Vibrational Mechanics

The following sections are included:

• Preface to the Russian Edition

• Preface to the English Edition

• Contents

The following sections are included:

• Oscillatory Processes in Nature and Engineering

• Damaging Vibration and Useful Vibration. Vibrational Engineering

• The Theory of Vibrational Processes and Devices — a New Section of the Applied Theory of Oscillations

• On the Effects Caused by the Action of Vibration in Non-linear Oscillatory Systems

• Vibrational Mechanics and Vibrational Rheology. Observer O and Observer V

• Vibrational Mechanics as a Section of the Mechanics of Systems with Hidden Motions

• Mistakes and Paradoxes, connected with the Interpretation of the Phenomena under Consideration. The Observer W

The following sections are included:

• General Statements and Main Equations; Theorem 1

• Special Cases, Examples
  • Mechanics of the Relative Motion
  • A Solid Body with the Internal (Hidden) Degrees of Freedom
  • Systems with Cyclic Coordinates
  • Motion of a Body in a Viscous Incompressible Fluid
  • Vibrational Mechanics — Systems with Hidden Fast Motions

The following sections are included:

• Vibrational Mechanics as the Mechanics of Systems with Hidden Motions

• Method of a Direct Separation of Motions as an Effective General Method of Solving Problems of Vibrational Mechanics
  • Preliminary Remarks
  • The Initial Equation and Its Reduction to a System of Integro-differential Equations for the Explicit and Hidden Components of Motions
  • The Case when a Separate Equation for the Explicit Component is Obtained: Theorem 2
  • The Main Assumption of Vibrational Mechanics, Its Formalization and Conditions of its Fulfillment; Theorem 3 and 4
  • 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, its Grounding; Theorem 5
  • An Important Special Case; Theorem 6
  • On the Simplifications of Solving Equations for the Fast Component of Motion. Purely Inertial Approximation
  • Additional Remarks, Certain Generalizations
  • Summary of the Main Relations and Results
  • On the Procedure of the Practical Use of the Method
  • Short Bibliographic Review

• 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 Similar Methods
  • Peculiarities and Limitations
  • Advantages
  • Final Remarks

• On Other Methods of Obtaining Expressions for the Vibrational Forces and the Main Equations of Vibrational Mechanics

The following sections are included:

• On Potentiality on the Average and on the Extremal Signs of Stability

• Systems with Synchronized Objects. Integral Signs of Stability (Extremal Properties) of Synchronous Motions
  • Systems with Almost Uniform Rotations, Synchronization of Vibroexciters; Theorems 7 and 8
  • Systems of Quasiconservative Objects, Canonical Systems; Theorem 9
  • Systems with Quasicyclic Coordinates; Theorem 10

• Systems with Kinematic Excitation of Vibration (the Minimax Sign of Stability); Theorem 11

• Systems with Dynamic Excitation of Vibration; Theorem 12

The following sections are included:

• Pendulum with a Vibrating Axis of Suspension
  • Brief Bibliographic Review
  • Equation of Motion
  • Solving the Problem by the Method of Direct Separation of Motions
  • The Use of the Mini-Max Criterion of Stability
  • Behavior of the Pendulum Depending on the Type of Vibration
  • Multilink Pendulums

• Chelomei's Pendulum with a Vibrating Axis of Suspension
  • Brief Bibliographical Review
  • Pendulum with an Absolutely Rigid Rod
  • The Behavior of the Washer on a Vibrating Deformable Rod

• The Follower-loaded Double Pendulum with a Vibrating Axis of Suspension

• Some Generalizations and Applications. On Vibrational Stabilization and Destabilization, Vibrational Shift (Drift)

The following sections are included:

• Unbalanced Rotor on a Vibrating Base — Vibrational Maintenance of Rotation
  • Brief Bibliography
  • Equation of Motion, the Main Equation of Vibrational Mechanics
  • Stationary Regimes of Rotation of the Rotor and Their Stability
  • Vibrational Maintenance of Rotation of the Rotor
  • Vibrational Capture of the Rotation of an Unbalanced Rotor of an Electric Engine. How the Efficiency of an Engine Can Be More than Unit

• Devices “A Roller (Ball) in a Vibrating Cavity” and “A Ring on a Vibrating Rod (Hula-Hoop)”. Vibrational Maintenance of the Planetary Motion
  • About the Devices under Consideration: a Brief Bibliographic Reference
  • Equation of Motion, the Main Equation of Vibrational Mechanics
  • The Discussion of Results, their Applications. The Critical Slot of Vibro-Inertial Crushing and Grinding Machines

• Unbalanced Rotor (Mechanical Unbalanced Vibro-Exciter) in the Oscillatory System — Vibrational Retardation of Rotation, Zommerfield's Effect
  • On the Effects under Consideration: Brief Bibliographic Reference
  • Simplest Case: the Oscillatory Part of the System is Linear and has One Degree of Freedom
  • General Case: the Oscillatory Part of the System is Non-linear and has Several Degrees of Freedom
  • On Certain Applications

• Machine Aggregates

The following sections are included:

• On the Phenomenon of Synchronization of Unbalanced Rotors (Mechanical Vibro-exciters). A Brief Review of Investigations

• The Simplest Case: Self-synchronization of Vibro-exciters in the Linear Oscillatory System with One Degree of Freedom
  • Equations of Motion and Setting up the Problem
  • Main Equations of Vibrational Mechanics
  • Analysis of the Main Equations. Vibrational Torques, Partial Angular Velocities; Vibrational coupling between the rotors
  • Stationary Regimes of Synchronous Rotation and Their Stability. The Integral Sign of Stability (Extreme Property) of Synchronous Motions
  • A Case of Two Exciters. Once Again about the Phenomenon of Vibrational Maintenance of Rotation
  • About the Case of Almost Identical Exciters

• The General Case. A Short Review of the Results of Investigations
  • Equations of Motion and the Main Equations of Vibrational Mechanics in the General Case
  • Potential Function and Integral Criterion of Stability
  • Short Review of the Results on the Theory of Synchronization of Mechanical Vibro-exciters

• Steadiness of Phasing the Vibro-exciters and the Adaptive Property of Vibrational Machines in case of Self-synchronization
  • On the Concept of the Steadiness of Phasing
  • Estimation of Steadiness. On the Relative Strength of Vibrational Coupling between the Unbalanced Rotors
  • The Adaptive Property of Vibrational Machines with Self-synchronized Vibro-exciters

• On Theoretic Investigation of the Devices with Self-synchronized Vibro-exciters. An Example of Using the Integral Criterion of Stability

• On the Synthesis of the Devices with Self-synchronized Vibroexciters

• Regularities and Paradoxes of Self-synchronization of Unbalanced Vibro-exciters

• Supplement: On the Phenomena of Synchronization of the Oscillatory and Rotational Motions in Nature and in Technology. The modern state of the problem. General Definition of synchronization

The following sections are included:

• Laval's Principle and its Generalization, Ensuing from the Theory of Self-synchronization of Mechanical Vibro-exciters and from Some Other Investigations

• Application to the Theory of Group Foundations for the Unbalanced Machines

• Supplement to the Theory of Auto-balance Devices

The following sections are included:

• On the Effect of Vibrational Displacement, its Theory and Applications

• Simplest Model of the Process of Vibrational Displacement
  • Motion of a Particle over a Rough Horizontal Plane under the Action of a Longitudinal Harmonic Force or the Longitudinal Vibration of the Plane
  • Analysis of the Solutions. Effect of the Seeming Vibrational Transformation of Dry Friction into Viscous. The Driving and the Vibro-transformed Vibrational Forces

• Physical Mechanisms and Main Types of Asymmetry of the System, Causing Vibrational Displacement

