Mechanical instability

This book presents a study of the stability of mechanical systems, i.e. their free response when they are removed from their position of equilibrium after a temporary disturbance. After reviewing the main analytical methods of the dynamical stability of systems, it highlights the fundamental difference in nature between the phenomena of forced resonance vibration of mechanical systems subjected to an imposed excitation and instabilities that characterize their free response. It specifically develops instabilities arising from the rotor-structure coupling, instability of control systems, the self-sustained instabilities associated with the presence of internal damping and instabilities related to the fluid-structure coupling for fixed and rotating structures. For an original approach following the analysis of instability phenomena, the book provides examples of solutions obtained by passive or active methods.

Author Notes

Tomasz Krysinski, a dynamics specialist, is Head of the vibration and internal noise department at Eurocopter engineering and design department.
Franois Malburet is an Associate Professor at ENSAM, Aix-en-Provence, France.

Philippe Roesch

Foreword	p. ix
Preface	p. xiii
Chapter 1 Notions of Instability	p. 1
1.1 Introduction	p. 1
1.1.1 Lyapunov's Direct Method	p. 3
1.1.2 Lyapunov's Indirect Method	p. 5
1.2 Comparison of Notions of Resonance and Instability	p. 8
1.2.1 Notion of Resonance	p. 8
1.2.2 Notion of Instability	p. 22
1.3 Instability Due to Self-Sustained Excitation	p. 23
1.3.1 Multiple-Degree-of-Freedom Systems	p. 24
1.3.2 Single-Degree-of-Freedom System	p. 46
1.4 Parametric Instability	p. 54
1.4.1 General Case	p. 54
1.4.2 Mathieu's Equation	p. 54
1.4.3 Typical Application	p. 57
1.5 Summary of Methods Used to Ensure or Increase the Stability of a System	p. 60
1.5.1 Notion of Degrees of Stability	p. 60
1.5.2 Main Corrector Systems	p. 67
Chapter 2 Rotor/Structure Coupling: Examples of Ground Resonance and Air Resonance	p. 91
2.1 Introduction to Ground Resonance	p. 91
2.2 Ground Resonance Modeling	p. 99
2.2.1 Minimum Degree-of-Freedom Model	p. 99
2.2.2 Stability Criteria	p. 110
2.2.3 Energy Analysis	p. 113
2.3 Active Control of Ground Resonance	p. 115
2.3.1 Active Control Algorithm	p. 115
2.3.2 Performance Indicators	p. 135
2.3.3 Implementation of Active Control	p. 137
2.4 Air Resonance	p. 143
2.4.1 Phenomenon Description	p. 143
2.4.2 Modeling and Setting Up Equations	p. 144
2.4.3 Active Control of Air Resonance	p. 149
Chapter 3 Torsional System: Instability of Closed-Loop Systems	p. 153
3.1 Introduction	p. 153
3.2 Governing Principle	p. 153
3.2.1 History and Sizing of Flyball Governor	p. 154
3.2.2 Simple Mathematical Sizing Criterion	p. 155
3.2.3 Physical Analysis of Criterion and Effect of Parameters	p. 164
3.3 Industrial Cases	p. 168
3.3.1 Case of Airplane With Variable-Setting Angle Propeller Rotor	p. 168
3.3.2 Case of Tiltrotor Aircraft	p. 175
3.3.3 Case of Helicopter	p. 176
Chapter 4 Self-Sustaining Instability for Rotating Shafts	p. 201
4.1 Introduction to Self-Sustaining Instability	p. 201
4.2 Modeling of Effect of Internal Damping on Rotating Systems	p. 206
4.2.1 Instability Origins	p. 206
4.2.2 Highlighting Instability	p. 207
4.2.3 Stability Criterion for a Flexible Shaft	p. 222
Chapter 5 Fluid-Structure Interaction	p. 245
5.1 Introduction	p. 245
5.1.1 Fluid-Structure Interaction Issues	p. 245
5.1.2 Instability and Energy Analysis	p. 246
5.1.3 Brief Description of Flutter	p. 248
5.2 Flutter of an Airfoil in an Airstream	p. 250
5.2.1 Setting Up Equations	p. 252
5.2.2 Industrial Examples	p. 259
5.3 Whirl Flutter	p. 312
5.3.1 Introduction to Convertible Aircraft Case	p. 313
5.3.2 Enhanced Convertible Aircraft Rotor Reed's Modeling - Stability	p. 315
5.3.3 Whirl Flutter Active Control: Case of Tilt Rotor	p. 326
Bibliography	p. 335
Index	p. 339

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