Self-Excited Vibration

Transcription

1 Wenjing Ding Self-Excited Vibration Theory, Paradigms, and Research Methods With 228 figures Ö Springer

2 Contents Chapter 1 Introduction Main Features of Self-Excited Vibration Natural Vibration in Conservative Systems Forced Vibration under Periodic Excitations Parametric Vibration Self-Excited Vibration Conversion between Forced Vibration and Self-Excited Vibration Excitation Mechanisms of Self-Excited Vibration Energy Mechanism Feedback Mechanism A Classification of Self-Excited Vibration Systems Discrete System Continuous System Hybrid System Outline of the Book 18 References 20 Chapter 2 Geometrical Method Structure of Phase Plane Phase Diagrams of Conservative Systems Phase Diagram of a Simple Pendulum Phase Diagram of a Conservative System Phase Diagrams of Nonconservative Systems Phase Diagram of Damped Linear Vibrator Phase Diagram of Damped Nonlinear Vibrator Classification of Equilibrium Points of Dynamic Systems Linear Approximation at Equilibrium Point Classification of Equilibrium Points Transition between Types of Equilibrium Points The Existence of Limit Cycle of an Autonomous System The Index of a Closed Curve with Respect to Vector Field Theorems about the Index of Equilibrium Point The Index of Equilibrium Point and Limit Cycle 39 Hi

3 2.5.4 The Existence of a Limit Cycle Soft Excitation and Hard Excitation of Self-Excited Vibration Definition of Stability of Limit Cycle Companion Relations Soft Excitation and Hard Excitation Self-Excited Vibration in Strongly Nonlinear Systems Waveforms of Self-Excited Vibration Relaxation Vibration Self-Excited Vibration in a Non-Smooth Dynamic System Mapping Method and its Application Poincare Map Piecewise Linear System Application of the Mapping Method 56 References 58 Chapter 3 Stability Methods Stability of Equilibrium Position Equilibrium Position of Autonomous System First Approximation Equation of a Nonlinear Autonomous System Definition of Stability of Equilibrium Position First Approximation Theorem of Stability of Equilibrium Position An Algebraic Criterion for Stability of Equilibrium Position Eigenvalues of Linear Ordinary Differential Equations Distribution of Eigenvalues of a Asymptotic Stable System Hurwitz criterion A Geometric Criterion for Stability of Equilibrium Position Hodograph of Complex Vector D(ico) Argument of Hodograph of Complex Vector D(ico) Geometric Criterion for Stability of Equilibrium Position Coefficient Condition corresponding to the Second Type of Critical Stability Parameter Condition for Stability of Equilibrium Position Stable Region in Coefficient Space Stable Region in Parameter Space Parameter Perturbation on the Boundaries of Stable Region A Quadratic Form Criterion for Stability of Equilibrium Position Linear Equations of Motion of Holonomic System Quadratic Form of Eigenvectors of a Holonomic System Quadratic Form Criterion for a Holonomic System 78 iv

4 -I Influence of Circulatory Force on Stability of Equilibrium Position 78 References 79 Chapter 4 Quantitative Methods Center Manifold Concept of Flow Hartman-Grobman Theorem Center Manifold Theorem Equation of Center Manifold Hopf Bifurcation Method Poincare-Birkhoff Normal Form Poincare-Andronov-Hopf Bifurcation Theorem Hopf Bifurcation Method Lindstedt-Poincare Method Formulation of Equations Periodic Solution of the van der Pol Equation An Averaging Method of Second-Order Autonomous System Formulation of Equations Periodic Solution of Rayleigh Equation Method of Multiple Scales for a Second-Order Autonomous System..' Formulation of Equation System Formulation of Periodic Solution Periodic Solution of van der Pol Equation 105 References 107 Chapter 5 Analysis Method for Closed-Loop System Mathematical Model in Frequency Domain Concepts Related to the Closed-Loop System Typical Components Laplace Transformation Ill Transfer Function Block Diagram of Closed-Loop Systems Nyquist Criterion Frequency Response Nyquist Criterion Application of Nyquist Criterion A Frequency Criterion for Absolute Stability of a Nonlinear Closed-Loop System Absolute Stability Block Diagram Model of Nonlinear Closed-Loop Systems 122

5 5.3.3 Popov Theorems Application of Popov Theorem Describing Function Method Basic Principle Describing Function Amplitude and Frequency of Self-Excited Vibration Stability of Self-Excited Vibration Application of Describing Function Method Quadratic Optimal Control Quadratic Optimal State Control Optimal Output Control Application of Quadratic Optimal Control 137 References 139 Chapter 6 Stick-Slip Vibration Mathematical Description of Friction Force Physical Background of Friction Force Three Kinds of Mathematical Description of Friction Force Stick-Slip Motion A Simple Model for Studying Stick-Slip Motion Non-Smooth Limit Cycle Caused by Friction First Type of Excitation Effects for Stick-Slip Motion Hunting in Flexible Transmission Devices A Mechanical Model and its Equation of Motion Phase Path Equations in Various Stages of Hunting Motion Topological Structure of the Phase Diagram Critical Parameter Equation for the Occurrence of Hunting Asymmetric Dynamic Coupling Caused by Friction Force Mechanical Model and Equations of Motion Stability of Constant Velocity Motion of Dynamic System Second Type of Excitation Effect for Stick-Slip Motion 164 References 166 Chapter 7 Dynamic Shimmy of Front Wheel Physical Background of Tire Force Tire Force Cornering Force Analytical Description of Cornering Force Linear Model for Cornering Force 172 VI

