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Part Ⅰ.Sources of Vibrations1

Chapter 1.Unbalance and Gyroscopic Effects5

1.1.Introduction5

1.1.1.Physico-mathematical model of a rotating system7

1.1.2.Formation of equations and analysis7

1.2.Theory of balancing10

1.2.1.Balancing machine or “balancer”12

1.2.1.1.The soft-bearing machine12

1.2.1.2.The hard-bearing machine17

1.2.2.Balancing in situ17

1.2.2.1.The method of separate planes19

1.2.2.2.The method of simultaneous planes - influence coefficients24

1.2.3.Example of application：the main rotor of a helicopter26

1.2.3.1.Bench test phase on the ground27

1.2.3.2.Test phase on a helicopter in flight30

1.3.Influence of shaft bending32

1.3.1.The notion of critical speed33

1.3.2.Forward precession of the flexible shaft38

1.3.2.1.Subcritical speed （Ω ＜ωcr）39

1.3.2.2.Resonance （Ω = ωcr）41

1.3.2.3.Supercritical speed （Ω ＞ ωcr）41

1.3.3.Balancing flexible shafts42

1.3.4.Example of application：transmission shaft of the tail rotor of a helicopter44

1.4.Gyroscopic effects44

1.4.1.Forward or backward motion44

1.4.2.Equations of motion47

1.4.2.1.Natural angular frequencies （shaft off motion）51

1.4.2.2.Critical speeds during forward precession51

1.4.2.3.Critical speeds during retrograde precession51

Chapter 2.Piston Engines53

2.1.Introduction53

2.2.Excitations generated by a piston engine54

2.2.1.Analytic determination of an engine torque55

2.2.2.Engine excitations on the chassis frame59

2.2.2.1.Knocking load60

2.2.2.2.Pitch torque63

2.2.2.3.Review of actions for a four phase cylinder engine64

2.2.3.The notion of engine balancing64

2.2.3.1.Balancing the knocking loads64

2.2.3.2.Balancing the galloping torque67

2.3.Line shafting tuning67

2.3.1.The notion of tuning67

2.3.2.Creation of the equations68

2.3.3.Line shafting optimization71

2.3.3.1.Results for a non-optimized line shafting71

2.3.3.2.Results for an optimized line shafting73

Chapter 3.Dynamics of a Rotor75

3.1.Introduction75

3.2.Description of the blade/hub relationship75

3.2.1.Some historical data75

3.2.2.Hinge link of the blade and the hub76

3.2.2.1.Formation of the equations for blade motion77

3.2.2.2.Homokinetic rotor86

3.3.Rotor technologies87

3.3.1.Articulated rotors88

3.3.1.1.Conventional articulated rotors88

3.3.1.2.Starflex？and Spheriflex？rotors89

3.3.2.Hingeless rotors91

3.3.3.Hingeless rotor92

3.4.Influence of alternate aerodynamic loads93

3.4.1.Load characterization94

3.4.1.1.Loads on a blade94

3.4.1.2.Dynamic response of a blade99

3.4.1.3.Loads transmitted by a mode i100

3.4.2.Analysis of loads transmitted to the rotor hub102

3.4.2.1.Loads transmitted to the rotor103

3.4.2.2.Synthesis of rotor loads on the rotor mast109

3.4.3.Dynamic optimization of a blade111

3.4.3.1.Introduction111

3.4.3.2.Study of the example of an optimized blade111

3.4.3.3.Contribution of the second flapping mode116

Chapter 4.Rotor Control119

4.1.Introduction119

4.2.Blade motions121

4.2.1.Flapping equation - general case121

4.2.2.The case of a rotor without eccentricity and flapping stiffness123

4.3.Control through cyclic and collective swashplates127

4.4.Control through flaps129

4.4.1.Description129

4.4.2.Modeling131

4.4.2.1.Flapping equation131

4.4.2.2.Torsion equation134

4.4.3.Ways to control the blade136

Chapter 5.Non-Homokinetic Couplings141

5.1.Introduction141

5.2.Analysis of operation142

5.2.1.Parametric transformation143

5.2.2.Effects of non-homokinetics：modulation of acceleration144

5.2.3.Effects of non-homokinetics：variation of the motor torque146

5.3.Solutions to make the link homokinetic150

5.3.1.Double Cardan150

5.3.2.Introduction of high flexibility151

5.3.3.Homokinetic drive system of a tilt rotor152

Chapter 6.Aerodynamic Excitations159

6.1.Introduction159

6.2.Excitations caused by the Karman vortices - fuselage effects160

6.3.Aerodynamic excitations generated by the main rotor of a helicopter164

6.4.Practical solutions for tail-shake168

PARTⅡ.Vibration Monitoring Systems171

Chapter 7.Suspensions177

7.1.Introduction177

7.2.Filtering effects of the interface link177

7.2.1.Stiffness modification for an excitation in force177

7.2.1.1.Modeling177

