Part Ⅰ.Sources of Vibrations 1
Chapter 1.Unbalance and Gyroscopic Effects 5
1.1.Introduction 5
1.1.1.Physico-mathematical model of a rotating system 7
1.1.2.Formation of equations and analysis 7
1.2.Theory of balancing 10
1.2.1.Balancing machine or “balancer” 12
1.2.1.1.The soft-bearing machine 12
1.2.1.2.The hard-bearing machine 17
1.2.2.Balancing in situ 17
1.2.2.1.The method of separate planes 19
1.2.2.2.The method of simultaneous planes - influence coefficients 24
1.2.3.Example of application：the main rotor of a helicopter 26
1.2.3.1.Bench test phase on the ground 27
1.2.3.2.Test phase on a helicopter in flight 30
1.3.Influence of shaft bending 32
1.3.1.The notion of critical speed 33
1.3.2.Forward precession of the flexible shaft 38
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 shafts 42
1.3.4.Example of application：transmission shaft of the tail rotor of a helicopter 44
1.4.Gyroscopic effects 44
1.4.1.Forward or backward motion 44
1.4.2.Equations of motion 47
1.4.2.1.Natural angular frequencies （shaft off motion） 51
1.4.2.2.Critical speeds during forward precession 51
1.4.2.3.Critical speeds during retrograde precession 51
Chapter 2.Piston Engines 53
2.1.Introduction 53
2.2.Excitations generated by a piston engine 54
2.2.1.Analytic determination of an engine torque 55
2.2.2.Engine excitations on the chassis frame 59
2.2.2.1.Knocking load 60
2.2.2.2.Pitch torque 63
2.2.2.3.Review of actions for a four phase cylinder engine 64
2.2.3.The notion of engine balancing 64
2.2.3.1.Balancing the knocking loads 64
2.2.3.2.Balancing the galloping torque 67
2.3.Line shafting tuning 67
2.3.1.The notion of tuning 67
2.3.2.Creation of the equations 68
2.3.3.Line shafting optimization 71
2.3.3.1.Results for a non-optimized line shafting 71
2.3.3.2.Results for an optimized line shafting 73
Chapter 3.Dynamics of a Rotor 75
3.1.Introduction 75
3.2.Description of the blade/hub relationship 75
3.2.1.Some historical data 75
3.2.2.Hinge link of the blade and the hub 76
3.2.2.1.Formation of the equations for blade motion 77
3.2.2.2.Homokinetic rotor 86
3.3.Rotor technologies 87
3.3.1.Articulated rotors 88
3.3.1.1.Conventional articulated rotors 88
3.3.1.2.Starflex？and Spheriflex？rotors 89
3.3.2.Hingeless rotors 91
3.3.3.Hingeless rotor 92
3.4.Influence of alternate aerodynamic loads 93
3.4.1.Load characterization 94
3.4.1.1.Loads on a blade 94
3.4.1.2.Dynamic response of a blade 99
3.4.1.3.Loads transmitted by a mode i 100
3.4.2.Analysis of loads transmitted to the rotor hub 102
3.4.2.1.Loads transmitted to the rotor 103
3.4.2.2.Synthesis of rotor loads on the rotor mast 109
3.4.3.Dynamic optimization of a blade 111
3.4.3.1.Introduction 111
3.4.3.2.Study of the example of an optimized blade 111
3.4.3.3.Contribution of the second flapping mode 116
Chapter 4.Rotor Control 119
4.1.Introduction 119
4.2.Blade motions 121
4.2.1.Flapping equation - general case 121
4.2.2.The case of a rotor without eccentricity and flapping stiffness 123
4.3.Control through cyclic and collective swashplates 127
4.4.Control through flaps 129
4.4.1.Description 129
4.4.2.Modeling 131
4.4.2.1.Flapping equation 131
4.4.2.2.Torsion equation 134
4.4.3.Ways to control the blade 136
Chapter 5.Non-Homokinetic Couplings 141
5.1.Introduction 141
5.2.Analysis of operation 142
5.2.1.Parametric transformation 143
5.2.2.Effects of non-homokinetics：modulation of acceleration 144
5.2.3.Effects of non-homokinetics：variation of the motor torque 146
5.3.Solutions to make the link homokinetic 150
5.3.1.Double Cardan 150
5.3.2.Introduction of high flexibility 151
5.3.3.Homokinetic drive system of a tilt rotor 152
Chapter 6.Aerodynamic Excitations 159
6.1.Introduction 159
6.2.Excitations caused by the Karman vortices - fuselage effects 160
6.3.Aerodynamic excitations generated by the main rotor of a helicopter 164
6.4.Practical solutions for tail-shake 168
PARTⅡ.Vibration Monitoring Systems 171
Chapter 7.Suspensions 177
7.1.Introduction 177
7.2.Filtering effects of the interface link 177
7.2.1.Stiffness modification for an excitation in force 177
7.2.1.1.Modeling 177