• More Complicated Main Models and Problems of the Theory of Vibrational Displacement
  • One-dimensional Motion of the Particle in the Resistant Medium under the Action of Vibration [95, 99, 198, 564, vol. 2 and 4]
  • The Problem in 9.4.1 in the Case of Harmonic Vibration
  • The Motion of a Particle over the Tilted Rough Plane, Performing Harmonic Oscillations in Two Mutually Perpendicular Directions
  • On Vibrational Transportation of a Body (Particle) Upwards along the Tilted Plane. The Limiting Angle of Rise
  • About Other Models and Problems

The following sections are included:

• Vibrational Transportation

• Vibrational Separation of the Components of Granular Mixtures
  • Factors, Determining the Efficiency of Using Vibration in the Processes of Separating the Components of Granular Mixtures
  • On Separation of Particles in a Layer of Granular Material under the Action of Vibration (Segregation, Stratification, Self-classification)
  • General Statement of the Problem on the Separation of Particles of the Granular Mixture in a Vibrating Vessel. Brief Characteristic of the State of the Problem
  • Motion of a Heavy Particle in the Medium with a Resistance of Dry Friction Type, Performing Circular Horizontal Oscillations. Pseudo-resonance Effect
  • Problem of 10.2.4 in the Case of Vertical Harmonic Oscillations of the Medium. The Effect of Floating up of a Heavy Particle in the Medium of Light Small Particles
  • Kinetics of Vibrational Separation of Multi-Component Mixture (the Continual Description)
  • Separation of Particles on the Vibrating Surfaces
  • On Separating the Particles in the Vibrational and Wave Fields, Created in the Rarefied Suspensions

• Vibrational Sinking and Intrusion, Vibrational Cutting

• Vibrational Transformation of Motion; Vibro-engines

• Vibrational Movement, Vibrational Coaches
  • Definitions. Preliminary Remarks
  • Vibrational Movement over a Rough Surface. Self-propelled Vibro-compactors of the Ground, Travel on Skateboards
  • Vibrational Movement in Fluid and in Gas. Vibro-flier, the movement of living organisms
  • Vibrational Movement in Heterogeneous Force Fields. Gravi-flier, Magneto-flier

• The Vibro-jet Effect, Vibrational Pumps; Some More about Vibrational Coaches
  • On Vibro-jet Effect, Vibrational Pumps
  • Lishansky's Vibrational Pump; The Simplest vibroflighter: An Approximate Theory
  • Generating the Flow of Fluid by Means of a Vibrating Plate with Holes of Changing Cross-sections
  • Five Principle of Generating Vibrational Flows and Forces
  • On Vibrational Convection
  • On the Main Equations of Vibrational Hydromechanics and Hydraulic

The following sections are included:

• On the Notion of Vibrational Shift (Drift)

• Effect of Vibrational Shift in Applications; Special Features of the Effect in Systems with Dry Friction

The following sections are included:

• Rheology as a Section of Mechanics

• Definition of Vibrorheology. Macro- and Microvibrorheology

• Vibrorheological Equations, Vibrorheological Properties and Effective Vibrorheological Characteristics

The following sections are included:

• Effective Coefficients of Dry Friction under the Action of Vibration or Shock; Some Applications
  • Effective Frictional Coefficients of Rest. The simplest Model —an Absolutely Solid Body under a Harmonic Action
  • A More Complicated Model is a Solid Body with an Internal Degree of Freedom
  • Several Other Models; Shock Effect; Experiment of Tolstoy
  • Vibrational Conception of the Sliding Friction; Vibrational Control of Dry Friction
  • Application to the Theory and to Projecting Designs, Operating in Conditions of Impacts and Vibration
  • Possible Applications to Seismology and to the Theory of Explosive Effects