6 7.2 Point Contact Theory Classification of Point Contact Theory Nonholonomic Constraint Potential Energy of a Rolling Tire Dynamic Shimmy of Front Wheel Isolated Front Wheel Model Stability of Front Wheel under Steady Rolling Stable Regions in Parameter Plane Influence of System Parameters on Dynamic Shimmy of Front Wheel Dynamic Shimmy of Front Wheel Coupled with Vehicle A Simplified Model of a Front Wheel System Mathematical Model of the Front Wheel System Stability öf Steady Rolling of the Front Wheel System Prevention of Dynamic Shimmy in Design Stage 189 References 190 Chapter 8 Rotor Whirl Mechanical Model of Rotor in Planar Whirl Classification of rotor whirls Mechanical Model of Whirling Rotor Fluid-Film Force Operating Mechanism of Hydrodynamic Bearings Reynolds' Equation Pressure Distribution on Journal Surface Linearized Fluid Film Force Concentrated Parameter Model of Fluid Film Force Linear Expressions of Seal Force Oil Whirl and Oil Whip Hopf Bifurcation leading to Oil Whirl of Rotor Threshold Speed and Whirl Frequency Influence of Shaft Elasticity on the Oil Whirl of Rotor Influence of External Damping on Oil Whirl Oil Whip Internal Damping in Deformed Rotation Shaft Physical Background of Internal Force of Rotation Shaft Analytical Expression of Internal Force of Rotation Shaft Three Components of Internal Force of Rotation Shaft Rotor Whirl Excited by Internal Damping A Simple Model of Internal Damping Force of Deformed Rotating Shaft Synchronous Whirl of Rotor with Unbalance Supersynchronous Whirl 236 vii

7 8.6 Cause and Prevention of Rotor Whirl Structure of Equation of Motion Common Causes of Two Kinds of Rotor Whirls Preventing the Rotor from Whirling 239 References 240 Chapter 9 Self-Excited Vibrations from Interaction of Structures and Fluid Vortex Resonance in Flexible Structures Vortex Shedding Predominate Frequency Wake Oscillator Model Amplitude Prediction Reduction of Vortex Resonance Flutter in Cantilevered Pipe Conveying Fluid Linear Mathematical Model Critical Parameter Condition Hopf Bifurcation and Critical Flow Velocity Excitation Mechanism and Prevention of Flutter Classical Flutter in Two-Dimensional Airfoil A Continuous Model of Long Wing Critical Flow Velocity of Classical Flutter Excitation Mechanism of Classical Flutter Influence of Parameters of the Wing on Critical Speed of Classical Flutter Stall Flutter in Flexible Structure Aerodynamic Forces Exciting Stall Flutter A Mathematical Model of Galloping in the Flexible Structure Critical Speed and Hysteresis Phenomenon of Galloping Some Features of Stall Flutter and its Prevention Schemes Fluid-Elastic Instability in Array of Circular Cylinders Fluid-Elastic Instability Fluid Forces Depending on Motion of Circular Cylinders Analysis of Flow-Induced Vibration Approximate Expressions of Critical Flow Velocity Prediction and Prevention of Fluid-Elastic Instability 298 References 299 Chapter 10 Self-Excited Oscillations in Feedback Control System Heating Control System Operating Principle of the Heating Control System 303 VIM

8 Mathematical Model of the Heating Control System Time History of Temperature Variation Stable Limit Cycle in Phase Plane Amplitude and Frequency of Room Temperature Derivation An Excitation Mechanism of Self-Excited Oscillation Electrical Position Control System with Hysteresis Principle Diagram Equations of Position Control System with Hysteresis Nonlinearity Phase Diagram and Point Mapping Existence of Limit Cycle Critical Parameter Condition Electrical Position Control System with Hysteresis and Dead-Zone Equation of Motion Phase Diagram and Point Mapping Existence and Stability of Limit Cycle Critical Parameter Condition Hydraulic Position Control System Schematic Diagram of a Hydraulic Actuator Equations of Motion of Hydraulic Position Control System Linearized Mathematical Model Equilibrium Stability of Hydraulic Position Control System Amplitude and Frequency of Self-Excited Vibration Influence of Dead-Zone on Motion of Hydraulic Position Control System Influence of Hysteresis and Dead-Zone on Motion of Hydraulic Position Control System A Nonlinear Control System under Velocity Feedback with Time Delay 338 References 344 Chapter 11 Modeling and Control Excitation Mechanism of Self-Excited Oscillation An Explanation about Energy Mechanism An Explanation about Feedback Mechanism Joining of Energy and Feedback Mechanisms Determine the Extent of a Mechanical Model Minimal Model and Principle Block Diagram 351 ix

9 First Type of Extended Model Second Type of Extended Model Mathematical Description of Motive Force Integrate the Differential Equations of Motion of Continuum Use of the Nonholonomic Constraint Equations Establishing Equivalent Model of the Motive Force Construct the Equivalent Oscillator of Motive Force Identification of Grey Box Model Constructing an Empiric Formula of the Motive Force Establish Equations of Motion of Mechanical Systems Application of Lagrange's Equation of Motion Application of Hamilton's Principle Hamilton's Principle for Open Systems Discretization of Mathematical Model of a Distributed Parameter System Lumped Parameter Method Assumed-Modes Method Finite Element Method Active Control for Suppressing Self-Excited Vibration Active Control of Flexible Rotor Active Control of an Airfoil Section with Flutter 384 References 387 Subject Index 390 x