Applied Methods in the Theory of Nonlinear Oscillations

Front Cover
Title Page
PREFACE
CONTENTS
PART 1 OSCILLATIONS IN LYAPUNOV SYSTEMS
CHAPTER 1 INTRODUCTION
§ 1. Transformation of Lyapunov Systems
1.1. General case
1.2. Systems of second-order equations
§ 2. On the Poincare Method of Finding Periodic Solutions of Nonautonomous Quasilinear Systems
2.1. Differential equations of the generating solution and firstc orrections.
2.2. Nonresonant case.
2.3. Resonant case.
2.4. Variational equations for periodic unperturbed motion.
2.5. Case of distinct multipliers of unperturbed system of variational equations.
2.6. Case of multiple multipliers
2.7. Examples.
§ 3. Forced Vibrations in Centrifuges Used for Spinning
3.1. Statement of the problem and equations of motion
3.2. Determination of a periodic solution.
3.3. Stability analysis.
CHAPTER 2 OSCILLATORY CHAINS
§ 1. Completely Elastic Free Oscillatory Chains
1.1. Definition of an oscillatory chain.
1.2. Determination of equilibrium positions.
1.3. Asymptotic stability in the large of the lower equilibrium position for distinct resistance forces.
1.4. Variational equations for vertical oscillations of the system.
1.5. Conservative case.
1.6. Stability of vertical vibrations of a spring-loaded
§ 2. Partly Elastic Free Oscillatory Chains
2.1. Statement of the problem.
2.2. Kinetic and potential energies.
2.3. Example.
2.4. Pendulum subject to elastic free suspension.
2.5. Pendulum subject to elastic guided suspension.
CHAPTER 3 APPLICATION OF THE METHODS OF SMALL PARAMETER TO OSCILLATIONS IN LYAPUNOV SYSTEMS
§ 1. Loss of Stability of Vertical Vibrations of a Spring-Loaded Pendulum
1.1. Step 1.
1.2. Step 2
1.3. Step 3.
§ 2. On Coupling of Radial and Vertical Oscillations of Particles in Cyclic Accelerators
2.1. Step 1.
2.2. Step
2.3. Step 3.
§ 3. Loss of Stability of Vertical Oscillations of a Pendulum Subject to Elastic Guided Suspension
3.1. Determination of nontrivial periodic modes (Step 2).
3.2. Transient process (Step 3)
§ 4. Periodic Modes of a Pendulum Subjectto Elastic Free Suspension
4.1. Transformation of equations of motion.
4.2. Periodic solutions.
CHAPTER 4 OSCILLATIONS IN MODIFIED LYAPUNOV SYSTEMS.
§ 1. Lyapunov Systems with Damping
1.1. Transformation of equations of motion.
1.2. Cnmplete system of variational equations in the Poincare parameter and its solution.
1.3. Vibrations in mechanical systems with one degree of freedom and different types of nonlinearity.
1.5. Spring-loaded pendulum with linear damping.
§ 2. On Lyapunov-Type Systems
2.1. Statement of the problem.
2.2. Transformation of Lyapunov-type systems.
PART 2 APPLICATION OF THE THEORY OF NORMAL FORMS TO OSCILLATION PROBLEMS
CHAPTER 5 ELEMENTS OF THE THEORY OF NORMAL FORMS OF REAL AUTONOMOUS SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS
§ 1. Introductory Information
1.1. Statement of the problem
1.2. The Fundamental Brjuno theorem
1.3. The Poincare theorem
§ 2. Additional Information
2.1. Some properties of normalizing transformations.
2.2. Classification of normal forms; integrable normal forms
2.3. Concept of power transformations
2.4. The Brjuno theorem on convergence and divergence of normalizingtrans formations
§ 3. Practical Calculation of Coefficients of Normalizing Transformation and Normal Form
3.1. Fundamental identities.
3.2. Computational alternative.
3.3. Fundamental identities in general form and their transformation.
3.4. Computational alternative in general case
3.5. Remark on the transition from symmetrized coefficients toordinary ones.
3.6. Formulas for coefficients of fourth-power variables
3. 7. Case of composite elementary divisors of the matrix of thelinear part.