• Effective Friction when the System with a Positional-Viscous Resistance is acted upon by Vibration
  • Vibrational Transformation of the Characteristic of the Resistance of the Oscillatory System with One Degree of Freedom
  • Special Cases — the Asynchronous Suppression and Excitation of Self-oscillations; Some Applications

• Equation of Reinolds' as a Vibrorheological Equation. Effective Viscosity of Fluid in Turbulent Motion; Effect of the External Vibrational Action

• On Other Cases of Using Concepts of Effective Viscosity under Vibration

The following sections are included:

• Vibrorheology of Systems with Dry Friction

• Vibrorheology of Systems with Periodic Collisions
  • On Vibrorheological Simulation of Vibroshock Interactions by Forces of Viscous Friction
  • Applications to the Calculation of the Capacity of Crushers

The following sections are included:

• Vibrorheological Models of a Layer of Granular Medium

• Some Applications
  • The Process of Vibrobunkering of Granular Materials
  • Motion of a Layer of Granular Material in a Rectangular Tray Whose Bottom is Vibrating Heterogeneously in the Transverse Direction (to the Theory of Vibrational Screens with an Elastic Resonant Sieve)
  • Motion of the Loading in Vibrational Mills and Instruments for the Volume Vibrational Treatment of Machinery Parts
  • On Supplements to the Theory of Crushers

• On the Behavior of the Granular Material in the Communicating Vibrating Vessels

The following sections are included:

• Preliminary Remarks

• Penetration of Vibration into a Viscous Fluid

• On the Penetration of Vibration into Suspensions

• Penetration of Vibration into Granular Material
  • The Case of Circular Oscillations of a Horizontal Plate in its Plane
  • The Case of Rectilinear Longitudinal Oscillations of a Horizontal Plate
  • The Case of Transverse Oscillations of the Plate

• On Penetration of Vibration into Concrete Mixes

• On the Theory of Vibro-conduction

The following sections are included:

• Preliminary Remarks

• Structureless Suspensions — Solid Particles in a Viscous Fluid

• Structurized Suspensions — Particles in the Medium with the Resistance of the Type of Dry Friction

The following sections are included:

• On the Problem of Forming the Properties of Nonlinear Mechanical Systems by Means of Vibration

• On Dynamics Materials
  • The Idea and Definition of Dynamic Materials
  • Two kinds of Dynamic Materials
  • About Some Possibilities of the Implementation of Dynamic Materials
  • On Some Potentialities Provided by Using the Dynamic Materials
  • On Dynamic Surfaces

The following sections are included:

• Vibrorheological Effects in Macroscopic Homogeneous Media (Turbulent Viscosity, Vibrocreep, Vibrorelaxation, Vibroplasticity, Fatigue of Materials)

• Certain General Vibrorheological Regularities

The following sections are included:

• The simplest Case: Rectilinear Motion in a Field of a Harmonic Standing Wave

• Special Case: a Pendulum with a Vertically Vibrating Axis of Suspension

• Generalization of the Problem

The following sections are included:

• Preliminary Remarks

• Main Results of the General Theory of Synchronization and of the Theory of Potential on the Average Dynamic Systems as Applied to the Problem under Consideration
  • Description of the System. The Initial System and the Frame System
  • The Potential Function and the Integral Sign of Stability (Extreme Property) of Resonant Motions of the System of Bodies
  • On the Theoretic Explanation of the Natural Prevalence of Resonance in the Solar System
  • On the “Principle of the Least Interaction” by Ovenden, Feagin and Graff and on the Role of Dissipative Forces in the System

• The Case when the Orbits of Bodies Lie in the Same Plane or in Proximate Planes and Have Small Eccentricities. On classification of resonances

• The Case of Two Revolving Bodies: Comparison with the Results of the Direct Analytical Investigation and with the Observed Data

• On Resonances in the Solar System. Hypothesis about Simple Resonancity

• On Other Problems (Motion of Celestial Bodies with the Internal Degrees of Freedom)

The following sections are included:

• Bibliography

• Index