, requiringabout1.7hours,or 1.1 hourson a PentiumII 333and0.8 hourson a PentiumII 450, for the fullconversionsimulation. Testsarebeingperformedon Digital workstationswithAlpha processor. Thespeedhasbeenincreasedof a factor4.5 for typical simula-tions.Thesenumbersmake this kind of analysisinterestingevenfor a large,timeconsumingparametricstudyfor rotorcraftdesign.


Chapter 12

Tiltr otor vibration control

Thepredictive controldescribedin Chapter7 andassessedin Section10.3is ap-plied to the tiltrotor modeldescribedandanalysedin Chapter11. This form ofcontrolmayberegardedasHHC, sincetheswashplateis usedto actuatetherotor;nevertheless,thereis noexploitationof theknowledgeof therotatingspeedof therotor, thecontrolledsystembeingusedasa blackbox.

12.1 Hover — harmonic excitation

The effectivenessof the GPCappliedto a realisticsystemhasbeenassessedbyperformingsimpleSISOcontrolanalysesof theWRATS modelin hover. Analo-gouspreliminaryhover testshave beenperformedby Bell HelicopterandLaRCpersonnelwith a prototype,proprietaryimplementationof theGPCthatrequiredoff-line identificationof thesystem.Wind tunneltestsarebeingscheduled.Therotor is rotatingat 888 rpm, andit is externally excited by a shaker with a har-monic load at 5 Hz, closeto the first wing out-of-planebendingfrequency, atabout5.5Hz. Thetime stepis 0.001s,andthecontrolsamplesaretakenevery 8steps,resultingin a frequency of 125Hz, which is higherthanthefirst torsionalfrequency of theblade,to avoid bladeresonance.The bendingstrain at the root of the wing is measured,filtered by a washout(band-pass)analogfilter to cut out of the measurethe static signal as well asthehigherfrequencies,andtherotor thrustis usedasactuatorby controlling thecollective pitch. A band-passfiltered measureof the vertical accelerationat thepylon is alternatively used. A goodcompromisefor the systemorderhasbeenfound in p r 60. The resultsin the two cases,comparedto a baselineanalysiswith harmonicexcitationbut without control,arepresentedin Figures12.1,12.2,for two differentvaluesof λ. They show thebendingout-of-planemomentat thewing root.

165

166 CHAPTER12. TILTROTORVIBRATION CONTROL

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baselinelambda = 1.0lambda = 0.1

Figure12.1:Hoverbendingmoment,strainmeasure

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lambda = 0.01

Figure12.2:Hoverbendingmoment,accelerationmeasure

12.2. FORWARD FLIGHT — HARMONIC EXCITATION 167

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t


Figure12.3:Hovercollective,strainmeasure

Thecontrolsignalsareshown in Figures12.3,12.4,while theverticalaccelerationat thepylon in thelattercaseis shown in Figure12.5;thehigh frequency noiseisthepersistentexcitationthatis usedto continuouslyidentify thesystem,while thecontrolof theharmonicmotiondeterminesthemain,low frequency oscillation.

12.2 Forward flight — harmonic excitation

Forward flight analyseshave beenperformedby controlling the collective andthe cyclic pitch of the bladesbasedon differentmeasuresof strainat the wingroot. The model is in airplaneconfiguration,at an airspeedof 100 ft/s, andtherotor is rotatingat 742 rpm. In this casethe order is p 20, sincethe numberof measuresis higher (3 vs. 1). First the wing out-of-planeexcitation force isoffset aft of the wing to obtain also a twisting excitation. The rotor haslittlecontrol authority in its planein termsof force, the flapping of the disk beingrequiredto tilt the thrust. Sincethe flappingresponsehasa delayof about90o,the accuracy of the predictionis key to the effectivenessof the control. In thiscasetheactuationforce,transverseto thewing, lies in theplaneof therotor, thusbeingnot directly controllableby a simplechangein thrust. Moreover, sincethemotionof thegimballedrotor is characterisedby awidespectrumdynamics,fromthehigh frequency vibrationsinducedby theadvancingblademodes,to thewingelasticmodes,down to thevery low frequency precessionmotion,a high numberof physicalandnumericalpolesarerequiredfor anadequateidentification. Theresultsof the simulationsarereportedin Figure12.6, that shows the wing root


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0

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0 0.5 1 1.5 2 2.5 3

deg

t

baselinelambda = 1.0

lambda = 0.01

Figure12.4:Hovercollective,accelerationmeasure

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-1

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0

0.5

1

1.5

0 0.5 1 1.5 2 2.5 3

ft/s^

2

t

lambda = 1.0lambda = 0.01

Figure12.5:Hoveraccelerometersignal

12.3. FORWARD FLIGHT — GUSTRESPONSE 169

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Figure12.6:Forwardflight bending

bendingmoment.Figure12.7showsthecontrolsignals.Theinitial low frequencyoscillationsin the control signalsare due to the precessionof the rotor duringthetransientfollowing theapplicationof theharmonicexcitation. Theuncertaininitial behaviour of the controller is relatedto a poor initial identificationof thelow frequency polesof thesystem.In fact,with λ 1 0 thecontrolauthorityislow, but with λ 0 1, aftera few cyclesthesystemgoesslightly unstable(afterabout1.5s),returningundercontrolassoonastheidentificationis improved.Thefollowingbehavior is definitelybetterthanthepreviouscase,ascanbeappreciatedin thelastpartof theplot.

12.3 Forward flight — gust response

A morerealisticcaseis considered,by usingthe control parameterstunedwiththe former case.A cosinusoidalvertical gust,with an amplitudeof 4 ft/s andawavelenghtof 20 ft, is encounteredby the modelwhile the control is working.The effect of the control on the wing bendingis shown in Figure12.8: the freeoscillationsresultingfrom the wind-up of the rotor are dampedas the controlstarts;whenthemodelencountersthegust,thepeakof themomentis attenuatedfirst, thenthe control overshootsdueto the needto re-identify the system.Thenewly identifiedsystembringsthebendingmoment,aswell astheothermeasuredinternalmoments,to a neglibible valuein a few cycles. Thecontrol signals,i.e.


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deg Fore/Aft cyclic

t

lambda=1.0lambda=0.1

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0

0.2

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0 0.5 1 1.5 2 2.5 3

deg Lateral cyclic

t

Figure12.7:Forwardflight controlsignals

thepitch controlsdeterminedby thecontroller, areparticularlymeaningful.Thecollective is negligible, sinceit mainly controlsthein-planebendingof thewing,that is not directly excited by the vertical gust. The cyclic controlsinsteadareheavily usedby thecontrollerto generatetherotoraerodynamicmomentrequiredto tilt the rotor disk. Sincethe disk tilts abouta horizontalaxis dueto thewingbendingandtorsionexcitedby thegust,therotor is mainly requiredto generateapitch moment(in airplanesense)thatcounteractsthis motion. In fact thehighercyclic controlsignalis thelateralpitch, abouttwice aslargeasthefore/aftpitch,which causesa fore/aft flappingof the rotor. Figure12.9 shows a detail of thecontrolsignalsacrossthegustinput.

Thegustproblemhasbeeninvestigatedfurther, to addresstheeffectof thechoiceof the measureson the quality of the performancesof the controlledsystem.Alongerlearningtime hasbeenused,togetherwith higherexcitationsignals.Fourcontrol configurationsare considered,whoseresultsare shown in figuresFig-ures12.10–12.13;the figuresreportthe wing root out-of-planebendingandthetransverseaccelerationat the wing tip. All the figurescompareto a referencemeasureaddressedasbaseline,that is a referencesimulationwith two consec-utive gustsandtotal absenceof disturbsandexcitation signals. The threebladepitchcontrolsareusedasactuators.In thefirst threecasesthemeasuresof thetwobendingandof the twisting strainsareused,thusresultingin a 3 input-3outputcontrolsystem.In thefourthcase,thefore-aftandtwo transverseaccelerationsatdifferentchordstations,measuredat thewing tip, havebeenaddedto themeasureset,resultingin a3 input-6outputcontrolsystem.

12.3. FORWARD FLIGHT — GUSTRESPONSE 171

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Figure12.8:Gustbending

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deg

t

Fore/AftLateral

Figure12.9:Gustcontrolsignals


Thefirst case,shown in Figure12.10,simply addressesthe persistentexcitationthe four casesare subjectto in order to allow the correct identificationof thesystem.Thesecondcase,consideredin Figure12.11,refersto non-adaptive control: thecontroller is designedoncefor all after the initial identification,andthereis nofurther control design. The responseto the gust is quite smooth,andthe initialpeakin thebendingmomentis cancelledveryquickly with nooscillations,but theresponsesto thetwo consecutivegustsarenearlythesame,with noattenuationoftheinitial peakin eithercase.The third case,Figure12.12,shows the reponsedueto theadaptive control: thesystemis continuouslyidentifiedon-line,andthecontrolleris redesignedto learnfrom the response.Thecontrol is very effective, but thecontinuouslearningre-sults in high frequency disturbanciesaddedto the responseof the system.Thisis apparentfrom theplot of theaccelerationsin Figure12.12.They arenot mea-sured,sothereis no attenuation;on thecontrary, thecontinuousattemptto coun-teracttheexcitation input magnifiesthem. Thebendingis dampedvery quickly,but thereis no peakattenuationat thefirst gust. Theadaptivity of thecontrollerresultsin the systemlearningaboutthe gust;asa consequence,the peakduetothesecondgustis neatlyattenuated.The fourth case,Figure12.13,shows how the additionof the accelerationmea-suresto the outputpool of signalsallows the systemto prevent the peakduetothegustfrom thevery first time. In fact, thesystemcandetectthearrival of thegustfrom theverybeginning,dueto theinitial acceleration,while in caseof strainmeasureit wasableto detecttheadverseconditiononly after thebendingof thewing. Thereis noappreciablechangein theresponseto thesecondgust,theatten-uationbeingsatisfactoryduringbothoccurrencesof thedisturbance.In this casethe accelerationsat the tip are neatlyattenuated,especiallyat low frequencies.As expected,thereis someincrementin the accelerationat higher frequencies,becausethecontrolmovesthepolesof thesystemtowardshigherfrequencies.

12.4 Forward flight — flutter suppression

As a final test,a flutter suppressioncasehasbeeninvestigated.The system,inthelocked-downstopconfiguration,is flown slightly above theflutter limit speed,at 174 Kts, and the controller basedon both strain and accelerationmeasuresdescribedin thefourthcaseof theprevioussectionis appliedattimet 1 0 s,asaverticalgustarrives.Figure12.14show thetwist andthebeamandchordbendingat the root of the wing, while Figure12.15shows the transverseaccelerationatthe pylon, andFigure12.16shows the pitch controls. This exampleshouldbeintendedasa demonstrationof thepossibilitiesof theproposedcontroltechnique

12.4. FORWARD FLIGHT — FLUTTERSUPPRESSION 173

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baselinecontrolled

Figure12.10:Baseline/persistentexcitation;wing root bending,top (lb-ft); wingtip acceleration,bottom(ft/s2).


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baselinecontrolled

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baselinecontrolled

Figure12.11:Baseline/non-adaptivecontrol;wing rootbending,top (lb-ft); wingtip acceleration,bottom(ft/s2).


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Figure12.12: Baseline/adaptive control; wing root bending,top (lb-ft); wing tipacceleration,bottom(ft/s2).


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0.0 1.0 2.0 3.0 4.0

baselinecontrolled

Figure12.13:Baseline/mixedstrain-accelerationcontrol;wing root bending,top(lb-ft); wing tip acceleration,bottom(ft/s2).


rather than a flutter suppressioninvestigation;however, it is apparenthow thecontrolis effectivebothin cancellingtheresponseto thegustandin stabilisingthesystem.Flutter suppressionrepresentsa future aim of this research,andcurrentresultsareencouraging.


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Cho

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t)

Time (s)


Figure12.14:mixedstrain-accelerationcontrol: flutter suppression;internalcou-plesat thewing root.


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20

25

0 0.5 1 1.5 2 2.5 3 3.5 4

Pyl

on tr

ansv

erse

acc

eler

atio

n (f

t/s^2

)

Time (s)


Figure12.15:mixedstrain-accelerationcontrol: flutter suppression;pylon trans-verseacceleration.

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0 0.5 1 1.5 2 2.5 3 3.5 4

Con

trol

sTime (s)

collectivefore/aftlateral

Figure12.16:mixedstrain-accelerationcontrol: flutter suppression;pitch controlsignals.


Chapter 13

Active Twist Rotor analysis

Theanalysisof anactivetwist rotor is presented.Themodelproposedin Ref.[90]for thepreliminaryinvestigationon thefeasibility of aninducedtwist control forhelicopterrotorshasbeenreproducedstartingfrom thepiezoelectricpropertiesofthe material. The equivalenthomogeneouspropertiesof the Active Fibre Com-positematerialhavebeendeterminedby three-dimensionalfinite elementanalysisof anactivefibre specimen.Thebladesectionpropertieshave beensubsequentlycomputedbasedonthegeometryandonthelaminationcharacteristicsproposedin[90], following theprocedureoutlinedin Section9.1. Finally, a multibodymodelof theactivetwist rotorhasbeeninvestigatedto assesstheauthorityof theinducedtwist actuationin hoverandin forwardflight conditions.Preliminaryresultshavebeenreportedin [36].