CHAPTER 6 NORMAL FORMS OF ARBITRARY-ORDER SYSTEMS IN THE CASEOF ASYMPTOTIC STABILITY IN LINEAR APPROXIMATION
§ 1. Damped Oscillatory Systems
1.1. Reduction to diagonal form
1.2. Calculation of coefficients of normalizing transformation.
1.3. General solution of the initial system. (general solution of the Cauchy problem).
§ 2. Examples
2.1. A system with one degree of freedom.
2.2. Oscillations of a spring-suspended mass with linear damping.
CHAPTER 7 NORMAL FORMS OF THIRD-ORDER SYSTEMS
§ 1. Case of Two Pure Imaginary Eigenvalues of the Matrix of the Linear Part
1.1. Reduction to normal form.
1.2. Calculation of coefficients of normalizing transformation and normal form.
1.3. Application of power transformation.
1.4. Free oscillations of an electric servodrive
§ 2. Case of Neutral Linear Approximation
2.1. Normal form.
2.2. Calculation of coefficients of normalizing transformation andnormal form.
2.3. Remark
2.4. Conclusions on stability
2.5. Integration of normal form in quadratic approximation
2.6. Example
§ 3. Case of a Zero Eigenvalue of the Matrix of the Linear Part
3.1. Normal form and normalizing transformation.
3.2. Integration of normal form.
3.3. Remark on convergence.
3.4. Free oscillations in a tracking system with a TV sensor.
CHAPTER 8 NORMAL FORMS OF FOURTH- AND SIXTH-ORDER SYSTEMSIN NEUTRAL LINEAR APPROXIMATION
§ 1. Fourth-Order Systems
1.1. Remark on coefficients of systems of diagonal form.
1.2. Reduction to normal form.
1.3. Calculation of coefficients of normalizing transformation andnormal forms.
1.4. The Molchanov criterion of oscillation stability.
1.5. The Bihikov-Piiss criterion.
§ 2. The Ishlinskii Problem
2.1. Reduction of equations of motion to the Lyapunov form.
2.2. Transformation of systems similar to Lyapunov.
2.3. Determination of periodic solutions.
2.4. Reduction of equations of motion to diagonal form and transformation to normal form.
2.5. General solution of the Cauchy problem.
2.6. Preliminary conclusions on stability.
2.7. Construction of the Lyapunov function.
§ 3. The Trajectory Described by the Centreof a Shaft's Cross Section in One Revolution
3.1. Statement of the problem and equations of motion.
3.2. Reduction to diagonal form.
3.3 Reduction to normal form.
3.4. General solution of the Cauchy problem.
§ 4. Sixth-Order Systems
4. 1. Solutions of the resonant equation.
4.2. Normal forms.
4.3. Calculation of coefficients of normalizing transformation and'normal forms.
4.4. Stability in the third approximation. The Molchanov criterion.
CHAPTER 9 OSCILLATIONS OF A HEAVY SOLID BODYWITH A FIXED POINT ABOUT THE LOWER EQUILIBRIUM POSITION
§ 1. Case of Centroid Located in a Principal Planeof the Ellipsoid of Inertia with Respect to a Fixed Point
1.1. Reduction to diagonal form.
1.2. Reduction to the Lyapunov form
1.3. Resonances.
1.4. Simplest motions.
1.5. Transformation of equations of diagonal form.
1.6. Possible generalizations.
1.7. Situation similar to the Kovalevskaya case.
1.8. Application of the method of successive approximations.
1.9. Remarks on the determination of the position of a solid bodywith a fixed point.
§ 2. The General Case
2.1. Base reference frame.
2.2. Special reference frame.
2.3. Equations of motion of a heavy solid body in the special r£'ferenceframe.
2.4. Reduction to the Lyapunov form.
2.5. Resonances.
2.6. Application of the method of successive approximalions.
BRIEF BIBLIOGRAPHICAL NOTES
Part One
Chapter I
Chapter II
Chapter III
Chapter IV
Part Two
Chapter V
Chapter VI
Chapter VII
Chapter VIII
Chapter IX
REFERENCES
A. Monographs, textbooks, reviews
B. Articles
SUBJECT INDEX