Blekhman I. Selected Topics in Vibrational Mechanics

World Scientific Publishing Co. Pte. Ltd., 2004. 438 p. — ISBN: 981-238-055-8 — (Series on Stability, Vibration and Control of Systems. Series A. Volume 11).Vibrational mechanics is a new, intensively developing section of nonlinear dynamics and of the theory of nonlinear oscillations. It presents a general approach to the study of the effects of vibration on nonlinear systems. This approach is characterized by simplicity of application and by physical clearness.ForewordThe Basis of Vibrational Mechanics
On Some Nonlinear Oscillatory Effects. The Main Idea of Vibrational Mechanics
Blekhman (Translated by M. Perelman)
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 Main Mathematical Apparatus of Vibrational Mechanics and of the Method of Direct Separation of Motions
Blekhman (Translated by M. Perelman)
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
On Other Methods of Obtaining Expressions for the Vibrational Forces and the Main Equations of Vibrational Mechanics
Blekhman (Translated by M. Perelman)
Well Known Methods
Two Other Methods
A Simplest Example: Solving the Problem about a Pendulum with a Vibrating Axis of Suspension by Different Methods of the Theory of Nonlinear Oscillations
Blekhman, D. A. Indeitsev
(Translated by M. Perelman)
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
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
Blekhman (Translated by M. Perelman)
Peculiarities and Limitations
Advantages
Final Remarks
References to PartPendulum and Pendulum Systems under High-Frequency Excitation — Non-Trivial Effects
Quasi-equilibrium Positions and Stationary Rotations of the Pendulums with a Periodically Vibrating Axis
Blekhman, H. Dresig, P. Rodionov
(Translated by M. Perelman)
Preliminary Remarks, Equation of Motion
Regimes of Quasi-Equilibrium
Regimes of RotationNon-Trivial Effects of High-Frequency Excitation for Pendulum Systems
S. Jensen, J. J. Thomsen, D. M. Tcherniak
Preliminary Remarks
Chelomei's Pendulum — Resolving a ParadoxHistory 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
ConclusionsOn the Theory of the Indian Magic Rope
Blekhman, H. Dresig, E. Shishkina
(Translated by M. Perelman)
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 RopeConjugate Resonances and Bifurcations of Pendulums under Biharmonical Excitation
Blekhman, P. S. Landa
(Translated by M. Perelman)
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 ResultsOn the Investigations of the Electromechanical Systems. On the Behavior of the Conductivity Bodies of Pendulum Types in High-Frequency Magnetic Fields
Blekhman (Translated by M. Perelman)
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 High-Current in Frequency Magnetic FieldProblems of the Theory of Sclfsynchronization
On General Definitions of Synchronization
Blekhman, A. L. Fradkov
Preliminary Remarks
Evolution of the Synchronization Concept
General Definition of Synchronization
Examples
DiscussionA Guide to Solving Certain Self-Synchronization Problems
L. Sperling, H. Duckstein
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
ExampleThe Setting Up of the Self-Synchronization Problem of the Dynamic Objects with Inner Degrees of Freedom and Methods of Its Solution
Blekhman, L. Sperling
(Translated by M. Perelman)
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 MotionsSystems 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 sectionOn 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
Blekhman, N. P. Yaroshevich
(Translated by M. Perelman)
Preliminary Remarks
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 BodyProblems of Creating Dynamic Materials
On Dynamic Materials
Blekhman (Translated by M. Perelman)
Briefly on the Idea of Dynamic Materials
On the Development of the Idea of Creating Dynamic MaterialsThe Active Control of Vibrations of Composite Beams by Parametric Stiffness Modulation
S. V. Grishina, O. A. Ershova, S. V. Sorokin
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
ConclusionsVibrational Hydrodynamics and Hydraulics
On Vibrational Hydrodynamics and Hydraulics
Blekhman (Translated by M. Perelman)
Reinolds' Equation as an Equation of Vibrational Mechanics
The Analog of Bernoulli's Equation for the Flows, Subjected to VibrationOn the Vibro-Jet Effect and on the Phenomena of Vibrational Injection of the Gas into Fluid
Blekhman, L. I. Blekhman., L. A. Vaisberg, V. B. Vasilkov, K. S. Yakimova
(Translated by M. Perelman)
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 ExperimentsSome Mathematical Supplements and Generalization
On Asymptotic Analysis of Systems with Fast Excitation
A. Fidlin
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
ConclusionsOn the Averaging of Discontinuous Systems
A. Fidlin
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 BookAppendix. Table of Contents of the Book Vibrational Mechanics {Nonlinear Dynamic Effects, General Approach, Applications) Iliya I. Blekhman, World Scientific,