13.1 Material characterisation

Thecharacterisationof a compositematerialis a complex topic. Dif ferenttech-niqueshavebeendevelopedto predicttheelasticpropertiesof afibrouscompositematerialstartingfrom theelasticpropertiesof its components,namelythefibresandthematrix [19]. Extensionsto thecaseof thermoelectroelasticmaterialshavebeenproposedin theliterature[22, 21]. Suchanalyticalmodelsrely on thesym-metryandthesimplicity of theanalyticalmodellingof a single,indefinitely longfibrousinclusion. Thecaseof theAFC with IDE is far morecomplicatedduetothepeculiargeometryof theelectrodes,thataddahigherdegreeof complexity tothebehaviour of thematerial.A ply of AFC materialis usuallymadeof a singlelayer of piezoelectricfibres, boundedto two thin films that carry the electrodepatterns.It representsa three-dimensional,discrete,repetitivestructurebothfromthemechanicalandtheelectricstandpoint.As aconsequence,theelasticandelec-tric fieldsarefully three-dimensionalandperiodical,dueto therepetitivity of the

181

182 CHAPTER13. ACTIVE TWIST ROTORANALYSIS

Electrodes PZT-5H

Epoxy - Glass Fiber

Figure13.1:FEmodelsof thetwo specimen

patternof thefibresin direction2 andof theelectrodesin direction1. Thepresenttechnologyallows themanufacturingof one-fibrethick pliesof theorderof 0.10mm [8]. Thegranularityof therepetitivepatternis suchthat,from amacroscopicpoint of view, an equivalent,homogeneousorthotropicpiezoelectricmaterialisexpectedto exist, andto beableto capturethefundamentalbehaviour of theAFCmaterial. Herean attemptis madeto predict the propertiesof suchcorrespond-ing continuum,by conventionalelectroelasticFiniteElementAnalysis(FEA). Byexploiting thesymmetryof thepiezoelectriccomponent,a finite portionof fibre,comprisedbetweenthe centerlinesof two rows of electrodes,is modelledby acommercialFEA code;theFE modelsareshown in Figure13.1.

Thefigurerefersto thecircularsectionfibrepresentlyinvestigatedatMIT [8] andto therectangularsectionfibreunderinvestigationatNASA LaRC[90]. Thelattergeometryseemsto be morepromisingin termsof easeof manufacturing,accu-racy in bondingtheIDEs to thefibres,andhomogeneityof properties.Thefibreis madeof raw piezoelectricmaterial(presentlyMorganMatroc PZT-5H is be-ing usedat MIT’ s ActiveMaterialsandStructuresLaboratory[80]). Thecircularfibre manufacturedby CeraNova Corp.hasa diameterof 0.13mm, anddimen-sionssmallerthan0.10mm areexpectedsoon;therectangularfibre, obtainedbycuttingmonolithicceramicsheets,canbemanufactureddown to 0.13 0.07mmsections[59]. Thepiezoelectricpropertiesof thespecimenaredeterminedby im-posingoneby onetheboundaryconditionspertainingto eachindependentstrainand electric field conditions(i.e. the threeunit extensions,the threeunit sheardeformationconditions,andtheconditionof unit electricfield betweentwo elec-trodesets)andcomputingthestressresultants.All theseboundaryconditionsarerequiredbecausethe beamsectionanalysispresentedin Section9.1 is basedonthe full three-dimensionalpropertiesof thematerial. The resultingpropertiesof

13.2. BLADE SECTIONCHARACTERISATION 183

Table13.1:EquivalenthomogeneouspiezoelectricmaterialpropertiesProperty Circular RectangularC11 10.43e6 12.11e6 [psi]C12 1.12e6 2.14e6 [psi]C13 1.25e6 2.93e6 [psi]C22 2.39e6 6.15e6 [psi]C23 0.91e6 2.26e6 [psi]C33 3.90e6 8.14e6 [psi]C44 1.16e6 2.36e6 [psi]C55 2.47e6 0.78e6 [psi]C66 2.47e6 0.29e6 [psi]e11 64.52e-6 104.48e-6 [psi in/V]e12 -5.71e-6 -17.13e-6 [psi in/V]e13 -6.85e-6 -25.69e-6 [psi in/V]

theequivalentmaterialarereportedin Table13.1.Figures13.2and13.3respectively show theVonMisesstressandtheelectricfielddueto unit axial strainappliedto theclampedandclose-circuitspecimenandtounit tensionappliedto thepairsof electrodesof thecompletelyclampedspecimen.

13.2 Bladesectioncharacterisation

Figure13.4showstheFEM modelof theNACA 0012airfoil thathasbeenusedinthis analysis,while Figure13.5refersto thewarpingdueto differentmechanicalandelectricloadconditionsof thebladesectionunderanalysis.Table13.2presentsthestructuralpropertiesresultingfrom thisanalysiscomparedto thosegivenin Reference[90]; threedifferentmeshesof increasingrefinementhavebeenconsidered.Thepropertiesin Table13.2convergeveryquickly to theirfinal value; the most challengingterm is the position of the shearcenter, thattravelsfrom negative to positivewith respectto thereferencepoint at 25%of thechordasthe meshis refined. Furtheranalyseswith finer meshesconfirmedthatthe presentedvaluehasreasonablyconvergedto the final value. The AFC withrectangularfibredescribedin Table13.1hasbeenused.Theothermaterialsused,namely 45o fiberglass(E-glass)for theouterskinandbetweenthepiezoelectricplies, andT300 unidirectionalgraphite-epoxyfor the inner part of the spar, aredescribedin Table13.5. Thelaminateof thesparhasbeenobtainedby stacking,from theinnerside,a ply of fiberglass,followedby 9 pliesof T300andby threesetsof substacksmadeof fiberglass, 45o AFC,fiberglassand 45o AFC.A finalply of fiberglasshasbeenusedto wrap the whole section,including the trailing


Figure13.2: Von Misesstress(top) andelectricfield (bottom)dueto axial strainin thesquarefibre.


Figure13.3:VonMisesstress(top)andelectricfield (bottom)dueto electricvolt-agebetweentheelectrodes.


E-glass skin

Foam[T300x9]

[E-glass/AFC 45/E-glass/AFC 45] x3

+-

E-glass skin

Figure13.4:Sketchof thebladesection,with detailof thepliesin thespar

Table13.2:Bladestiffnesspropertieswith rectangularfibre.Stiffness Mesh1 Mesh2 Mesh3 Ref. [90]Axial [lb] 1.79e7 1.85e7 1.86e7 n.a.Sheari.p. [lb] 1.83e6 1.81e6 1.79e6 n.a.Shearo.o.p. [lb] 2.48e5 2.06e5 1.96e5 n.a.Twist [lb ft2] 2.94e4 2.80e4 2.77e4 2.61e4Bendingo.o.p. [lb ft2] 6.89e4 7.07e4 7.11e4 9.16e4Bendingi.p. [lb ft2] 5.36e5 5.92e5 6.08e5 n.a.Axial forcecenter [% chord] 2.28 2.79 2.96 0.00Shearcenter [% chord] -1.68 1.26 2.18 0.00Nodes 175 362 662 —Elements 162 336 618 —


X

Y

Z

a)

b)

c)E

Figure13.5: Warpingsdueto (a) out-of-planebending, (b)torsion,(c) electrictensionthatgivesinducedtwist

part,thathasbeenfilled with foam,withoutany structuralfunction;aschematicisshown in thedetailof Figure13.4.Thethicknessesof thesinglepliesarereportedin Table13.5. The spargoesfrom the leadingedgeto 40% of the bladechord;thepiezoelectricpliesareappliedon the top andbottomsurfacesapproximatelyfrom 5% to 40% of the chord; in the web andin the nosethe fiberglassis usedinsteadof theAFC. Thefull stiffnessandpiezoelectricmatricesobtainedwith themostrefinedmesharereportedin Table13.3.Noticethatthecouplingcoefficientsthataremarked in italics cannotbecapturedby a conventional,geometry-basedbeamcross-sectionanalysismethod.In thepresentwork, many of themarenulldue to geometricor materialsymmetry, but thereare significantcasesof non-null couplingcoefficients,astheaxial strain/twistcoupling,the axial strain/out-of-planeshearcoupling,the twist/out-of-planebendingcouplingthat areduetotheanisotropy of thematerial,andthepiezoelectrictwist andout-of-planeshearcouplingsthataredueto theanisotropy of the inversepiezoelectriceffect of thematerial.


Table13.3:Sectionstiffnessandpiezoelectricmatrices,ref. 25%chord;units: lb,ft andV.

Axial Shear Twist Bendingi.p. o.o.p. i.p. o.o.p.

Axial 1.86e+7 0.00e+0 1.91e+2 4.29e+2 0.00e+0 -7.53e+5Sh.i.p. 1.79e+6 0.00e+0 0.00e+0 -7.44e+1 0.00e+0Sh.o.o.p. 1.96e+5 5.85e+3 0.00e+0 -3.16e+2Twist 2.77e+4 0.00e+0 -8.47e+1B. o.o.p. sym. 7.11e+4 0.00e+0B. i.p. 6.08e+5Piezo 0.00e+0 0.00e+0 1.29e-1 1.86e-1 0.00e+0 0.00e+0

13.3 ActiveTwist Rotor modeldescription

The numericalanalysisis basedon an analyticalmodelof the active twist rotorpresentedin Reference[90]. This is theanalyticalbenchfull-scalehelicopterro-tor theNASA Langley ResearchCenterATR wind tunnelmodelrefersto [88,91].It is representativeof a largeclassof mediumweighthelicopterrotors.Thebasicpropertiesof the rotor aredescribedin Table13.4. Two analyticalmodelshavebeenstudiedin this paper, both basedon the main propertiesof the mentionedrotor. Thefirst oneusesthesameelasticandpiezoelectriccoefficientsdescribedin thementionedpaper, for directcomparisonandmodelvalidationpurposes.Thesecondmodelis basedontheproposedbladesectioncharacterisationmethod.Thenumericalresultscannotbe consideredcompletelyrepresentative of the modelthey refer to, becausethe final active twist fitting of the ATR model is still un-derrefinement.Theseresultsshouldbeconsideredasasampleapplicationof theproposedanalysisprocedure.A flexible modelof thebenchmarkactive twist ro-tor hasbeenimplemented.The modelconsistsin the hub,a rigid body rotatingat constantspeedwith respectto theground,andcarryingthehingesof the fourblades.Coincidenthingesfor flap andlag have beenconsidered,at 0.027R fromtherotationaxis.A pitchbearingis placedright outwardsof thehinges.Theflex-ible bladesaremodelledby four three-nodebeamelements,with lumpedinertia.Theswashplateis modelled,with scissorsto preventtherelativerotationabouttherotor axisbetweenthefixedplateandtheground,andbetweentherotatingplateandthehub,with thethreevariabledistanceactuatorsfrom thegroundto thefixedplate,andwith fixeddistancelinks from therotatingplateto eachbladeto imposethebladepitch.

13.4. MODEL VALIDATION 189

Table13.4:ActiveTwist Rotorgeometricanddynamicproperties(Ref. [90])Symbol Property baseline AFC blade

R bladeradius[in] 336 (same)Ω rotationspeed[rad/s] 22.25 (same)m sectionmass,[lb-s2/in/in] 1.11e-3 1.03e-3

c R nondimensionalchord 0.0488 (same)θ1 linearpretwist[deg/R] -8 (same)

γ Lock number 9.77 (same)

Table13.5:BladesectionmaterialsProperty glass-epoxy graphite-epoxyc11 2.15e6 26.36e6 [psi]c12 0.19e6 0.42e6 [psi]c22 1.98e6 1.50e6 [psi]c66 0.28e6 1.04e6 [psi]ρ 6.50e-2 5.78e-2 [lbs/in3]thicknessa h 5.10e-3 4.92e-3 [in]

aThethicknessof theAFC pliesis 5.50e-3in.

13.4 Model validation

Themodelhasbeenfirst validatedfrom a dynamicstandpointby comparingthein vacuorotatingfrequenciesto thoseobtainedin Ref.[90]. Thefrequencieshavebeenobtainedby systemidentificationfrom thetimedomainresponseof therotorsubjectto randomexcitation.Theresultsarepresentedin Table13.6.No lead-lagfrequenciesarepresented,becauseno referencedatais availablefor comparison.The first, rigid lead-lagfrequency, due to an hinge offset of 9 in (0.027R), is0 20 Ω. The resultsfrom the modelwith beampropertiestaken from Ref. [90]arepresentedfirst, followedby thoseobtainedwith thebeampropertiesestimatedby theproposedprocedure.Theagreementof the frequenciesobtainedwith thegiven,diagonalstiffnessmatrix is excellent;thefrequenciesobtainedby comput-ing thestiffnessof thesectionwith theproposedprocedureis good,especiallyonthelowerflapfrequenciesbecausethey aredominatedby thecentrifugalstiffness;the frequency of the fourth flappingmode,obtainedfrom the modelwith calcu-latedstiffnessproperties,is appreciablylower thanin Ref. [90], dueto thehighercontributionof thestructuralstiffnessto higherfrequency flappingmodes.Thesimplified,diagonalstiffnessmatrix commonlyused,asin Ref. [90], repre-sentsa first approximationof the true stiffnesspropertiesof sucha complicatedbeam,that canbeusefulfor preliminaryperformanceevaluation,but maycause


Table13.6:Comparisonof in-vacuorotatingfrequencies[1/rev]Mode AFC blade referenceblade

MBDyn MBDyna [90] MBDyn [90]1stflap 1.02 1.02 1.02 1.02 1.022ndflap 2.59 2.58 2.62 2.61 2.623rd flap 4.69 4.55 4.79 4.67 4.794th flap 7.31 6.89 7.85 7.20 7.851sttwist 3.31 3.38 3.38 6.48 6.162ndtwist 9.65 10.0 9.78 18.4 18.3

aStiffnesspropertiesfrom beamsectionanalysis

Table13.7:Hoversimulationswith differentstiffnesspropertiesRef. [90] diagonal coupled

Tip verticaldisplacement [ft] 1.1053 1.1057 1.0343Flapat flaphinge [deg] 2.275 2.276 2.129Elastictwist from root to tip [deg] -0.006 -0.005 -0.120Thrust [lb] 5.119e+03 5.133e+03 4.906e+03

underestimationof cross-couplingeffectsduefor instanceto theanisotropy of thematerials.The importanceof the cross-couplingcoefficients,with particularre-gardto thedynamicsof compositerotorblades,hasbeenhighlightedin [29]. Thestiffnessmatrix reportedin Table13.3showssignificantcouplings;someof themcanbeaccountedfor, in a conventionalbeamanalysis,by applyingoffsetsto theshearcenteror to the normalstraincenter, namelythe couplingsbetweentwistandout-of-planeshear, or the couplingsbetweenaxial force andin-planebend-ing. Othercouplingsarestronglyrelatedto theanisotropy of thematerial,namelythe couplingbetweenaxial andout-of-planeshearforces,andthat betweenax-ial force andtwist. A simplehover simulation,with 4o of collective pitch, hasbeenperformedfirst with thepropertiesform Ref. [90], thenwith propertiesfromthe proposedanalysis,both completeand“diagonalised”by neglectingthe cou-pling terms. A comparisonof somesignificantresultsis reportedin Table13.7.It clearlyshows how thestiffnesscouplingtermsheavily affect theconfigurationof thesystem,while by usingthediagonalcoefficientsonly, nearlythesamebe-haviour of thereferencecaseis obtained.

