Contents -- Foreword -- Preface -- Part I. The Basis of Vibrational Mechanics -- Chapter 1. On Some Nonlinear Oscillatory Effects. The Main Idea of Vibrational Mechanics -- 1.1 On the Effects Caused by the Action of Vibration in Nonlinear Oscillatory Systems -- 1.2 The Main Idea of Vibrational Mechanics. Observer O and Observer V -- Chapter 2. The Main Mathematical Apparatus of Vibrational Mechanics and of the Method of Direct Separation of Motions -- 2.1 Preliminary Remarks -- 2.2 The Initial Equation and Its Reduction to a System of Integro-differential Equations -- 2.3 The Case When a Separate Equation for the Slow Component is Obtained -- 2.4 The Main Assumption of Vibrational Mechanics Its Formalization and Conditions of its Fulfillment -- 2.5 The Main Equation of Vibrational Mechanics. Vibrational Forces Observers O and V -- 2.6 Method of an Approximate Derivation of the Expression of Vibrational Forces and of Composing the Main Equation of Vibrational Mechanics -- 2.7 Important Special Case -- 2.8 On the Case of a Mechanical Systems with Constraints -- 2.9 On the Simplifications of Solving Equations for the Fast Component of Motion. Purely Inertial Approximation -- 2.10 Additional Remarks Certain Generalizations -- 2.11 Summary: On the Procedure of the Practical Use of the Method -- Chapter 3. On Other Methods of Obtaining Expressions for the Vibrational Forces and the Main Equations of Vibrational Mechanics -- 3.1 Well Known Methods -- 3.2 Two Other Methods -- Chapter 4. A Simplest Example: Solving the Problem about a Pendulum with a Vibrating Axis of Suspension by Different Methods of the Theory of Nonlinear Oscillations -- 4.1 Preliminary Remarks -- 4.2 Equation of Motion -- 4.3 The Poincare-Lyapunov Method of Small Parameter.

4.4 The Use of Floquet-Lyapunov's Theory and of Ince-Strutt's Diagram -- 4.5 Asymptotic Method -- 4.6 Method of Multiple Scales -- 4.7 Methods of Harmonic Balance and of Bubnov-Galerkin -- 4.8 Method of Direct Separation of Motions -- 4.9 Discussion -- Chapter 5. Conclusion: On the Main Peculiarities and Advantages of the Approaches of Vibrational Mechanics and of the Method of Direct Separation of Motions as Compared to Other Methods of Nonlinear Mechanics -- 5.1 Peculiarities and Limitations -- 5.2 Advantages -- 5.3 Final Remarks -- References to Part 1 -- Part II. Pendulum and Pendulum Systems under High-Frequency Excitation - Non-Trivial Effects -- Chapter 6. Quasi-equilibrium Positions and Stationary Rotations of the Pendulums with a Periodically Vibrating Axis -- 6.1 Preliminary Remarks Equation of Motion -- 6.2 Regimes of Quasi-Equilibrium -- 6.3 Regimes of Rotation -- References -- Chapter 7. Non-Trivial Effects of High-Frequency Excitation for Pendulum Systems -- 7.1 Preliminary Remarks -- 7.2 Chelomei's Pendulum - Resolving a Paradox -- 7.3 Nonlinear Dynamics of the Follower-Loaded Double Pendulum with Added Support-Excitation -- 7.4 Articulated Pipes Conveying Fluid Pulsating with High Frequency -- References -- Chapter 8. On the Theory of the Indian Magic Rope -- 8.1 Preliminary Remarks -- 8.2 Equation of Oscillations and Its Consideration -- 8.3 Solving the Problem by Method of Direct Separation of Motions -- 8.4 Analysis of the Result. Physical Explanation of the Effect of the Indian Rope -- References -- Chapter 9. Effect of Conjugate Resonances and Bifurcations at the Biharmonic Excitation of Nonlinear Systems (By an Example of Pendulum with a Vibrating Axis) -- 9.1 Preliminary Remarks -- 9.2 Equation of Motion -- 9.3 Equation of Slow Motions.