7.2.1.2.Response to a harmonic excitation179

7.2.1.3.Response to an unbalanced excitation183

7.2.2.Stiffness modification for displacement excitation185

7.2.2.1.Modeling186

7.2.2.2.Analysis of the results187

7.2.2.3.Example：vehicle suspension188

7.2.3.Damping modification190

7.2.3.1.Principle190

7.2.3.2.Modeling191

7.2.4.Complex case of the rotor/fuselage link of a helicopter195

7.3.Acting on the interface through kinematic coupling202

7.3.1.The example of the DAVI system202

7.3.1.1.Principle202

7.3.1.2.Formulation of the equations203

7.3.1.3.Implementation206

7.3.1.4.Experimental analysis207

7.3.2.Example of the Aris system209

7.3.2.1.Mechanical system209

7.3.2.2.Hydraulic system210

7.3.3.Example of a fluid inertia resonator214

7.3.3.1.Principle214

7.3.3.2.Formation of the equations214

7.3.3.3.Example of application：integration of the system on a helicopter216

Chapter 8.Self-Tuning Systems219

8.1.Introduction219

8.2.Modification of link characteristics （stiffness or damping）220

8.3.Modification of the kinematic coupling：example of self-tuning Sarib？221

8.3.1.Modeling of the suspension behavior222

8.3.1.1.Degrees of freedom of the system222

8.3.1.2.Formulation of the equations224

8.3.1.3.Analysis of the general behavior of the suspension225

8.3.1.4.Conclusion227

8.3.2.Presentation of the control algorithm228

8.3.3.Performances231

8.3.3.1.Simulation and behavior analysis231

8.3.3.2.Tests conducted on a model234

8.3.3.3.Flight tests on a real structure237

Chapter 9.Active Suspensions239

9.1.Principle239

9.2.Formulation of system equations and analysis of the system240

9.3.Technological application244

Chapter 10.Absorbers253

10.1.Introduction253

10.2.Optimization of the structure253

10.3.Dynamic absorbers254

10.3.1.Coupling with preponderant stiffness255

10.3.1.1.Translation system255

10.3.1.2.Rotating system：torsion resonator264

10.3.2.Coupling using damping and stiffness266

10.3.2.1.Operation of the equations266

10.3.2.2.Tuning method270

10.3.2.3.Industrial application：resonator used on a helicopter for the tail boom vibrations272

10.3.2.4.Industrial application：resonator for torsion movements274

10.3.3.Coupling with preponderant damping274

Chapter 11.Self-Adjusting Absorbers279

11.1.Introduction279

11.2.Implementation279

11.3.System coupling281

11.3.1.Analog algorithm281

11.3.2.Digital algorithm282

Chapter 12.Active Absorbers289

12.1.Introduction289

12.2.Active control with a resonator289

12.2.1.Electromagnetic actuator290

12.2.1.1.Single stage resonator290

12.2.1.2.Two-stage electromagnetic resonator295

12.2.2.Hydraulic actuator300

12.2.2.1.Technological principle300

12.2.2.2.Control algorithm303

12.2.2.3.Results of lab tests304

12.3.Active control through external loads305

12.3.1.Mechanical load generator305

12.3.1.1.Description of the mechanism305

12.3.1.2.Positioning of the generator307

12.3.2.Active control through the anti-torque rotor309

Chapter 13.Resonators319

13.1.Introduction319

13.2.Kinematic coupling319

13.2.1.Pendular masses319

13.2.1.1.Principle319

13.2.1.2.Modeling320

13.2.1.3.Analysis of the results323

13.2.2.Coplanar resonators323

13.3.Stiffness coupling325

13.3.1.Principle325

13.3.2.Modeling327

13.3.3.Forced response of the system331

13.3.4.Analysis of the results332

Chapter 14.Self-Adapting Resonators335

14.1.Introduction335

14.2.Acting near the source：hub resonator335

14.2.1.Principle335

14.2.2.Control algorithm339

14.2.2.1 Type 1 controller339

14.2.2.2 Type 2 controller339

14.2.3.Experiment340

Chapter 15.Active Systems343

15.1.Introduction343

15.2.Principle of the active system in the fixed frame of reference345

15.2.1.Principle345

15.2.2.Control algorithm346

15.2.3.Experiment349

15.2.4.Conclusions350

15.3.Principle of the active system in a rotating frame of reference350

15.3.1.Introduction350

15.3.2.Individual blade control352

15.3.2.1.Principle352

15.3.2.2.Design352

15.3.2.3.Hydraulic actuators of the IBC system353

15.3.2.4.Implementation353

15.3.3.Individual control by servo-flaps354

15.3.3.1.Principle of the rotor with blade flaps operated by piezoelectric actuators354

15.3.3.2.Technological solutions355

Bibliography359

Index365