13.5. HOVER HARMONIC ACTUATION 191

00.5

11.5

22.5

33.5

0 2 4 6 8 10A

mpl

., de

g1/rev.

Blade twistBlade flap

Blade twist, Wilkie et al.

-630-540-450-360-270-180-90

0

0 2 4 6 8 10

Pha

se, d

eg1/rev.

Blade twistBlade flap

Blade twist, Wilkie et al.

Figure13.6: Frequency responseof root-to-tip twistandrootflapat maximumcontrolvoltage

13.5 Hover harmonic actuation

The effectivenessof the active twist of the rotor bladehasbeeninvestigatedbyperforminganalysesof therotor in air with harmonicactuationof theblade.Thehover condition is consideredfirst. Root to tip twist, and root flap anglesdueto harmonicactuationof theactive twist bladearereportedin Figure13.6in thefrequency rangefrom 0.25 to 10 Ω. 250 V areappliedto the electrodesof theAFC, correspondingto an averageelectric field of 850 V/mm. The frequencyresponseshows a peakat about3.5/rev. correspondingto the first twist modeofthe blade,that is slightly dampedby the aerodynamics.Anotherpeakat about2/rev. resultsfrom theexcitationof thefirst flexible flap modeof theblade,thatresonatesat a lower frequency dueto thehigh aerodynamicdampingof theflapmodes.Thebladetwist from Ref. [90] doesshow thepeakat 2/rev., but thefirsttwist modeis missing,possiblydue to the unsteadyaerodynamicsmodel usedin the mentionedreference,that addsconsiderableaerodynamicdampingto thepitchmovement;in thepresentwork, thepitchmovementis mainlydampenedbystructuraldamping.As expected,thefrequency responseof thethrust(notshown)shows a peakat zerofrequency, dueto the steadychangein pitch, andthe twoabovementionedpeaksat2/rev. and3.5/rev.; it is nearlyzeroat1.02/rev., becauseat thefirst, rigid flap frequency, all thework madeby theaerodynamicforcesisspentin rigidly flappingthe bladesat resonance.A primary goal for the activecontrolof rotorcraftby controllingthepitch of thebladesis to obtainaninducedtwist of theorderof 2o in therangeof 0–5Ω [45,90]; Fig. 13.6clearlyshowsthatthegoalis achievable.


-500

-250

0

250

500

750

1000

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Ml,

lb ft

t, s

baselinecontrolled, 100 V

-1000

-750

-500

-250

0

250

500

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Mm

, lb

ft

t, s

baselinecontrolled, 100 V

Figure13.7: Bendingmomentsaboutx andy axesatthemast;advancingratioµ 0 20;actuatingtension:100V (40%of allowable)at 2/rev.; 131o phaseshift

13.6 Forward flight open-loopcontrol

Forward flight conditionsat different advancingratios have beensubsequentlyconsidered;open-loop2/rev. actuationhasbeenperformedwith the objective ofreducingthe vibrationsat the mastinducedby the steadyforward flight. Thesevibrationsmainlyare4/rev. dueto theperiodicforcesgeneratedby thefour bladesof the rotor; they can be cancelledby using 2/rev. actuationsince2/rev. pitchchangecauses2/rev. flappingof therotor, which in turn resultsin Coriolis forcesin theplaneof therotor at twice thefrequency of theflappingmotion.Openloopcontrolresultsarereportedin Figures13.7,13.8,13.9atdifferentadvancingratiosrangingfrom 0.20to 0.30;convergenceat higheradvancingratiosis difficult.

13.6. FORWARD FLIGHT OPEN-LOOPCONTROL 193

-500-250

0250500750

1000

0 0.5 1 1.5 2

Ml,

lb ft

t, s

baselinecontrolled, V=137.5

-1000-750-500-250

0250500

0 0.5 1 1.5 2

Mm

, lb

ft

t, s


Figure13.8: Bendingmomentsaboutx andy axesatthemast;advancingratioµ 0 25;actuatingtension:137.5V (55%of allowable)at2/rev.; 128o phaseshift

-500-250

0250500750

1000

0 0.5 1 1.5 2

Ml,

lb ft

t, s


-1000-750-500-250

0250500

0 0.5 1 1.5 2

Mm

, lb

ft

t, s


Figure13.9: Bendingmomentsaboutx andy axesatthemast;advancingratioµ 0 30;actuatingtension:187.5V (75%of allowable)at2/rev.; 127o phaseshift


Chapter 14

Conclusionsand futur e research

Theanalysisandthedesignof activecontrolsfor rotorcraftapplicationsrepresentvery complex anddifficult tasks,involving many analyticalandexperimentalas-pects.In this work a multibodyanalyticalframework for thedynamicanalysisofrotorcraft(Chapters1–4), includinga self-containedformulationfor theanalysisanddesignof rotor blades(Chapter5), andthe implementationof a general,dis-crete,predictivecontrol(Chapter7), hasbeenpresented.Theformulationis basedonthedirectwriting of theequilibriumequationsof discretebodies,connectedbyholonomicconstraintsdescribedby meansof algebraicconstraintequationsandby flexible elements,mainlymodelledasbeams.Thestate-of-the-artfor rotorcraftactuationmeansis discussed(Chapter6), andtherelatedextensionof theformu-lation, including a detailed,comprehensive piezoelectricbeamanalysisscheme,is presented(Chapters8–9).Thefoundationsof theproposedapproachhavebeenvalidated(Chapter10); its applicationto rotorcraft analysishasbeenvalidatedby analysingthe wind-tunnelmodelof the V–22 tiltrotor aircraft (Chapter11).TheGeneralisedPredictiveControlhasbeenconsideredto evaluatethefeasibilityof vibration reductionof the V–22 model(Chapter12). An advancedactuationscheme,basedon the conceptof inducedbladetwist by meansof piezoelectricembeddeddeviceshasbeeninvestigated,by comparingthe resultsof open-loopcontrol analysesto the state-of-the-artin suchresearchfield (Chapter13). Al-thoughdirectcomparisonof resultsis oftendifficult or even impossiblebecauseof restrictionsontheavailability of dependableandcomprehensivedata,theworkshows thefeasibility of theproposedapproachfor theanalysisandthedesignofrotorcraftaswell asthe feasibility of the proposedcontrol schemes,andclearlyoutlinestheopportunityof prosecutingtheresearchin this direction.Very different interestingfields still have to be explored. Dif ferent actuationschemesandcontrol strategiesmustbe implementedto allow anextensive eval-uationof thefeasibility andof theefficiency of controllinga rotorcraftin a widespectrumof operationalconditions. Oneof the conditionswe expectto investi-

195

196 CHAPTER14. CONCLUSIONSAND FUTURERESEARCH

gateis thedynamicsof thefreehelicopter, with anadaptivecontrollerworkingasan autopilot that flies the rotorcraft following somedesiredpath. A completelydifferent,but really interestingfield is thatof thesimulationof groundandair res-onancephenomenarelatedto thetiltrotor, thatmayoccurwhenasoft-in-planehubis mounted.Plansin this directionhave beenmadeat NASA Langley, fundedbyBell Helicopter, in orderto obtainusefulinformationon a possiblesoft-in-planerotordesignfor thecivil tiltrotor Bell-Agusta609.Fromaformulationstandpoint,nearfutureplansinvolvetheintegrationinto themultibodycodeof modalflexibil-ity to allow themodellingof theflexibility of thebodyof therotorcraft,hydraulicandelectrohydrauliccircuitry for aeroservoelasticityanalysis,modellingof land-ing relateddevices for helicopterlanding and taxiing analysesand for a bettermodelling of the contactwith the groundfor groundresonanceanalyses.Thecouplingwith an uncompressible,unsteadywake analysiscodeis being imple-mented.From a computationalstandpoint,a simplebut effective parallelisationschemebasedon distributedcomputingandmessagepassing,with local matrix-freesolutionof subsystems,is beinginvestigated.A significantreductionin timefor typical runsis expected.Someof thementionedimprovementshave alreadybeenseparatelyimplemented;their integrationin oneanalysistool will allow tomeetthefundamentalgoalthatinspiredtheresearchprojectthiswork contributedto: theintegratedanalysisof theentirerotorcraftin freeflight.

Appendix A

Rigid body momenta

Therigid body momentaresultfrom the integrationof the momentumof a con-tinuousbodywhosemotioncanbedescribedby meansof a rigid rototranslation.Thepositionof anarbitrarypoint p in theglobalframe,is madeof areferencepo-sition x anda relative position f thatcanbeexpressedasa local vector f rotatedin the global frameby an orthonormalrotationmatrix R, definedin Chapter3,resultingin p x f . The momentumandmomentummomentdensitiesof amaterialpoint, referredto thereferencepoleof theglobalframe,O, are

β 2 ρp γ O ρp p

whereρ representsthedensityat point p. Thelinearandangularmomentaof therigid bodyresultfrom theintegrationof themomentumdensitiesover thevolumeof thebody, asfollows:

β V ρp dV γO V ρp p dV

By noting that p x ω f , andconsideringthat the referencepoint positionandvelocitycanbecarriedoutof theintegral, themomentabecome

β V ρ dV x ω h V ρ f dV 9γO x β V ρ f dV x j V ρ f f dV ω

and the integrals involve only the densityof the body and the relative positionvector f ; noticethat theangularmomentumis madeof a contribution relatedtothe velocity andthe angularvelocity of the referencepoint, plus a contributiondueto thetransportof themomentumβ; themomentummomentreferredto pointx is simply γ γO x β, asdirectlydeterminedin thesecondof Eqs.1.1.When

197

198 APPENDIXA. RIGID BODY MOMENTA

expressedin thelocal frame,theinertial invariantsresultin

m V

ρ dV S

Vρ f dV R V

ρ f dV RSJ

Vρ f f dV R V

ρ f f dVRT RJRT wherethetilde ˜J refersto thelocal frameof thebody.

Appendix B

Integration formulas

Theproposedintegrationschemeis describedin detail,by formulatingsomein-terestinghigher-orderformulas.

B.1 Definitions

ConsideranOrdinaryDifferentialEquation(ODE) in explicit form, andanInitialValue(IV) problem:

y f y t y t0 y0 (B.1)

Theunknown y is approximatedby apolynomialinterpolationrangingfrom timestepk r to thecurrentonek, beingr thenumberof stepsof theformulaandyk

theunknown solution:

y r

∑i 0 mi ξ yk ¡ i hni ξ yk ¡ i 9 (B.2)

whereh is the time step,ξ is thenon-dimensionaltime; theorigin for theξ is atthecurrenttimestep,i.e.ξ ¢ t tk ` h; mi , ni arethepolynomialshapefunctionsrelatedto yi , yi .

B.2 Numerical integration

Theintegrationis carriedonbyweightingthevalueof thederivativeatappropriatepoints:

yk yk ¡ r 3 tk

tk £ r

f y t dt (B.3)¤ yk ¡ r h∑i

wi f y ti 9 ti ¥ (B.4)

199

200 APPENDIXB. INTEGRATION FORMULAS

wheretheproblem f y t , Eq.B.1,hasbeenusedin lieu of y. In thefollowing, aone-stepformulais consideredwithout any lossin generality.

B.3 Solution

Theperturbationof thenumericalsolution,Eq.B.4,

yk ∆yk yk ¡ 1 h∑i

wi ¦ f y ti 9 ti ) ∂ f y ti 9 ti ∂y

∂y ti ∂yk

∆yk § where

∂y ti ∂yk

m0 ξi 9 hn0 ξi ∂ f yk tk ∂y

descendsfrom theinterpolationformulaof Eq.B.2,aftersubstitutionbecomes:¨I h∑

iwi

∂ f y ti 9 ti ∂y ¦ m0 ξi I hn0 ξi ∂ f yk tk

∂y §6© ∆yk

ª yk yk ¡ 1 h∑i

wi f y ti 9 ti Thekey ideais thatgiventheorderof theinterpolationfor they, they is oneorderless,thusdeterminingtheminimumrequiredorderfor thenumericalintegration.However, the higherthe orderof the numericalintegration,the higherthe orderof accuracy, aswill beshown in thefollowing section.This interpretationis onlyeuristic;oneshouldbeawareof the fact that thegoal is not to achieve theexactintegrationof theapproximatedform of thesolution,but ratherto determinethecoefficientsthatmake thenumericalmethodstableaswell asaccurate.

B.3.1 Cubic interpolation

Shapefunctions:

m0 1 ξ2 3 2ξ 9n0 ξ 1 ξ 2 m1 ξ2 3 2ξ 9

n1 ξ2 1 ξ 9

B.3. SOLUTION 201

GaussQuadrature

Two pointsarerequired,but threeJacobiansmustbe computed,respectively atthetwo integrationpointsandat theendof thestep.Two Jacobianmultiplicationsmustbe performed,makingthis choiceinefficient with respectto the followingone.Pointsandweightsare:

point,ξi weight,wi« 1 1¬ 3® 2 1/2« 1 1 ¬ 3® 2 1/2

Trapezoid rule

Threepoints are used,but only two Jacobiansmust be computed,at mid- andend-point;only oneJacobianmultiplicationis required.Pointsandweightsare:

point,ξi weight,wi

-1 1/6-1/2 2/30 1/6

Theintegrationbecomes:

yk yk ¡ 1 h ¦ 16

f ¯ ξ ¡ 1 23

f ¯ ξ ¡ 1° 2 16

f ¯ ξ 0 § andthesolutionis:¨

I h

¨23

∂ f∂y ±±±± ξ ¡ 1° 2 ¨ 1

2I h

8∂ f∂y ±±±± ξ 0

© 16

∂ f∂y ±±±± ξ 0

©²© ∆yk ª yk yk ¡ 1 h ¦ 16

f ¯ ξ ³¡ 1 23

f ¯ ξ ³¡ 1° 2 16

f ¯ ξ 0 § B.3.2 Parabolic interpolation

Shapefunctions:

m0 1 ξ2

n0 ξ 1 ξ m1 ξ2

n1 0


GaussQuadrature

Onepoint is required,at ξ w 1 2, with unit weight,but two Jacobiansoughttobecomputed,andoneJacobianmultiplicationis required.