9.4 The Case of "Ordinary" Resonance. Conjugate Resonances and Bifurcations -- 9.5 The Case of Parametric Resonances -- 9.6 The Biharmonic Effect on the System Described by Duffing's Equation -- 9.7 Some Remarks on the Application of the Results -- References -- Chapter 10. On the Investigations of the Electromechanical Systems. On the Behavior of the Conductivity Bodies of Pendulum Types in High-Frequency Magnetic Fields -- 10.1 On Some New Results of the Theory of Electro-Mechanical Systems -- 10.2 The Problem about a Passive (Resonant) Electrostatic Suspension -- 10.3 The Problem about the Motion of the Pendulum with the Closed Circuit of High-Current in Frequency Magnetic Field -- References -- Part III. Problems of the Theory of Selfsynchronization -- Chapter 11. On General Definitions of Synchronization -- 11.1 Preliminary Remarks -- 11.2 Evolution of the Synchronization Concept -- 11.3 General Definition of Synchronization -- 11.4 Examples -- 11.5 Discussion -- References -- Chapter 12. A Guide to Solving Certain Self-Synchronization Problems -- 12.1 Preliminary Remarks -- 12.2 Unbalanced Rotors on an Oscillatory System -- 12.3 Application of the Method of Direct Separation of Motion -- 12.4 Harmonic Influence Coefficients and Vibrational Moments -- 12.5 Conditions for the Existence of Synchronous Motions -- 12.6 Conditions of Stability -- 12.7 Two Rotors -- 12.8 Example -- 12.9 Summary -- References -- Chapter 13. The Setting Up of the Self-Synchronization Problem of the Dynamic Objects with Inner Degrees of Freedom and Methods of Its Solution -- 13.1 Preliminary Remarks -- 13.2 Statement of the Problem -- 13.3 Structure of the Kinetic and Potential Energy of the System. The Generalized Forces.

13.4 Integral Criterion (Extreme Property) of the Stability of Synchronous Motions -- 13.5 Method of Direct Separation of Motions. Methods of Small Parameter -- 13.6 Self-Synchronization of Two Identical Vibro-Exciters with the Internal Degrees of Freedom Whose Axes Pass through the Center of Gravity of a Solid Body (Plane Motion) -- References -- Chapter 14. On the Expansion of the Field of Applicability of the Integral Signs (Extreme Property) of Stability in Problems on the Synchronization of the Dynamic Objects with Almost Uniform Rotations -- 14.1 Preliminary Remarks -- 14.2 On the Integral Sign (Extreme Property) of the Stability of Synchronous Motions -- 14.3 The Statement of the Problem on the Synchronization of Objects with Almost Uniform Rotations -- 14.4 Solution of the Problem by Method of Direct Separation of Motions -- 14.5 Extended Formulation of Integral Signs -- 14.6 Examples Comparison to the Results Obtained by Other Methods -- References -- Part IV. Problems of Creating Dynamic Materials -- Chapter 15. On Dynamic Materials -- 15.1 Briefly on the Idea of Dynamic Materials -- 15.2 On the Development of the Idea of Creating Dynamic Materials -- References -- Chapter 16. The Active Control of Vibrations of Composite Beams by Parametric Stiffness Modulation -- 16.1 Preliminary Remarks -- 16.2 The Governing Equations -- 16.3 Modal Analysis of Vibrations -- 16.4 Direct Partition of Motions -- 16.5 The Eigenvalue Problem for a Beam with the Resonant Parametric Stiffness Modulation -- 16.6 Forced Vibrations. An Influence of Internal Damping -- 16.7 Vibrations of a Beam with Parametrically Modulated Stiffness in Heavy Fluid Loading Conditions -- 16.8 Discussion of the Parametric Stiffness Modulation.

16.9 A Modal Formulation of the Control Problem for a Sandwich Beam -- 16.10 Analysis of Vibration Control for a Model Two-Degrees of Freedom Mechanical System -- 16.11 Conclusions -- References -- Part V. Vibrational Hydrodynamics and Hydraulics -- Chapter 17. On Vibrational Hydrodynamics and Hydraulics -- 17.1 Reinolds' Equation as an Equation of Vibrational Mechanics -- 17.2 The Analog of Bernoulli's Equation for the Flows Subjected to Vibration -- References -- Chapter 18. On the Vibro-Jet Effect and on the Phenomena of Vibrational Injection of the Gas into Fluid -- 18.1 On Phenomenon under Consideration -- 18.2 Common Expression for the Gas or Fluid Discharge through a Hole in Vibrating Vessel -- 18.3 On the Theory of Vibro-jet Effect -- 18.4 On the Theory of the Vibrational Injection of Gas into the Fluid -- 18.5 Results of the Experiments -- References -- Part VI. Some Mathematical Supplements and Generalization -- Chapter 19. On Asymptotic Analysis of Systems with Fast Excitation -- 19.1 Introduction. Classification of Systems with Fast Excitation. Weakly Excited Systems -- 19.2 Systems with Strong Excitation. General Analysis -- 19.3 Systems with Very Strong Excitation in a Special Case of Fast Oscillating Inertial Coefficients -- 19.4 Two Mathematical Examples of Systems with Strong Excitation -- 19.5 Response of a One Degree of Freedom System to Strong and Very Strong High Frequency External and Parametric Excitation -- 19.6 Conclusions -- References -- Chapter 20. On the Averaging of Discontinuous Systems -- 20.1 Introduction. Types of Discontinuities in Oscillating Systems. Short Review of the Investigations -- 20.2 Averaging of Constant Order Discontinuous Systems -- 20.3 Averaging of Variable Order Discontinuous Systems.

20.4 On the Averaging of Discontinuous Systems in the Vicinity of a Strong Nonlinear Resonance.