Tomasz Krysinski, Consultant.Francois Malburet is an Associate Professor at ENSAM, Aix-en- Provence, France.

• Foreword ixPhilippe ROESCHPreface xiiiChapter 1. Notions of Instability 11.1. Introduction 11.1.1. Lyapunov’s Direct Method 31.1.2. Lyapunov’s Indirect Method 51.2. Comparison of Notions of Resonance and Instability 81.2.1. Notion of Resonance 81.2.2. Notion of Instability 221.3. Instability Due to Self-Sustained Excitation 231.3.1. Multiple-Degree-of-Freedom Systems 241.3.2. Single-Degree-of-Freedom System 461.4. Parametric Instability 541.4.1. General Case 541.4.2. Mathieu’s Equation 541.4.3. Typical Application 571.5. Summary of Methods Used to Ensure or Increase the Stability of a System 601.5.1. Notion of Degrees of Stability 601.5.2. Main Corrector Systems 67Chapter 2. Rotor/Structure Coupling: Examples of Ground Resonance and Air Resonance 912.1. Introduction to Ground Resonance 912.2. Ground Resonance Modeling 992.2.1. Minimum Degree-of-Freedom Model 992.2.2. Stability Criteria 1102.2.3. Energy Analysis 1132.3. Active Control of Ground Resonance 1152.3.1. Active Control Algorithm 1152.3.2. Performance Indicators 1352.3.3. Implementation of Active Control 1372.4. Air Resonance 1432.4.1. Phenomenon Description 1432.4.2. Modeling and Setting Up Equations 1442.4.3. Active Control of Air Resonance 149Chapter 3. Torsional System: Instability of Closed-Loop Systems 1533.1. Introduction 1533.2. Governing Principle 1533.2.1. History and Sizing of Flyball Governor 1543.2.2. Simple Mathematical Sizing Criterion 1553.2.3. Physical Analysis of Criterion and Effect of Parameters 1643.3. Industrial Cases 1683.3.1. Case of Airplane With Variable-Setting Angle Propeller Rotor 1683.3.2. Case of Tiltrotor Aircraft 1753.3.3. Case of Helicopter 176Chapter 4. Self-Sustaining Instability for Rotating Shafts 2014.1. Introduction to Self-Sustaining Instability 2014.2. Modeling of Effect of Internal Damping on Rotating Systems 2064.2.1. Instability Origins 2064.2.2. Highlighting Instability 2074.2.3. Stability Criterion for a Flexible Shaft 222Chapter 5. Fluid-Structure Interaction 2455.1. Introduction 2455.1.1. Fluid-Structure Interaction Issues 2455.1.2. Instability and Energy Analysis 2465.1.3. Brief Description of Flutter 2485.2. Flutter of an Airfoil in an Airstream 2505.2.1. Setting Up Equations 2525.2.2. Industrial Examples 2595.3. Whirl Flutter 3125.3.1. Introduction to Convertible Aircraft Case 3135.3.2. Enhanced Convertible Aircraft Rotor Reed’s Modeling – Stability 3155.3.3. Whirl Flutter Active Control: Case of Tilt Rotor 326Bibliography 335Index 339

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