Pointsandweightsare:

point,ξi weight,wi

-1/2 1


yk yk ¡ 1 h f ¯ ξ ¡ 1° 2andthesolutionis:¨

I h∂ f∂y ±±±± ξ ¡ 1° 2 ¨ 3

4I h

4∂ f∂y ±±±± ξ 0

©´© ∆yk

µ yk yk ¡ 1 ) h f ¯ ξ ³¡ 1° 2Trapezoid rule

Two pointsarerequired,thestart-andtheend-point.OneJacobianmustbecom-putedandno Jacobianmultiplication is required.This integrationis degenerate,anda higher-ordernumericalintegrationschemeis requiredto achievehigherac-curacy. Pointsandweightsare:

point,ξi weight,wi

-1 1/20 1/2


yk yk ¡ 1 h ¶ f ¯ ξ ¡ 1 f ¯ ξ 0 ·andthesolutionis:¨

I h∂ f∂y ±±±± ξ 0

© ∆yk

µ yk yk ¡ 1 ) h ¶ f ¯ ξ ³¡ 1 f ¯ ξ 0 ·

B.3. SOLUTION 203

Ad hoc integration

Two pointsareused,oneinsidethetime step,theotherat theend;two Jacobiansandonly oneJacobianmultiplicationarerequired.Pointsandweightsare:

point,ξi weight,wi

ζ 1¸ 2ζ 0 2ζ 1`¸ 2ζ

They allow to exactlyintegratealinearfunction,regardlessof thevalueof ζ; how-ever, noticethatfor ζ V 2 3 theexactintegrationof asecond-orderpolynomialis achieved.Theintegrationbecomes:

yk yk ¡ 1 h ¦ 12ζ

f ¯ ξ ζ 2ζ 12ζ

f ¯ ξ 0 §andthesolutionis:¨

I h

¨ 12ζ

∂ f∂y ±±±± ξ ζ

¨ «1 ζ2 I ζ 1 ζ h ∂ f

∂y ±±±± ξ 0©

2ζ 12ζ

∂ f∂y ±±±± ξ 0

©¹© ∆yk ª yk yk ¡ 1 h ¦ 12ζ

f ¯ ξ ζ 2ζ 12ζ

f ¯ ξ 0 §B.3.3 Linear interpolation

Shapefunctions:

m0 1 ξn0 0

m1 ξn1 0

GaussQuadrature

Onepoint is required,ξ V 1 2, with unit weight,alongwith oneJacobianeval-uationatmidpoint.No Jacobianmultiplicationis neededsincen0, theshapefunc-tion of yk, is identicallyzero.Theintegrationbecomes:

yk yk ¡ 1 h f ¯ ξ ¡ 1° 2andthesolutionis:¨

I h12

∂ f∂y ±±±± ξ ¡ 1° 2 © ∆yk µ yk yk ¡ 1 ) h f ¯ ξ ³¡ 1° 2


Trapezoid rule

Two pointsareused,at thebeginningandat theendof thetimestep,weightedby1 2. Only oneJacobianis required,at theendof thetimestep.Pointsandweightsare:

point,ξi weight,wi

-1 1/20 1/2


yk yk ¡ 1 h ¦ 12

f ¯ ξ ¡ 1 12


I h12

∂ f∂y ±±±± ξ 0

© ∆yk µ yk yk ¡ 1 h ¦ 12

f ¯ ξ ³¡ 1 12

f ¯ ξ 0 §Backward Differ ences

Thefunction is evaluatedonly at theendof thetime step,with unit weight. Theintegrationbecomes:

yk yk ¡ 1 h f ¯ ξ 0

andthesolutionis:¨I h

∂ f∂y ±±±± ξ 0

© ∆yk ª yk yk ¡ 1 h f ¯ ξ 0

B.4 Stability

Considera lineardifferentialequationy λy, with λ complex.

B.4.1 Cubic interpolation

Trapezoid rule

With thethree-pointtrapezoidrule theintegral is:

yk yk ¡ 1 12

λhyk 12

λhyk ¡ 1 112 λh 2yk 1

12 λh 2yk ¡ 1

B.4. STABILITY 205

yk ¶ 1 12λh 1

12 λh 2 ·¶ 1 12λh 1

12 λh 2 · yk ¡ 1

The integratoris A-stable;it hasunit spectralradiusnormregardlessof the timestepratio h T. Figure B.1 shows the error in phasefor undampedsystems(λimaginary).Thesolutionis exact(fourth-order)in thedamping,sinceno numer-ical dampingis added. Accuracy is fourth-orderfor the phaseerror regardlessof thephysicaldampingof the problem(real partof λ), seeFiguresB.1, 4.3. Itdecreasesasthetimestepincreases.

Gaussintegration

Thesameexpressionfor theasymptoticradiusis foundwith thepreviously men-tionedtwo-pointGaussquadrature.

B.4.2 Parabolic interpolation

Gaussintegration

In caseof single-pointGaussintegration,theintegral is:

yk yk ¡ 1 34

λhyk 14

λhyk ¡ 1 14 λh 2yk

yk «1 1

4λh¶ 1 34λh 1

4 λh 2 · yk ¡ 1

This integratoris L-stable.

Trapezoid rule

In caseof end-pointstrapezoidrule integration,theintegral is:

yk yk ¡ 1 12

λhyk 12

λhyk ¡ 1

yk «1 1

2λh«1 1

2λh yk ¡ 1

This integratoris A-stableandloseshigher-orderaccuracy, resultingin theCrank-Nicholsonrule.


Ad hoc integration

In caseof thead hoc integrationformula,with moving mid-point,theintegral is:

yk yk ¡ 1 ζ 22

λhyk ζ2

λhyk ¡ 1 ζ 12

λh 2yk

yk ¶ 1 ζ2λh·¶ 1 ζ º 2

2 λh ζ º 12 λh 2 · yk ¡ 1

The integrator is L-stable(Figure4.1), except for ζ » 1, whereit yields oncemorethe Crank-Nickolsonrule. Figure4.2 shows the error in dampingfor un-dampedsystems(λ imaginary).Theaccuracy is cubicfor thenumericaldamping,(FigureB.7), andquarticfor thephaseerror (FigureB.1) whenthesystemis un-damped,otherwiseit is cubic for both errors(Figure 4.3). It decreasesas thetime stepincreases,(FigureB.8), andwith thevalueof therealpartof λ. This istruewhenthe“optimal” point ζ w 2 3, which allows theexact integrationof asecond-orderpolynomial, is used. The samespectralradiuscanbe obtainedbyusingthesymmetriccombinationof collocationpoints,namelythestartingpointof the time step,ξ » 1, andthe point at ξ » 1 ζ; the latter form is not asmuchappealingasthe former onebecausein this casethe conditionsfor stifflyaccurate integrationareviolated.Whena valueof ζ otherthan 2 3 is used.theaccuracy is quadratic,andtheformulalosesits appealinghigher-orderproperties.

B.4.3 Linear interpolation

Trapezoidand Gaussintegration

In caseof bothGaussandtrapezoidrule integration,theintegral is:

yk yk ¡ 1 12

λhyk 12

λhyk ¡ 1

yk «1 1

2λh«1 1

2λh yk ¡ 1

Thetwo integrationtechniquesareequivalentfrom thelinearstability standpoint.Theintegratoris A-stable,with unit spectralradius.Indeed,it representsthewell-known Crank-Nicholsonsecond-orderintegrationtechnique.

B.5. TUNABLE ALGORITHMIC DAMPING 207

Backward differences

In caseof BackwardDifferencestheintegral is:

yk yk ¡ 1 λhyk

yk 1 1 λh yk ¡ 1

which representsthe well-known Implicit Euler formula, the single-stepcaseoftheBackwardDifferentiationFormulas[17]:

n

∑i 1

1i∇iyk f yk

B.5 TunableAlgorithmic Damping

Considera cubic interpolation,andan integrationby meansof the threepointstrapezoidrule that is requiredto exactly integratethe second-orderpolynomialonly. Pointsandweightsare:

point,ξi weight,wi 1 ¼ 2º 3ζ ½6 ¼ 1º ζ ½

ζ ¡ 16ζ ¼ 1º ζ ½

0 ¼ 1º 3ζ ½6ζ

whereζ is the positionof themid-point,which becomesa parameter. The inte-grationbecomes:

yk yk ¡ 1 h ¦ 2 3ζ 6 1 ζ f ¯ ξ ³¡ 1 1

6ζ 1 ζ f ¯ ξ ζ 1 3ζ 6ζ


I h

¨ 16ζ 1 ζ ∂ f

∂y ±±±± ξ ζ

¨ «1 ζ2 3 2ζ % I h ¶ ζ 1 ζ 2 · ∂ f

∂y ±±±± ξ 0©

1 3ζ 6ζ

∂ f∂y ±±±± ξ 0

©´© ∆yk µ yk yk ¡ 1 h ¦ 2 3ζ 6 1 ζ f ¯ ξ ¡ 1 1

6ζ 1 ζ f ¯ ξ ζ 1 3ζ 6ζ

f ¯ ξ 0 §


Whenappliedto thelineardifferentialproblemy λy, this formulagives:

yk yk ¡ 1 2 ζ 3

λhyk 1 ζ 3

λhyk ¡ 1 1 ζ 6

λh 2yk ζ6 λh 2yk ¡ 1

which,by collectingyk andyk ¡ 1, yields

yk ¶ 1 V¼ 1 ¡ ζ ½3 λh ζ

6 λh 2 ·¶ 1 ¾¼ 2º ζ ½3 λh V¼ 1º ζ ½

6 λh 2 · yk ¡ 1 The asymptoticradiusis ρ∞ ¿ ζ ¸ 1 ζ , whereρ∞ is the absolutevalue ofthe asymptoticradius. It vanishesfor ζ 0, which meansthat the mid-pointmoved to the endof the time step. Notice that for ζ ÀV 1 2 the formula is nolongerunconditionallystable.It shouldbenotedthatthelasttwo weightsbecomesingular, but sincethey weigh the function f y t evaluatedat the samepoint,their summustbeconsidered,whoselimit is 2 3; theJacobianproductdoesnotvanish,reducingto the squareof the Jacobianevaluatedat the endof the timestep.Theasymptoticradiusat ζ 0 is exactly thesamethat is obtainedwith theparabolicinterpolation. For ζ » 1 2 the previously describedfourth-orderA-stableintegratoris obtained.Soasthemid-pointmovestowardstheendpoint, theasymptoticradiusof themethodgoesfrom 1 to 0, thusallowing thetuningof thealgorithmicdissipation.Thepositionof themid point canbewritten asfunctionof thedesiredradius:

ζ ρ∞ ¸ 1 ρ∞ FiguresB.1 and4.3show thecomparisonof thephaseerrorfor anundampedanda dampedsystemwhenthemidpoint is movedfrom ξ w 1 2, correspondingtounit asymptoticradius,to ξ W 3 8, correspondingto anasymptoticradiusρ∞ 0 6. In caseof ζ 0 theintegratorbecomesL-stable,requiringtheevaluationofonly oneJacobian,andof its square.Theintegrationin this caseis:

yk yk ¡ 1 h ¦ 13

f ¯ ξ ¡ 1 23


I h∂ f∂y ±±±± ξ 0

¨23

I h16

∂ f∂y ±±±± ξ 0

©´© ∆yk µ yk yk ¡ 1 9 h ¦ 13

f ¯ ξ ³¡ 1 23

f ¯ ξ 0 §

B.6. CONCLUDINGREMARKS 209

which is obtainedas the limit for ζ Á 0 of the generalexpression. It is worthrecallingthat the threepoints integrationrule usedhereinallows oneto exactlyintegratepolynomialsup to thethird degreewhenthemid-point is in themiddleof thestep,otherwiseonly second-degreepolynomialsareexactlyintegrated.Thisexplainsthechangein orderof accuracy whenthemid-pointis moved.

