Selected Topics in Vibrational Mechanics

The following sections are included:

• Foreword

• Preface

• Contents

The following sections are included:

• On the Effects Caused by the Action of Vibration in Nonlinear Oscillatory Systems

• The Main Idea of Vibrational Mechanics. Observer O and Observer V

The following sections are included:

• Preliminary Remarks

• The Initial Equation and Its Reduction to a System of Integro-differential Equations

• The Case when a Separate Equation for the Slow Component is Obtained

• The Main Assumption of Vibrational Mechanics, its Formalization and Conditions of its Fulfillment

• The Main Equation of Vibrational Mechanics. Vibrational Forces, Observers O and V

• Method of an Approximate Derivation of the Expression of Vibrational Forces and of Composing the Main Equation of Vibrational Mechanics

• Important Special Case

• On the Case of a Mechanical Systems with Constraints

• On the Simplifications of Solving Equations for the Fast Component of Motion. Purely Inertial Approximation

• Additional Remarks, Certain Generalizations

• Summary: On the Procedure of the Practical Use of the Method

The following sections are included:

• Well Known Methods

• Two Other Methods

The following sections are included:

• Preliminary Remarks

• Equation of Motion

• The Poincare-Lyapunov Method of Small Parameter

• The use of Floquet-Lyapunov's Theory and of Ince-Strutt's Diagram

• Asymptotic Method

• Method of Multiple Scales

• Methods of Harmonic Balance and of Bubnov—Galerkin

• Method of Direct Separation of Motions

• Discussion

The following sections are included:

• Peculiarities and Limitations

• Advantages

• Final Remarks

• References to Part 1

The following sections are included:

• Preliminary Remarks, Equation of Motion

• Regimes of Quasi-Equilibrium

• Regimes of Rotation

• References

The following sections are included:

• Preliminary Remarks

• Chelomei's Pendulum – Resolving a Paradox
  • Introduction
  • History of the Problem and the Controversy
  • Experimental Observations
  • A Theoretical Model
  • Predicting Stationary Solutions
  • Analysing the Results
  • Conclusions

• Nonlinear Dynamics of the Follower-Loaded Double Pendulum with Added Support-Excitation
  • The Model and Model Equations
  • Direct Partition of Motion
  • Linear Stability of the Upright Pendulum Position
  • Global Dynamic Behavior
  • Effects of Bi-Directional Support-Excitation
  • Conclusions

• Articulated Pipes Conveying Fluid Pulsating with High Frequency
  • The Model and Model Equations
  • Autonomous Model Equations
  • Linear Stability of the Hanging Position
  • Periodic Motion of the Autonomous System
  • Conclusions

• References

The following sections are included:

• Preliminary Remarks

• Equation of Oscillations and Its Consideration

• Solving the Problem by Method of Direct Separation of Motions

• Analysis of the Result. Physical Explanation of the Effect of the Indian Rope

• References

The following sections are included:

• Preliminary Remarks

• Equation of Motion

• Equation of Slow Motions

• The Case of "Ordinary" Resonance. Conjugate Resonances and Bifurcations

• The Case of Parametric Resonances

• The Biharmonic Effect on the System, Described by Duffing's Equation

• Some Remarks on the Application of the Results

• References

The following sections are included:

• On Some New Results of the Theory of Electro-Mechanical Systems

• The Problem about a Passive (Resonant) Electrostatic Suspension

• The Problem about the Motion of the Pendulum with the Closed Circuit of Current in High-Frequency Magnetic Field

• References

The following sections are included:

• Preliminary Remarks

• Evolution of the Synchronization Concept

• General Definition of Synchronization

• Examples

• Discussion

• References

The following sections are included:

• Preliminary Remarks

• Unbalanced Rotors on an Oscillatory System

• Application of the Method of Direct Separation of Motion

• Harmonic Influence Coefficients and Vibrational Moments

• Conditions for the Existence of Synchronous Motions

• Conditions of Stability

• Two Rotors

• Example

• Summary

• References

The following sections are included:

• Preliminary Remarks

• Statement of the Problem

• Structure of the Kinetic and Potential Energy of the System. The Generalized Forces

• Integral Criterion (Extreme Property) of the Stability of Synchronous Motions
  • Introduction
  • Systems with almost uniform rotations, non-quasiconservative idealization of rotation
  • Systems with the objects, idealized as quasiconservative
  • Comparison of the results at different types of idealization. Some conclusions

• Method of Direct Separation of Motions. Methods of Small Parameter

• 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)
  • Description of the system
  • Synchronization of the system in the absence of inner degrees of freedom of the exciters
  • Equations of motion of the system
  • Stationary solutions and their stability
  • Discussion of the results of the section

• References

The following sections are included:

• Preliminary Remark

• On the Integral Sign (Extreme Property) of the Stability of Synchronous Motions

• The Statement of the Problem on the Synchronization of Objects with Almost Uniform Rotations

• Solution of the Problem by Method of Direct Separation of Motions

• Extended Formulation of Integral Signs

• Examples, Comparison to the Results, Obtained by Other Methods
  • Double Synchronization of Two Vibro-Exciters on a Platform with One Degree of Freedom
  • Double Synchronization of Three Unbalanced Vibro-Exciters, Located Symmetrically on a Softly Vibro-Isolated Flatly Oscillating Solid Body

• References

The following sections are included:

• Briefly on the Idea of Dynamic Materials

• On the Development of the Idea of Creating Dynamic Materials

• References

The following sections are included:

• Preliminary Remarks

• The Governing Equations

• Modal Analysis of Vibrations

• Direct Partition of Motions

• The Eigenvalue Problem for a Beam with the Resonant Parametric Stiffness Modulation

• Forced Vibrations. An Influence of Internal Damping

• Vibrations of a Beam with Parametrically Modulated Stiffness in Heavy Fluid Loading Conditions

• Discussion of the Parametric Stiffness Modulation

• A Modal Formulation of the Control Problem for a Sandwich Beam

• Analysis of Vibration Control for a Model Two-Degrees of Freedom Mechanical System
  • Analysis by the method of multiple scales
  • Analysis by the method of direct partition of motions
  • 'Standard' dynamical absorber

• Conclusions

• References

The following sections are included:

• Reinolds' Equation as an Equation of Vibrational Mechanic

• The Analog of Bernoulli's Equation for the Flows, Subjected to Vibration

• References

The following sections are included:

• On Phenomenon under Consideration

• Common Expression for the Gas or Fluid Discharge through a Hole in Vibrating Vessel

• On the Theory of Vibro-jet Effect

• On the Theory of the Vibrational Injection of Gas into the Fluid

• Results of the Experiments

• References

The following sections are included:

• Introduction. Classification of Systems with Fast Excitation. Weakly Excited Systems

• Systems with Strong Excitation. General Analysis

• Systems with Very Strong Excitation in a Special Case of Fast Oscillating Inertial Coefficients

• Two Mathematical Examples of Systems with Strong Excitation

• Response of a One Degree of Freedom System to Strong and Very Strong, High Frequency External and Parametric Excitation

• Conclusions

• References

The following sections are included:

• Introduction. Types of Discontinuities in Oscillating Systems. Short Review of the Investigations

• Averaging of Constant Order Discontinuous Systems

• Averaging of Variable Order Discontinuous Systems

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

• Constant Order Discontinuous Averaging in Systems with Quasi-Elastic Collisions

• Resonant Motions in a Quasi-Elastic Colliding Oscillator Excited by an Inertial Source of Limited Power
  • The problem statement
  • Dimensionless formulation. Small parameters
  • Discontinuous variables transformation. Main resonance
  • First step of the hierarchical averaging. Stationary resonance solution
  • Second step of the hierarchical averaging. Attraction area of the resonant solution
  • Concluding remarks

• Variable Order Discontinuous Averaging in Systems with Inelastic Collisions

• Conclusions

• Concluding Remarks of the Editor of the Book

• References

The following sections are included:

• Index