7.2.1.2.Response to a harmonic excitation 179
7.2.1.3.Response to an unbalanced excitation 183
7.2.2.Stiffness modification for displacement excitation 185
7.2.2.1.Modeling 186
7.2.2.2.Analysis of the results 187
7.2.2.3.Example：vehicle suspension 188
7.2.3.Damping modification 190
7.2.3.1.Principle 190
7.2.3.2.Modeling 191
7.2.4.Complex case of the rotor/fuselage link of a helicopter 195
7.3.Acting on the interface through kinematic coupling 202
7.3.1.The example of the DAVI system 202
7.3.1.1.Principle 202
7.3.1.2.Formulation of the equations 203
7.3.1.3.Implementation 206
7.3.1.4.Experimental analysis 207
7.3.2.Example of the Aris system 209
7.3.2.1.Mechanical system 209
7.3.2.2.Hydraulic system 210
7.3.3.Example of a fluid inertia resonator 214
7.3.3.1.Principle 214
7.3.3.2.Formation of the equations 214
7.3.3.3.Example of application：integration of the system on a helicopter 216
Chapter 8.Self-Tuning Systems 219
8.1.Introduction 219
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 behavior 222
8.3.1.1.Degrees of freedom of the system 222
8.3.1.2.Formulation of the equations 224
8.3.1.3.Analysis of the general behavior of the suspension 225
8.3.1.4.Conclusion 227
8.3.2.Presentation of the control algorithm 228
8.3.3.Performances 231
8.3.3.1.Simulation and behavior analysis 231
8.3.3.2.Tests conducted on a model 234
8.3.3.3.Flight tests on a real structure 237
Chapter 9.Active Suspensions 239
9.1.Principle 239
9.2.Formulation of system equations and analysis of the system 240
9.3.Technological application 244
Chapter 10.Absorbers 253
10.1.Introduction 253
10.2.Optimization of the structure 253
10.3.Dynamic absorbers 254
10.3.1.Coupling with preponderant stiffness 255
10.3.1.1.Translation system 255
10.3.1.2.Rotating system：torsion resonator 264
10.3.2.Coupling using damping and stiffness 266
10.3.2.1.Operation of the equations 266
10.3.2.2.Tuning method 270
10.3.2.3.Industrial application：resonator used on a helicopter for the tail boom vibrations 272
10.3.2.4.Industrial application：resonator for torsion movements 274
10.3.3.Coupling with preponderant damping 274
Chapter 11.Self-Adjusting Absorbers 279
11.1.Introduction 279
11.2.Implementation 279
11.3.System coupling 281
11.3.1.Analog algorithm 281
11.3.2.Digital algorithm 282
Chapter 12.Active Absorbers 289
12.1.Introduction 289
12.2.Active control with a resonator 289
12.2.1.Electromagnetic actuator 290
12.2.1.1.Single stage resonator 290
12.2.1.2.Two-stage electromagnetic resonator 295
12.2.2.Hydraulic actuator 300
12.2.2.1.Technological principle 300
12.2.2.2.Control algorithm 303
12.2.2.3.Results of lab tests 304
12.3.Active control through external loads 305
12.3.1.Mechanical load generator 305
12.3.1.1.Description of the mechanism 305
12.3.1.2.Positioning of the generator 307
12.3.2.Active control through the anti-torque rotor 309
Chapter 13.Resonators 319
13.1.Introduction 319
13.2.Kinematic coupling 319
13.2.1.Pendular masses 319
13.2.1.1.Principle 319
13.2.1.2.Modeling 320
13.2.1.3.Analysis of the results 323
13.2.2.Coplanar resonators 323
13.3.Stiffness coupling 325
13.3.1.Principle 325
13.3.2.Modeling 327
13.3.3.Forced response of the system 331
13.3.4.Analysis of the results 332
Chapter 14.Self-Adapting Resonators 335
14.1.Introduction 335
14.2.Acting near the source：hub resonator 335
14.2.1.Principle 335
14.2.2.Control algorithm 339
14.2.2.1 Type 1 controller 339
14.2.2.2 Type 2 controller 339
14.2.3.Experiment 340
Chapter 15.Active Systems 343
15.1.Introduction 343
15.2.Principle of the active system in the fixed frame of reference 345
15.2.1.Principle 345
15.2.2.Control algorithm 346
15.2.3.Experiment 349
15.2.4.Conclusions 350
15.3.Principle of the active system in a rotating frame of reference 350
15.3.1.Introduction 350
15.3.2.Individual blade control 352
15.3.2.1.Principle 352
15.3.2.2.Design 352
15.3.2.3.Hydraulic actuators of the IBC system 353
15.3.2.4.Implementation 353
15.3.3.Individual control by servo-flaps 354
15.3.3.1.Principle of the rotor with blade flaps operated by piezoelectric actuators 354
15.3.3.2.Technological solutions 355
Bibliography 359
Index 365