B.6 Concluding remarks

Accuracy andstability resultsaresummarisedin TableB.1. Thehigher-orderfor-mulasareillustratedfirst, followedby second-andfirst-orderformulasthatresultfrom thedegenerationof thegeneralscheme.At thebottomof thetable,thetwodegeneratemethodsthataresecond-orderbut requiretheoverheadof a mid-stepJacobiancomputationareillustratedfor completeness.Figures4.1–4.3andB.1–B.9 show someof thepropertiesof theproposedformulas.Figure4.1 shows thespectralradii of the formulas. Notice the very steepdescentof the higherorderformulas,with a large initial plateauat ¯ρ ¯ ¤ 1; asa comparison,theBDF algo-rithm descendsvery smoothly, while theImplicit Eulerdoesnot really show anyplateau,thusexplainingthefirst orderaccuracy. Theproposedtwo-stepalgorithmdescendssmoothly, but it shows anappreciablywide plateau.FigureB.9 showsthecomplex planeplot of thespectralradii; noticethat theasymptoticphasean-gleof theformulasis almostalwaysproportionalto theaccuracy, e.g.theImplicitEulerhasa-90deg. phase,while theCrank-Nicholsonrule, theBDF andthepro-posedtwo-stepformula all have a -180 deg. asymptoticphase;the higherorderalgorithmswith no dissipationor with tunabledissipationhave a phaseangleof-360 deg. andthe fully dissipative oneshows an angleof -270 deg. FigureB.8showssomeirregularbehaviour of theaccuracy orderfor thedampingof thetwo-stepformulas,but thelimit valuefor h T Á 0 is clearly2. Theerrorsin damping(Figures4.2 andB.5) andphase(FiguresB.2 andB.3) offer a very goodinsightinto thepropertiesof thedifferentalgorithms.While theCrank-Nicholsonandthefourth-orderformulasdo not introduceany error in damping,theproposedtwo-stepformulais comparableto thethird-orderonewith thesameasymptoticradiusin termsof dampingerror, andbotharefarbetterthantherespectiveL-stablecoun-terparts,theBDF andthe third-orderRadauIIA schemes.This is no longertruefor higheralgorithmicdissipationlevels, wherethe higher-orderformula showsa remarkablysmallererror, thusconfirming the optimality of ρ∞ 0 6 for thetwo-stepformula. Wherethe higher-orderformulasreally show their advantageis in thephaseerror: while theImplicit EulerandtheBDF havecomparable,verypoor accuracy (rememberthat the Implicit Euler is nothingbut a BDF of order1), the proposedtwo-stepformula doesnot performtoo badwhencomparedtotheCrank-Nicholsonrule,but theirerroris at leastoneorderof magnitudehigher


TableB.1: Summaryof propertiesof someintegrationformulasIntegrator Stability Phase Damp. IRK

Cubic A-stable 4 4 LobattoIIIACubicadhoc( 1 2 Â ζ Â 0) A L-stable 3 3Parabolicad hoc(ζ V 2 3) L-stable 3 3 RadauIIA

Crank-Nicholson A-stable 2 2 LobattoIIIAImplicit Euler L-stable 1 1

Multistep(ρ∞ Â 1) A L-stable 2 2Two-StepBDF L-stable 2 2

Parabolic(Gauss) L-stable 2 2Parabolicad hoc(ζ ÃV 2 3) L-stable 2 2

thanthatof thethird- andfourth-orderschemes.FiguresB.6,B.4 show theeffectof the asymptoticspectralradiuson the dampingandphaseerrorsfor the pro-posedtwo-stepformulaandfor thethird-orderformulaath T 0 01. FigureB.6clearlyshows that for thedamping,thetwo-stepformula introducesanerror thatis aboutquadraticwith the algorithmicdissipation,while the third-orderone isaboutlinear; the two lines crossfor ρ∞

¤ 0 6, yielding for higher dissipationsa dramaticallymoreaccuratebehaviour of the third-orderformula. FigureB.4,on thecontrary, clearlyshows that thephaseerrorof thehigher-orderformula isordersof magnitudesmallerthanthatof the second-orderone,regardlessof thevalueof theasymptoticradius.


0

1

2

3

4

0 0.02 0.04 0.06 0.08 0.1

log(

e(2h

))/lo

g(e(

h))Ä

h/T




two-step BDF

FigureB.1: Phaseerrororder, log2

«εϕ 2hS εϕ h (undampedsystem)

0

0.01

0.02

0.03

0.04

0.05

0 0.02 0.04 0.06 0.08 0.1

phas

eÅ

h/T




two-step BDF

FigureB.2: Phaseerror, εϕ (undampedsystem)


0

0.01

0.02

0.03

0.04

0.05

0 0.02 0.04 0.06 0.08 0.1

phas

eÅ

h/T




two-step BDF

FigureB.3: Phaseerror, εϕ (slightly dampedsystem)

0

0.0002

0.0004

0.0006

0.0008

0.001

0 0.2 0.4 0.6 0.8 1

phas

e er

rorÆ

asymptotic rho (h/T=0.01)

cubic, trapezoidmultistep

FigureB.4: Phaseerror, εϕ, at h T 0 01(undampedsystem)


0

0.002

0.004

0.006

0.008

0.01

0 0.02 0.04 0.06 0.08 0.1

diss

ipat

ionÇ

h/T




two-step BDF

FigureB.5: Dampingerror, εξ (dampedsystem)

0

2e-06

4e-06

6e-06

8e-06

1e-05

0 0.2 0.4 0.6 0.8 1

dam

ping

err

orÈ

asymptotic rho (h/T=0.01)

cubic, trapezoidmultistep

FigureB.6: Dampingerror, εξ, at h T 0 01(undampedsystem)


0

1

2

3

4

0 0.02 0.04 0.06 0.08 0.1

log(

e(2h

))/lo

g(e(

h))Ä

h/T




two-step BDF

FigureB.7: Dampingerrororder, log2

«εξ 2hÉ εξ h (undampedsystem)

0

1

2

3

4

0 0.02 0.04 0.06 0.08 0.1

log(

e(2h

))/lo

g(e(

h))Ä

h/T




two-step BDF

FigureB.8: Dampingerrororder, log2

«εξ 2hÉ εξ h% (dampedsystem)


-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

Im(r

ho)Ê

Re(rho)




two-step BDF

FigureB.9: Spectralradii, ρ, polarplot (undampedsystem)


Appendix C

Self-starting algorithm: convergence

Theconvergenceof theBackwardsDifferentiationFormulas(BDF) andanalogousintegration algorithmsappliedto Differential-Algebraicproblemsis key to theimplementabilityof anumericalsolution.A numericallyconsistentto orderk 1setof startingvalues,introducedin [17], p. 56, is needed,wherek is thenumberof stepsof the method. It is definedas a set of valuesat steps j 0 $$$. k 1, whosedifferentialunknownsmustbe known to O

«hk º 1 , ansmustsolve the

algebraicequationsto O«hkº 2 . Thebasicconceptis thatwhenthesolutionstarts

with an error, thereis a boundarylayer betweenthe exact and the numericalsolution,in which theerrorspropagate.Accordingto [17], theorem3.2.3,p. 57,thealgebraicequationsneedbesolved to O

«h4 regardlessof k 1 2 to obtain

a solutionthat convergeswith k-th orderaccuracy afterk 1 steps.So thefirst,self-startingCrank-Nicholson stepis usedto provide a numericalsolution thatformsanumericallyconsistentsetof startingpointsfor thefollowing stiffly stableintegration.TheCrank-Nicholsonstepensuresthedifferentialunknownsarek 1accuratefor k 2, but theaccuracy on thesatisfactionof thealgebraicconstrainscan be achieved only by a Newton-Raphsoncorrection. Considerthe simpleconstraineddynamicsproblem

y v 0 v λ 0

y cos t )whichis completelydeterminedby thealgebraicequationy cos t , andsupposeto solve it asa DAE. A setof startingvaluesthat arenumericallyconsistent toorder3 requirestheknowledgeof thevaluesat time t Ë 0, supposedlygiven,andat time t Ë h. The lattercanbedeterminedby a Crank-Nicholson stepfollowedby a Newton-Raphsoncorrection,which allows to solve the algebraicequationto the requiredlevel of accuracy, while the differential unknowns are assumed

217

218 APPENDIX C. SELF-STARTING ALGORITHM: CONVERGENCE

to satisfytherequiredaccuracy criteriabecausetheCrank-Nicholson formula issecond-orderaccurate.Thesolutionpropagationis

∆y Ë cosÌ t ÍÏÎ P Ì yÍ∆y Ë ∆yÐ c∆v Ë ∆y Ñ P Ì yÍÒÎ P Ì vÍ∆v Ë ∆vÐ c∆λ Ë Î ∆v Î P Ì vÍÒÎ P Ì λ Í

∆I Ì λ ÍÓË c∆λ

wherethepredictionoperatorP Ì.ÔÕÍ hasbeenused;I Ì λ Í is theintegral of thereac-tion λ, while c Ë hÐ 2 is thecoefficientof theCrank-Nicholson formula.Thesolu-tion is obtainedby addingtheperturbationsto thepredictedvalues.Thesecond-orderBDF is appliedto the previously definednumericallyconsistent startingvaluesfor 3 steps,to reacha solutionthatconvergeswith second-orderaccuracyto theexactsolution.Noticethat thecorrectionwith theBDF requiresc Ë 2hÐ 3.Theresults,for exactinitial conditions,arereportedin TableC.1on theleft. Theinitial conditionsarey Ì 0ÍÖË 1, v Ì 0Í®Ë×Î 1, λ Ì 0Í®Ë 1,all theotherunknownsbeingzero.Thedifferentialunknownsarealwayssecond-orderaccurate,while thereac-tionsarefirst-orderaccurate.Considernow themodifiedstartingvaluesthatresultfrom violating thesecondderivativeof theconstraints,namelyv Ì 0ÍØË λ Ì 0ÍÙË 0.Theresults,reportedin TableC.1 on theright, differ from thoseon the left onlyfor thefirst step,afterwhichthesolutionexactlymatchesthepreviousone.In thiscase,duringthefirst steptheerrorin thereactionunknown is O Ú h0 Û .

219

TableC.1: Crank-Nicholson/Backward Differencesolutionof constrainedprob-lem.

Initi

alco

nditi

onss

atis

fyin

gthe

seco

ndde

rivat

ive

ofth

eco

nstr

aint

step

time

yv

λ0

01

01

1h

cos

Ü h

Ý

2

Þ h

Ü cos

Ü h

Ýàß 1Ý

ß 4Þ h2Ü co

s

Ü h

Ý ß 1Ý ß 12

2hco

s

Ü 2h

Ý

1

ÞÜ 2h

ÝÜ 3cos

Ü 2h

Ý ß 4cos

Ü h

Ý á 1

Ý

ß 1ÞÜ 2h

Ý 2Ü 9cos

Ü 2hÝ ß 28co

sÜ h

Ý á 19

Ý

33h

cos

Ü 3h

Ý 1

ÞÜ 2h

ÝÜ 3cos

Ü 3h

Ý ß 4cos

Ü 2h

Ý á cos

Ü h

ÝÝ

ß 1ÞÜ 2h

Ý 2Ü 9cos

Ü 3h

Ý ß 24cosÜ 2h

Ý á 23co

sÜ h

Ýàß 8Ý

44h

cos

Ü 4h

Ý 1

ÞÜ 2h

ÝÜ 3cos

Ü 4h

Ý ß 4cos

Ü 3h

Ý á cos

Ü 2h

ÝÝß 1ÞÜ 2h

Ý 2Ü 9cos

Ü 4h

Ý ß 24cosÜ 3h

Ý á 22co

sÜ 2h

Ý ß 8cos

Ü h

Ý á 1

Ý

55h

cos

Ü 5h

Ý 1

ÞÜ 2h

ÝÜ 3cos

Ü 5h

Ý ß 4cos

Ü 4h

Ý á cos

Ü 3h

ÝÝ ß 1

ÞÜ 2h

Ý 2Ü 9cos

Ü 5h

Ýàß 24cosÜ 4h

Ý á 22co

sÜ 3h

Ý ß 8cosÜ 2h

Ý á cos

Ü h

ÝÝ

Initi

alco

nditi

onsn

otsa

tisfy

ingt

hese

cond

deriv

ativ

eof

the

cons

trai

ntst

eptim

ey

vλ

00

10

01

hco

s

Ü h

Ý

2

Þ h

Ü cos

Ü h

Ýàß 1Ý

ß 4Þ h2

Ü cos

Ü h

Ýàß 1Ý

22h

cos

Ü 2h

Ý

1

ÞÜ 2h

ÝÜ 3cos

Ü 2h

Ý ß 4cos

Ü h

Ý á 1

Ý

ß 1ÞÜ 2h

Ý 2Ü 9cos

Ü 2h

Ý ß 28cosÜ h

Ý á 19

Ý

33h

cos

Ü 3h

Ý 1

ÞÜ 2h

ÝÜ 3cos

Ü 3h

Ý ß 4cos

Ü 2h

Ý á cos

Ü h

ÝÝ

ß 1ÞÜ 2h

Ý 2Ü 9cos

Ü 3h

Ý ß 24cosÜ 2h

Ý á 23co

sÜ h

Ýàß 8Ý

44h

cos

Ü 4h

Ý 1

ÞÜ 2h

ÝÜ 3cos

Ü 4h

Ý ß 4cos

Ü 3h

Ý á cos

Ü 2h

ÝÝß 1ÞÜ 2h

Ý 2Ü 9cos

Ü 4h

Ý ß 24cosÜ 3h

Ý á 22co

sÜ 2h

Ý ß 8cos

Ü h

Ý á 1

Ý

55h

cos

Ü 5h

Ý 1

ÞÜ 2h

ÝÜ 3cos

Ü 5h

Ý ß 4cos

Ü 4h

Ý á cos

Ü 3h

ÝÝ ß 1ÞÜ 2h

Ý 2Ü 9cos

Ü 5h

Ýàß 24cosÜ 4h

Ý á 22co

sÜ 3h

Ýàß 8cos

Ü 2h

Ý á cos

Ü h

ÝÝ

220 APPENDIX C. SELF-STARTING ALGORITHM: CONVERGENCE

Appendix D

Constraints

Two classesof constraintscan be clearly identified: thosethat expressa kine-matic relationshipbetweennodes,andthosethatexpressa dynamicrelationshipbetweenthe kinematicsof the nodesandthe forcesthat act on the nodesthem-selves. In thefirst casetheindex threedifferential-algebraicsystemof equationsdescribedearlierin thedissertationresults.Thesecondcase,which refersto elas-tic constraints,mayresultin index onedifferential-algebraicequationsif thecon-straintrelationshipis explicitly written, or the systemcanbe castin the normalordinarydifferentialform if theconstraintforcesareexplicitatedasfunctionsofthecoordinatesof thenodes.The latterapproachhasbeenbasicallyfollowedinthis work, the former onegiving a higheraccuracy anda betterbehaviour onlywhentheconstraintdegeneratesin averystiff, rigid kinematicrelationship[49].

D.1 Kinematic constraints

Kinematicconstraints,asdescribedin Chapter2, consistin writing analgebraicrelationshipamongsomekinematicunknowns, namelypositionandvelocity ofnodes,resultingin holonomicor non-holonomicconstraintequations.As anex-ample,two basicconstraintsareoutlined.Mostof thealgebraicconstraintscanbeobtainedasa combinationof the coincidenceandorthogonalityconditionsherepresented.

D.1.1 Coincidence

Let xi and fi , i Ë 1 â 2, representthepositionof two independentnodes1 â 2 andtheoffsetsfrom thenodesto thepositionof the joint, both in theglobal frame. Forsakeof simplicity, theoffsets fi areassumedto beconstantin thelocal frame.The

221

222 APPENDIX D. CONSTRAINTS

constraintequationis:Ì x2 Ñ f2 ÍÒÎjÌ x1 Ñ f1 ÍãË 0 äThisconstraintgeneratesareactingforcer at thecoincidencepoint,which, trans-portedto thenodes,resultsin a forceanda coupleappliedto eachnode,duetotheoffsets:

F Ë å r âC Ë å fi æ r ä

Thedifferentiationof theconstraintgivesÌ ∆x2 Î f2 æ θ∆2 ÍÒÎjÌ ∆x1 Î f1 æ θ∆1 ÍhË ÎªÌ x2 Ñ f2 Í)ÑÌ x1 Ñ f1 Íandthatof theforcesandcouplesat thenodesyieldså ∆r Ë ç r âåèÌ fi æ ∆r Î r æ fi æ θ∆i ÍéË ç fi æ r âwhereθ∆ refersto theperturbationof rotation,namelyG∆g.

D.1.2 Orthogonality

Let ei , i Ë 1 â 2, representtheunit vectorsof somecoordinatedirectionreferredtotwo independentnodes1 â 2 andexpressedin theglobal frame.Theorthogonalityconstraintequationis

eT2 e1 Ë 0 ä

This constraintgeneratesa reactingcoupler that is scalar, andactsin directione2 æ e1:

C ËêåèÌ e2 æ e1 Í r äThedifferentiationof theconstraintequationgivesÌ e2 æ e1 Í T θ∆2 ÎjÌ e2 æ e1 Í T θ∆1 Ë Î eT

2 e1

andthatof thecoupleat thenodesyieldså r Ì e1 æ e2 æ θ∆2 Î e2 æ e1 æ θ∆1 Í9åÌ e2 æ e1 Í ∆r ËêçjÌ e2 æ e1 Í r äBy combining1, 2 and3 constraintsof this kind, universal,planeandprismatichingescanberespectively obtained.

D.2. DYNAMIC CONSTRAINTS 223

D.2 Dynamic Constraints

Dynamicconstraints,asdescribedin Chapter5, canbewrittenbothin termsof or-dinarydifferentialandof algebraic-differentialequations,the latter form usuallyyielding only a betterconditioningof theproblemandthepossibilityof account-ing for ill-posedconfigurations,i.e. extremelyvaryingstiffnessof flexible com-ponents.An arbitraryrelationshipbetweenforceandconfigurationis assumed,intheform

s Ì σ â ε ÍË 0 âwhereσ andε respectlyreferto somestressandstrainmeasure,regardlessto theirdimension.If therelationshipcanbedirectly written, theusualform

σ Ë σ Ì ε Ícanbe directly usedto addthe contribution of the constraintto the equilibriumequationsof thenodesit is appliedto. Otherwise,thestressesσ becomethecon-straintreactionunknowns,andtheirdefinitionin termsof constitutiverelationshipis addedto thesystem.In a generalsense,theσ maydependon therelative con-figurationof thesystem,thusinvolving thestraintime rateaswell to accountforsomeviscouseffects,i.e.

σ Ë σ Ì ε â ε ÍThelinearisationof thedirectconstitutivelaw yields

∆σ Ë ∂σ∂ε

∆ε Ñ ∂σ∂ε

∆ε;

this is trueregardlessof thedimensionsandof thenatureof stressesandstrains.This fact hasbeenexploited by introducingthe conceptof template1 constitu-tive law, namelya formal separationof the constitutive law propertiesfrom itsuse in writing the equilibrium equations. In the following, threesimple elas-tic/viscoelasticconstraintsaredescribed.It is interestingto noticethat thefinitevolumebeamformulationpresentedin Section5.4mayberegardedasa general-isationof thedynamicconstraintsherepresented.

D.2.1 Rods

A rod is definedasanelastic/viscoelasticelementthatgeneratesaninternalforcebetweentwo points,orientedasthe line thatconnectsthepointsthemselves,and

1SeeSectionH.1.2for somedetailson theimplemenetationof theconstitutive laws.


whoseamplitudedependson the distance/relative velocity of the points them-selves. Considertwo points, 1 and 2, whosepositionspi Ë xi Ñ fI are rigidlyoffset from the correspondingnodepositionsxi by fi . The relative distanceisd Ë p2 Î p1. Thestrainis definedas

ε Ë ll0Î 1 â

being l Ëìë d ë and l0 thecurrentandthereferencelengthof the rod, respectively.Thestraintime rateis

ε Ë 1l0

dT

ld

which correspondsto thenormalised(i.e. dividedby l0) projectionof therelativevelocity (i.e. d) alongthedirectionof therod (i.e.d Ð l ). In this casethestrain,thestrainrateandthestressarescalars.Their linearisationyields

∆ε Ë 1l0

dT

l∆d â

∆ε Ë 1l0 í dT

l∆d Î dT

ldl æ d

l æ ∆d îµâwhere

∆d Ë Ì ∆x2 Î f2 æ ∆θ2 ÍÒÎjÌ ∆x1 Î f1 æ ∆θ1 Í9â∆d Ë Ì ∆x2 Î f2 æ ∆ω2 Î ω2 æ f2 æ ∆θ2 ÍÎjÌ ∆x1 Î f1 æ ∆ω1 Î ω1 æ f1 æ ∆θ1 Í9ä

Theforcesandthecouplestherod appliesto thenodesare

Fi Ë å dl

σ âMi Ë å fi æ d

lσ ä

Their linearisationyields

∆Fi Ë å í Î σl

dl æ d

l æ ∆d Ñ ∂σ∂ε

∆ε Ñ ∂σ∂ε

∆ε î(â∆Mi Ë Fi æ fi æ ∆θi Ñ fi æ ∆Fi ä

Noticethattheformulasdonot rely on any specificconstitutive law, providedthevalueof theforceandthetangentstiffnessanddampingmatricesareavailable.

D.2. DYNAMIC CONSTRAINTS 225

D.2.2 Springs

Three-dimensionalspringscanbeformulatedaswell; indeedtheirformulais evensimpler. Considera flexible elementthatgeneratesa forcerelatedto therelativepositionandvelocityof twopointsin somereferenceframe,whichcanbeattachedto point 1 without any lossin generality. Theinternalforceis σ Ë σ Ì d â ˙d Í , whered and ˙d havethemeaningdefinedfor therods,but arerotatedbackin thematerialframeby matrixR1. In this casethe“strains”, “strain rates”andthe“stresses”arethree-dimensionalvectors.Their linearisationyields

∆d Ë RT1 Ì%Ì x2 Ñ f2 Î x1 Í æ ∆θ1 Ñ ∆x2 Î f2 æ ∆θ2 Î ∆x1 Í9â

∆ ˙d Ë RT1 Ì%Ì x2 Ñ ω2 æ f2 Î x1 Î ω1 æ f1 Í æ ∆θ1ÑÌ ∆x2 Î f2 æ ∆ω2 Î ω2 æ f2 æ ∆θ2 ÍÎjÌ ∆x1 Î f1 æ ∆ω1 Î ω1 æ f1 æ ∆θ1 Í+Í9ä

Theforcesandthecouplesactingat thenodesare

Fi Ë å R1σ âMi Ë å fi æ R1σ ä

Thelinearisationof theinternalforceyields

∆σ Ë ∂σ∂d

∆d Ñ ∂σ∂ ˙d

∆ ˙d âwherethe partial derivativesof the stresses,expressingthe tangentstiffnessanddampingmatrices,canrepresentany arbitraryanisotropiccostitutive law; thelin-earisationof nodalforcesandcouplesyields

∆Fi Ë ∆θ1 æ Fi Ñ R1∆σ â∆Mi Ë Mi æ fi æ ∆θ1 Ñ fi æ ∆Fi ä

D.2.3 Rotational springs

Therotationalelementcorrespondingto thespringappliesaninternalcouplebe-tweentwo nodesdependingon therelative rotationandrotationspeed.Thecon-stitutivelaw is σ Ë σ Ì ψ â ˙ψ Í , wheretheangleθ is relatedto therelativerotationofthetwo nodes,i.e. ψ Ë ax Ú RT

1 R2Û . Its linearisationis

∆ψ Ë ax ï ∆ Ú RT1 R2

Û Ú RT1 R2

Û T ðË ax ï Ú RT1 ∆θ1 æ TR2 Ñ RT

1 ∆θ2 æ R2Û Ú RT

1 R2Û T ðË ax Ú RT

1 Ì ∆θ2 Î ∆θ1 Í æ R1ÛË RT

1 Ì ∆θ2 Î ∆θ1 Í9ä


In analogousmanner, thederivativeof theangleis ˙ψ Ë RT1 Ì ω2 Î ω1 Í ; its lineari-

sationyields

∆ ˙ψ Ë RT1 Ì+Ì ω2 Î ω1 Í æ ∆θ1 Ñ ∆ω2 Î ∆ω1 Í)ä

Thecouplesappliedto thenodesare

Mi Ë2å R1σ

andtheir linearisationyields

∆Mi Ë ∆θ1 æ Mi Ñ R1∆σ äD.2.4 Remarks

The above describeddynamicconstraintsshouldbe usedcarefully. Apart fromthe relatively simplerod element,the springandthe rotationalspringrepresentan idealisationof morecomplex elasticandviscouselements.Thepossibility touseanisotropic,viscoelastic,viscoplasticconstitutive laws in a largestrain,largerotationenvironmentrequiresthe userto have a deepknowledgeof the systemunderinvestigation.In facttheidealisedmodelsheredescribedarebasedonsomeunderlyingassumption,i.e. in caseof the rotationalspringthat the rotationsaresphericalandindependentonthepaththatis followed.Thismightnotholdtrueforsomecases,e.g.for elastomericbearingsusedin someadvancedarticulatedrotors.In somecasesonemight want to combinekinematicanddynamicconstraintstoobtainthedesiredbehaviour; for instance,arotationalspringcanbesuperimposedto aplanehingejoint to addsomerootstiffnessandviscousdampingto a lead-laghinge;this solutionis definitelypreferableto usinga rotationalspringwith veryhighstiffnessesaboutflap andpitchdirections.

Appendix E

Beamsectionanalysis

Thischapterdescribesin detailsomeaspectsof thebeamsectionanalysis.

E.1 Inter nal work per unit volume

The internalwork submatrices,from Section5.3.2,describethe contribution ofeachindependentcoordinateto theinternalwork of thebeamsection.Thelinear,or linearisedconstitutive matrix of thematerialis partitionedin two blocks,thatseparatethecoefficientsaffectingthestrainsandthestresseson thesectionfromtheothers,asdescribedfor thestrains.It is

D Ëòñ DI I DIS

DSI DSS óThesubmatricesareôöõ Ë DSSâ÷ õ Ë ø DSI DSS ù í ñ 0

ρ æúó Ñüû3ÌÔJÍ î âý õ Ë DSS ø I Î t æéù âþ õ Ë í ñ 0ρ æ ó Ñüû3Ì.ÔJÍ+î T ñ DI I DIS

DSI DSS ó í ñ 0ρ æ ó ÑüûªÌ+ÔJÍ.îµâÿ õ Ë í ñ 0

ρ æéó Ñüû3Ì.ÔJÍ î T ñ DIS

DSS ó ø I Î t æÓù Ñ ø 0 σS æÓù â õ Ë ø I Î t æéù T DSS ø I Î t æéù ä227

228 APPENDIX E. BEAM SECTIONANALYSIS

RecallingthatσS Ë 0 â I σ, thegeneralisedstressesthatwork againstthedifferentpartsof thestrainsare

ΣõtξË σSâ

Σõt Ë í ñ 0

ρ æéó ÑüûªÌ+ÔJÍ.î T

σ âΣõψ Ë ø I Î t æÓù σSä

E.2 Discretisation

Regardlessof the natureof the shapefunctionsthat are usedto discretisethenodalwarping,they arewrittenast Ë N Ì η â ζ Í u Ì ξ Í . Theirgradientresultsin ∇t ËBu Ñ Nu ξ, whereB Ì η â ζ Í Ëwû3Ì N Í is the derivative of the shapefunctions. Theinternalwork submatricesthat resultafter discretistionand integrationover thebeamsectionareô Ë

SNTDSSN JdSâ÷ Ë

SNT ø DSI DSS ù í ñ 0

ρ æòó N Ñ Bî JdSâý Ë S

NTDSS ø I Î t æúù JdSâþ Ë S í ñ 0

ρ æéó N Ñ Bî T ñ DI I DIS

DSI DSS ó í ñ 0ρ æúó N Ñ Bî JdSâÿ Ë

S í ñ 0ρ æéó N Ñ Bî T ñ DIS

DSS ó ø I Î t æ ù Ñ ø 0 σS æ ù JdSâ Ë Sø I Î t æéù T DSS ø I Î t æéù JdSä

Theright-handarraysare

Σtξ Ë S

NT σS JdSâΣt Ë

S í ñ 0ρ æéó N Ñ Bî T

σ JdSâΣψ Ë

Sø I Î t æúù σS JdSä

E.3. FINITE ELEMENTS 229

E.3 Beampropertiesfr om finite elements

It is interestingto noticethatthematricesfor thecurvedandtwistedbeamcanbeobtainedfrom thoseof thestraightbeam(i.e. thepreviouslydefinedmatriceswithρ Ë 0) by consideringthetransformationô Ë ˆô â÷ Ë ˆ÷ Ñ ˆô ρ æ T âý Ë ˆý âþ Ë ˆþ Ñ ˆ÷ T ρ æ T Ñ ρ æ ˆ÷ Ñ ρ æ ˆô ρ æ T âÿ Ë ˆÿ Ñ ρ æ ˆý â Ë ˆ âwherethe hat symbol Ì ÔJÍ refersto the straight/untwistedbeammatrices.This istrue underthe assumptionthat the discretisationand the multiplication by ρ æarecommutable,i.e. Nρ æ Ë ρ æ N. Thusthe technique,proposedin Ref. [31]by Ghiringhelli andMantegazza, to computethebeamsectionpropertiesby con-ventionalfinite-elementcodes,in principlecanbeextendedto curvedandtwistedbeams.Thecitedapproachis basedonusingaconventionalfinite-elementmodelof a slice of beamto determinematrices

ô,÷

andþ

by condensingthe nodeson onefaceof the slice. The othermatricesresult from reducingthe first threeby rigid rototraslationdisplacements.Thecurvature/twistcorrectioncanbeaddedby modellinga slicewith linearly varyingthickness,to accountfor thechangeinvolumerelatedto J, andthencorrectingthematricesrelatedto theaxialderivativeof thewarpingwith thecurvature/twistterm,asshown above.

E.4 Distrib uted external loads

Theform of thedistributedexternalloadshasbeenintroducedin Section5.3.3.Itis presentedherefor thecaseof a straight,untwistedbeam.Theunknown nodalstressescanbediscardedby differentiatingthefirst block row of theproblemandbysubstitutingit in thesecondblockrow, thusyieldingthefollowingsecond-orderdifferentialproblemÎ ñ ô

0symä 0 ó u

ψ ξξÎ ñ ÷ Î ÷ T ý

skwä 0 ó uψ ξ

Ñ ñ þ ÿsymä ó u

ψ Ë Quqϑ â


whereQuq arethenodalforcesresultingfrom thedistributedexternalsurfaceandvolumeloads,andϑ arethe internalforces,respectfulof thedifferentialequilib-rium equationsof thebeam;q Ë q Ì ξ Í is a scalarfunctionthatdescribestheaxialdistributionof theload.BeingH thematrixof thenodaldisplacementsthatcorre-spondto thesix rigid displacementandrotationmotions,theforceandthecouplethatresultfrom thedistributedloadsareQr Ë HTQu. Thedifferentialequilibriumequationof the beamis ϑ ξ Î TTϑ Ñ Qrq Ë 0, matrix T beingthe derivative ofthearmsmatrix definedin Section5.4. A functiong is sought,whosesecondderivative is function q, i.e. g ξξ Ë q. Considerfor the internal forcesthe formϑ Ë ϑ0g Ñ ϑ1g ξ; theequilibriumequationyields ϑ1 Ë Î Qr , ϑ0 Ë Î TTQr . Bydefiningthematrices

2 Ë ñ ô0

symä 0 ó â 1 Ë ñ ÷ Î ÷ T ý

skwä 0 ó â 0 Ë ñ þ ÿ

symä ó â2 Ë Qu

0 â 1 Ë 0Î Qr â

0 Ë 0Î TTQr âandby collecting the nodaldisplacementsand the generaliseddeformationun-knownsin Ë u

ψ âtheproblemcanbewritten asÎ 2 ξξ Î 1 ξ Ñ 0 Ë

2g ξξ Ñ 1g ξ Ñ 0g äBy seekingasolutionof theform

Ë ∞

∑i 0 ig ξi â

thecoefficients i assumetherecursive form

i Ë 10 2

∑j 0

jδ j i Ñ 1 i 1 Ñ 2 i 2 â

wherenegative index entitiesare assumedto be null, and δi j is the Kroneckeroperator. Notice the load function g is requiredto be indefinitely differentiable,andthusregular. This is consistentwith the assumptionof smoothvariationofquantitiesalong the beamaxis that is implicit in the beammodel. Impulsive,concentratedloadscannotbehandledby thisapproach,andrequiretheevaluationof an extremity problem,as mentionedin [44] and in subsequentworks. The

E.4. DISTRIBUTED EXTERNAL LOADS 231

regularity propertiesthat are requiredfor the load function are exemplified byanalysingthe convergenceof the recursionfor an exponentialload distribution.After thesolutionsrelatedto arrays

i arecomputed,thehigher-ordertermsof the

solutioninvolve thepropagationof thelocaleffects.Therecursive form resultsin

λ i 1 i Ë ñ 0 I 10

2 1

0

1 ó i 1 i ;

asaconsequence,thesolutionatξ Ë 0 takestheform

hÌ 0ÍË 0

∞

∑i 0

λig ξi Ì 0Í9äTheexponentialloaddistributiong eξ σ gives

hÌ 0Í Ë 0

∞

∑i 0

λi Ð σi âwhich is finite for λ Ð σ 1. The eigenvalueλ representsthe largestpropaga-tion lengthof thebeamsection,which, in aheuristicsense,measuresthedistancefrom theextremitieswherethebeammodelcorrectlyrepresentsthebehaviour ofa beam-like structuralcomponent.Themodelhereconsideredis a finite-elementdiscretisationof thebeamsection,so theeigenvaluesmight berelatedto thenu-merical discretisationof the section;however, for practicalmodels,the largestpropagationlengthsshouldbeproperlycapturedevenby verycoarsemeshes,if areasonablyaccuratecharacterisationis required;eventualmeshrefinementshouldnot significantlyaltersucheigenvalues.For practicalpurposes,polynomialloaddistributionsbestfit with thepresentedmodel,becausetheir derivativesnaturallyvanishatsomeorder.


Appendix F

Platefibr eanalysis

Theentitiesdefinedin Section5.6aredetailedbelow.

F.1 Inter nal work per unit volume

Being Ξ Ë(â#â! " , and rangingboth i and j from 1 to 2, the matricesthatdefinetheinternalwork perunit volumeareô õ

i j Ë ΞTi DΞ j â÷ õ

i Ë ΞTi D ï ρξ æ Ñ# ρη æ Ñ$ 3Ì.ÔJÍ ζ ð âý õ

i j Ë ΞTi DΞ j ø I Î t æéù âþ õ Ë ï ρξ æ Ñ# ρη æ Ñ$ ªÌ+ÔJÍ ζ ð T

D ï ρξ æ Ñ# ρη æ Ñ$ 3Ì.ÔJÍ ζ ð âÿ õi Ë ï% ρξ æ Ñ# ρη æ Ñ$ ªÌ+ÔJÍ ζ ð T

DΞi ø I Î t æÓù â õi j Ë Ú Ξi ø I Î t æ ù Û T DΞ j ø I Î t æ ù ä

F.2 Discretisation

Thediscretisedwarpingunknownsaredefinedast Ë N Ì ζ Í u Ì ξ â η Í ; their gradientthusresultsin ∇t Ë Nu ξ Ñ Nu η Ñ Bu, whereB Ë N ζ. Thematricesthatdescribethediscretisedinternalwork perunit surfaceareô

i j Ë S

NTΞTi DΞ jN JdSâ÷

i Ë S

NTΞTi D Ú ρξ æ N Ñ# ρη æ N Ñ$ BÛ JdSâ

233

234 APPENDIXF. PLATE FIBREANALYSISýi j Ë

SNTΞT

i DΞ j ø I Î t æéù JdSâþ Ë S

Ú& ρξ æ N Ñ# ρη æ N Ñ$ BÛ TD Ú& ρξ æ N Ñ# ρη æ N Ñ$ BÛ JdSâÿ

i Ë SÚ ρξ æ N Ñ# ρη æ N Ñ$ BÛ T DΞi ø I Î t æÓù JdSâ

i j Ë SÚ Ξi ø I Î t æÓù Û T DΞ j ø I Î t æÓù JdSâ

beingJ Ë detÌ ∇s0 Í themeasureof theintegrationvolume.

Appendix G

Piezoelectricbeamsectioncharacterisation

Thepiezoelectricmatricesusedin thecharacterisationof apiezoelectricbeamaredetailed.

G.1 Inter nal work per unit volume

Thepiezoelectriclinearconstitutivematricesarepartitionedin internalandsection-facingcoefficientsas

D ' E ( Ë ñ DI I DIS

DSI DSS ó âe Ë ñ eSI eSS

eI I eIS ó âε ' ε ( Ë ñ εSS εSI

εIS εI I ó äTheinternalwork matrices,apartfrom thealreadydefinedstructuralones1, areôöõ

se Ë eTSSâô õ

ee Ë Î εSSâ÷ õse Ë ø eT

SS eTIS ù û e Ì+ÔÕÍ)â÷ õ

es Ë ø eSI eSS ù í ñ 0ρ æéó ÑüûªÌ+ÔJÍ.îµâ÷ õ

ee Ë Î¾ø εSS εSI ù û e Ì+ÔJÍ)â1SeeAppendixE

235

236 APPENDIX G. PIEZOELECTRICBEAM ANALYSISý õe Ë eSS ø I Î t æÓù âþ õse Ë í ñ 0

ρ æúó ÑüûªÌ.ÔÕÍ î T ñ eTSI eT

I IeT

SS eTIS ó û e Ì+ÔJÍ)âþ õ

ee Ë Î û e Ì.ÔJÍ T ñ εSS εSI

εIS εI I ó û e Ì+ÔÕÍ)âÿ õe Ë û e Ì+ÔÕÍ T ñ eSS

eIS ó ø I Î t æÓù äTheelectricdisplacementtermsthatwork for theelectricfield termsare

ΣõVξ Ë )* 1

00 +,

T

De âΣõV Ë û e Ì+ÔJÍ T De ä

G.2 Discretisation

The discretisationof the electricunknowns is straightforward. The electricpo-tential is discretisedasV Ë N Ì η â ζ Í ue Ì ξ Í , whereelectricpotentialnodalvaluesdependingontheaxialpositiononly areusedto interpolateby conventionalshapefunctionsthevalueof thepotentialin thesection.Noticetheshapefunctionsarethe sameasusedfor the structuraldiscretisationwithout any loss in generality.The electricfield resultsin E Ë»Î Nue ξ Î Beue, with Be Ë û eN. The discrete,integralmatricesthatdescribetheinternalwork perunit lengthare:ô

se Ë Se

NTeTSSN JdSâô

ee Ë Î Se

NTεSSN JdSâ÷se Ë

Se

NT ø eTSS eT

IS ù Be JdSâ÷es Ë

Se

NT ø eSI eSS ù í ñ 0ρ æéó N Ñ Bî JdSâ÷

ee Ë Î Se

NT ø εSS εSI ù Be JdSâýe Ë Î

Se

NTeSS ø I Î t æÓù JdSâþse Ë

Se í ñ 0ρ æúó N Ñ Bî T ñ eT

SI eTI I

eTSS eT

IS ó Be JdSâ

G.2. DISCRETISATION 237þee Ë

Se

BTe ñ εSS εSI

εIS εI I ó Be JdSâÿe Ë Î#

Se

BTe ñ eSS

eIS ó ø I Î t æÓù JdSäTheright-handarraysare

ΣVξ Ë Se

NT )* 100 +,

T

De JdSâΣV Ë

Se

BTe De JdSä

238 APPENDIX G. PIEZOELECTRICBEAM ANALYSIS

Appendix H

Implementation notes

A few implementationnotesarepresented.Theirmotivationis relatedto thelargeamountof codingthathasbeenperformedduringthisresearchproject.Themulti-body analysisprogramplayedan importantrole, andallowed to acquirea deepknowledgeof someprogrammingtechniquesthat led to the realisationof an in-terestingresearchtool evenfrom animplementationstandpoint.Thesenoteswantto beareminderfor subsequentimplementors,andoffer suggestionsfor develop-ing a policy for the implementationof medium/large projects,involving severalpersonswith seriousportability, maintenabilityandreusabilityissues.The phy-losophyfollowed in writing suchcodewasto privilege safeprogrammingstyleeven at the costof sacrificingperformanceissues.As a result, lots of codearededicatedto debuggingpurposes.Optimisationandperformanceenhancementsmightbeobtainedby tuningsomepartsof thecodeonceit is stabilised;however,sinceit is a researchcodeparticularlyintendedto investigatingnew problems,itis likely to neverstabiliseto adefinitive form!

H.1 Object Oriented programming

H.1.1 Dri ves

Many itemsmay requiresophisticatedparameter-dependentvaluesto be easilyassigned,anda commoninterfaceat input-file level. A basicissueis theassign-mentof thetime historyof a forcing term; in very simplecasesconstant,steporrampfunctionsmay suffice, but in rathersophisticatedsituationsonerequiresawideflexibility in definingsuchhistory. Moreover, from theprogrammer’sstand-point,reusabilityof thecodeis akey issue,aswell asacommoninput interfaceisfundamentalto allow theuserto quickly gainconfidencein theanalysiscode.Asolutionto suchrequirementshasbeenfoundby introducinga family of Drive-

239

240 APPENDIXH. IMPLEMENTATION NOTES

Callers, classesthat basicallyprovide a numericvaluebasedon someinputparameter. They canbeusedasnormalrealconstants,anddonotactuallyaddanyoverheadwhenthe constantvaluedrive type is used. More sophisticateddrivesallow theuserto exploit variousbasicmathfunctions,interpolationof datafromexternalfiles, linearcombinationof basicdrives,run-timeparsingandevaluationof symbolicmathandlogical expressions,evaluationof the currentvalueof de-greesof freedom.In normaluse,the input is representedby thecurrenttime; incaseof degreeof freedomevaluation,it canbefilteredthrougha drive,sotheac-tual resultingvalueis theresultof evaluatingthedrivewith thedof valueasinputinsteadof thetime. This allows to easilyimplementa simpledirect feedbackforbasiccontrolissues,e.g.stabilisingtherotationalspeedof a rotorcraft.

H.1.2 Templateconstitutive laws

TheC++template programmingtechniquehasbeenexploitedto implementageneralconstitutive law family independentof theuseandof thedimensionality.It hasbeenusedto characteriseelementsrangingfrom rods and abstractgen-eralpurposeelements(one-dimensional)to springsandrotationalsprings(three-dimensional)to viscoelasticfinite-volumebeamelements(six-dimensional).Be-ing N thedimensionalityof theproblem,the ideais to provide theN æ 1 dimen-sionalvalueof theforce,theN æ N dimensionaltangentstiffnessmatrix andop-tionally theN æ N dimensionaltangentdampingmatrix asfunctionsof theN æ 1dimensionalvalue of the strainsand optionally of the strain rates. The tem-plate mechanismrequiresall theentititesit is generatedfor to have a commonsetof functionsor methodsthey canwork with. Without excessive detail, inputmethods,overloadedmathoperatorsandgeneralpurposefunctionshavebeenim-plementedto meetthis requirement.As a result,basicconstitutiv laws like linearelasticisotropicandanisotropic,linearviscousisotropicandanisotropic,andlin-earviscoelasticisotropicandanisotropicrequiredto be implementedonly once,with codeandimplementationtime savingsbut with no computationaloverhead,sincethetemplateis resolvedat compiletime.

H.2 Genericprogramming

In many partsof the code,the genericprogrammingparadigmhasbeenusedasmuchaspossibleto easetheabstractionfrom thedetailsof theimplementationofspecificparts.As a result,a simpleinterfacebetweentheroutinesspecificto thecode,e.g.computationof elements’contribution to Jacobianmatrix andresidualcomputation,global Jacobianandresidualassembly, Jacobianmatrix factorisa-tion andbackwardssubstitution,hasbeenestablished.This allowedto introduce

H.3. REUSEOFCODE 241

differentdataorganisationschemesandlinearalgebraroutineswith very limitedeffort, with particularregardto thedebugging.

H.3 Reuseof code

A setof basicC++ librarieshasbeenimplementedto supportthe writing of thecode,MBDyn, usedfor this researchproject. Most of themarerelatedto linearalgebrawith the aim of simplifying the writing of problems,freeingthe imple-mentorfrom low-leveldetailsandallowing to focusonhigh-level,problemrelatedissues.

H.3.1 Matrix Handling

A commoninterfaceto datahandlinghasbeenimplemented,basedonanApplica-tion ProgrammingInterfacethatstandardizestheproceduresto instantiate,resizeanddeletevectorsandmatrices.Basicoperations,suchassinglecoefficient in-put andoutput,higher-level structuresinput, addition,scalarmultiplication andmore,areprovided for every memberin form of generalprogrammingroutines.Specialisedroutines,whenever convenient,areprovidedaswell in casecompile-time type detectionis possible. Substructures(i.e. subvectorsandsubmatrices)have beenadded,to easetheassemblyprocesswithout allowing theelementstodirectlydealwith theglobaldatastoragestructures.

H.3.2 Linear Algebra

Theoperationsrelatedto matrix factorisationandbacksubstitutionin solvinglin-earproblemsarehiddenin the typedefinition. Thereis no explicit factorisationof matrices,but only backsubstitutionwhich causesthesystemto befactoredifthematrixhasbeenchangedsincethelastsubstitution.In suchcase,for instance,a Newton-Raphsonsolutionschemerequiresto call anassemblyanda solverou-tine, i.e. Assumethe matrix is ownedby a SolutionManager, who knownshow to solveit, andlendsit to aDataManager thatknowshow to fill it (andtheresidual,aswell); then:

SolutionManager sm;DataManager dm;// ...dm.AssJac(sm);sm.Solve();


A modifiedNewton-Raphsoncanbesimplyobtainedby calling theassemblyrou-tineonly whenrequired,e.g.afteracertainnumberof iterations,i.e.

SolutionManager sm;DataManager dm;// ...if ( iter++ > maxiter )

dm.AssJac(sm);iter = 0;

sm.Solve();

Most of the linear algebradatastructuresand routineshave beenobtainedbywrappingexisiting C, andin somecasesFortran77,codeinto C++ classesthusobtaininga simple,practicaland“safe” interfaceat thecostof limited overhead.Moreover, the useof a safeinterfacefreesthe implementorfrom taking careofbasicissueslike memoryallocationandhandlingduringthefactorisationandso-lution phase,datainitialisation andrelatedissues,resultingin a better, andlesserrorprone,codingstyle.

H.3.3 Thr ee-dimensionaldata structur es

A setof datastructuresandmethodsfor the handlingof three(andsix) dimen-sionalentitieshasbeendeveloped.It allows thewriting of mostof thecommonoperationsinvolving three-dimensionalvectorsandmatricesin a formal way thatdirectly resemblesnormalhand-writingof formulas,which hasbeenobtainedbyproperlyoverloadingmostof themathematicaloperators.As a result,operationsarithmeticoperationsarestraightforward,i.e.

Vec3 v, w;Mat3x3 m;Vec3 x = m * v + w;

To avoid excessivesophistication,noparticularcarehasbeenput in handlingtem-poraries;a carefulprogrammingstylecantake advantageof theeaseof notationwithoutincurringin excessiveoverheadrelatedto temporaries.Operationssuchascreatinga vectorproductmatrix, or executinga vectorproductcanbeperformedasfollows:

Vec3 v;Mat3x3 m(v); // overload of constructor accepting a vector

// to yield m = v xVec3 w, z;z = w.Cross(v); // cross product between vectors

H.4. DEBUG/RELEASEAPPROACH 243

Many other operationsare available. Furtherdetailscan be obtainedby self-generatingthecodedocumentation:

$ make doc

H.3.4 Input handling

Particularcarehasbeendedicatedto implementinga safe,reliable,flexible inputhandling,which resultedin an input library independentof the multibody pro-gram.It is dedicatedto theacquisitionof card-like input lines,soit is notflexiblein a generalsense. In the future it might be modified to include somegeneralfeaturesof generalparsingroutinesandsomesyntaxinterpretationfeatures.It isbasedon a low-level parsingtool thatinterpretstheinput,generatingtokens.Thehigher-level interpreterextractsthe meaningfultokenstakingcareof separators,punctuation,file inclusion, input-relatedcrdsandso on. Whena specifictokentypeis expected(i.e. requiredby theuser),therelatedinput routinesareinvoked,otherwisesometokentyperecognitionis attempted.Basictokensarekeywords,whichcanbebuilt-in anduser-defined,numericvalues,andstrings.Numericval-uesare parsedby a simple mathematicalinterpreter, that parsesa sequenceofmathematicalexpressionsandreturnsthevalueof the last one. Thereis supportfor types(integer andreal), variabledeclaration,andfor a numberof built-in mathematicalfunctions.Thismakesparametricinput verysimpleandreliable.Extensionsallow MBDyn to consideralsoenvironmentvariabledefinition.A spe-cial drive (seeSectionH.1.1)returnsa valuebasedon evaluating,by meansofthesameparser, astringcontainingamathematicalexpressionwhichmaycontainvariablesdefinedfuring theinput,andthecurrenttime in built-in variableTime.For adetaileddescriptionof theinput format,type

$ make man

orseehtml://diampp1.aero.polimi.it/mbdyn/mbdyn-index.html.

H.4 Debug/releaseapproach

A fundamentaltaskof implementinga complicatedprojectis representedby de-bugging. For this purposea debug/releaseapproachhasbeenfollowed, by in-troducinga lot of redundantdebugging codethat can be activatedat compile-time by appropriatecompiler flags. The phylosophyis that of building a self-debugging program,basedon performingautomaticconsistencechecksbeforeandaftermany crucialoperations,withoutexcessivel reducingperformancesoncethe codeis released.It is fundamentalto follow someguidelinesto make this


strategy usefulandreliable. The fundamentalissueis to make the releasecode“smaller” thanthedebug code,but not different. In otherwords,thedebug ver-sion of the codecanaddlines of code,but it must interfereaslittle aspossiblewith thereleasepartof thecode,i.e. it mustnotaltervariables,streamsotherthanthe standardoutput/error, it must resortas little aspossibleto routinecalls thatmight alter the statusof the stack,it mustnot createdifferent logical paths,i.e.addconditionalstatements,or alter the logical pathby alteringconditionalcodeexecution. For this reasonthe bestthing is to usemacrosthatdirectly addcodeinsteadof functioncalls.After somecodegrowth, thedebugoutputbecamereallyhuge,so the approachhasbeenchangedby addingsomecommand-linetunableoutputselectionto thedebug version,to allow aselectivedebuggingoutput.

H.5 Safe-pointerprogramming

Pointerscanbe a nightmarein C programming. C++ suggestsa programmingstyle thatallows a saferuseof pointers,but a lot of caremustbe taken anyway.To freethe implementorfrom excessive attentionto memoryallocationandhan-dling, andto easedynamicallocationrelateddebugging,a collectionof memoryallocationmacrosandroutineshasbeenimplemented.Someof themhave beenrecentlyobsoletedby thefinal ANSI C++ draft, andthushave beenrecodedandsimplified. Theideais to provide macrosthat take careof allocatingandfreeingmemory, which do only little morethanstandardmemorymanagementroutineswhencompiledin debug version,but performredundantandtediousconsistencycheckswhencompiledin debug version. For instance,during allocationa goodideais to testwhetherthetargetpointeris NULL or not, in theformercasetracinga possiblememoryleakdueto thecancellationof a possiblereferenceto dynam-ically allocatedmemory. This test requiresthe implementoralways to resetapointerbeforeallocatingmemory, reulting in a goodprogrammingstyle,at leastin ourhumbleopinion.Analogousoperationshuldbedonewhenfreeingmemory,i.e. a pointershouldberesetto NULL afterdeletion,to ensurenobodycanrefer-encedeletedmemory. If it is done,it will resultin asegmentviolationsignalto beraisedat run-time,whichwill suggesttheimplementorthatthereis somememoryleakage.Whenthememorydebugflag is activated,aparallelmemorymanagerismaintained,which storesdynamicallyallocatedmemoryinformation,andallowsthe implementorto addconsistency checkson pointerswhenrequired.A setofchecksareavailable, rangingfrom the strict validity of a memorychunk, i.e. apointeris thebaseof a chunkof a prescribedsize,to a weakvalidity of a pointerpointingto asub-chunkentirelycomprisedin theallocatedchunk.Thefollowingexample,involving a simpleallocationroutine, that canbe usedfor typeswitha default constructor, e.g.the fundamentalC andC++ types,illustratesboth the

H.5. SAFE-POINTERPROGRAMMING 245

debug/releaseprogrammingparadigmandthesafeallocationconcept.ConsidertheSAFENEW macro:

#ifdef DEBUG# ifdef DEBUG_MEMMAN# define SAFENEW( pnt, item, memman ) \

do \assert( !( pnt ) ); \assert( sizeof( item ) ); \( pnt ) = new item; \if( !( pnt ) ) \

_Safenew( __FILE__, __LINE__ ); \ \( memman ).add( (void *)( pnt ), sizeof( item ) ); \

while( 0 )# else // !DEBUG_MEMMAN# define SAFENEW( pnt, item, memman ) \

do \assert( !( pnt ) ); \assert( sizeof( item ) ); \( pnt ) = new item; \if( !( pnt ) ) \

_Safenew( __FILE__, __LINE__ ); \ \

while( 0 )# endif // !DEBUG_MEMMAN#else // !DEBUG# define SAFENEW( pnt, item, memman ) \

( pnt ) = new item#endif // !DEBUG

It is importantto remarkthat:- the macrotakesthreearguments:the target pointer, pnt, the type of theobjectto create,item, andamemorymanager, memman;- the body of the macrois encapsulatedin a do ." while () loop, sothatit canbeusedasafunction,i.e.endedby asemicolon:SAFENEW( p,int, m );. However, sinceit expandsto a portionof code,it cannotbeusedinsideaconditionalstatement,i.e.anif clause,or astheargumentofa functioncall;

246 APPENDIXH. IMPLEMENTATION NOTES- theuser-definedmemorymanageris usedonly if theappropriatecompilerflag,DEBUG MEMMAN, is set,which on turn requirestheflagDEBUG to beset;- a function, Safenew(), is invokedin caseof failure;this is doneto senda standarderror messagewith limited codesizeoverhead. In fact, if theerror messagewere written directly inside the macrodefinition, it wouldexpandto a constantstring every time the macro is used,and only fewcompilersareableto recognisewhenaconstantstringis repeatedto storeitonly once.TheC preprocessor-definedmacros FILE and LINE areusedto customizetheerrormessageandmake it a little morehelpful. Theyareexpandedto thefile nameandtheline numberthey wereused;- thestandardC functionassert is usedto assesswhetherthepointeris nullor thesizeof thetypeis legal. Sotherearesomefunctioncallsin thedebugversionof thecode,but they areknown not to alter any data;they simplywrite to the standarderror stream.It is a goodpracticeto forcea flush ofthe streambeforeandafter usingit, to ensurethe messagesareprintedinthecorrectsequence;- finally, this macrois particularlydangerousbecausebothpnt anditemareevaluatedmorethanonce,so if non-appropriatevaluesareused,a dif-ferent (anderroneous)behaviour resultswhenusing the debug insteadofthereleaseversionof thecode.

A completesetof macrosis provided,to accountalsofor theallocationof objectsrequiringaconstructor, for theallocationof arrays,andfor theirdestruction.

H.6 Conclusion

Thisdigressionon implementationissuesdoesnot pretendto beexaustive,not tointroducenovel ideasin programmingtechniques.Its aim is to focusthereader’sattentionon someprogrammingtopicsthatdeserve a lot of attentionevenwhenwriting verysimpletoy-codes,especiallywhenotherpeoplewill haveto dealwiththosecodesbothasusersandasmaintainers.Big projectsrequirethinkingahead,to saveimplementationtime,cost,andto preservefrom unpleasantsurprises.Dur-ing my shortcareerasGraduateStudent,Ph.D.StudentandStaff ResearcherI hadto maintainandmodify many codes,mainly written in Fortran,startingfrom theearly Eightiesandsubsequentlymaintained,improved, modified, rewritten (notvery often!), resultingin somethingreally looking like unreadblesequencesofrandomchars! Of coursetherewasa logic in all thoselines of code,but it was

H.6. CONCLUSION 247

perfectlyclearonly to the implementor. Maybeat the momentI’m writing thissentencesomeoneis thinking thesameof my code,but my aim wasbeingsafe,redundantandhopefully readable.I hopesomeonewill considerthe importanceof theseissueswhencontinuingthis work.


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