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2017 | Book

Physical Fundamentals of Oscillations

Frequency Analysis of Periodic Motion Stability

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About this book

The book introduces possibly the most compact, simple and physically understandable tool that can describe, explain, predict and design the widest set of phenomena in time-variant and nonlinear oscillations. The phenomena described include parametric resonances, combined resonances, instability of forced oscillations, synchronization, distributed parameter oscillation and flatter, parametric oscillation control, robustness of oscillations and many others. Although the realm of nonlinear oscillations is enormous, the book relies on the concept of minimum knowledge for maximum understanding. This unique tool is the method of stationarization, or one frequency approximation of parametric resonance problem analysis in linear time-variant dynamic systems. The book shows how this can explain periodic motion stability in stationary nonlinear dynamic systems, and reveals the link between the harmonic stationarization coefficients and describing functions. As such, the book speaks the language of control: transfer functions, frequency response, Nyquist plot, stability margins, etc. An understanding of the physics of stability loss is the basis for the design of new oscillation control methods for, several of which are presented in the book. These and all the other findings are illustrated by numerical examples, which can be easily reproduced by readers equipped with a basic simulation package like MATLAB with Simulink. The book offers a simple tool for all those travelling through the world of oscillations, helping them discover its hidden beauty. Researchers can use the method to uncover unknown aspects, and as a reference to compare it with other, for example, abstract mathematical means. Further, it provides engineers with a minimalistic but powerful instrument based on physically measurable variables to analyze and design oscillatory systems.

Table of Contents

Frontmatter

Amplitude–Phase–Frequency Characteristics of Linear Steady-State Systems

Frontmatter
Chapter 1. Continuous Systems
Abstract
Let us take the differential equation in its operator form as a base definition of the dynamic system.
Leonid Chechurin, Sergej Chechurin
Chapter 2. Discrete Systems
Abstract
The chapter deals with discrete-time processes and discrete modeling.
Leonid Chechurin, Sergej Chechurin
Chapter 3. Experimental and Numerical Evaluations of Frequency Response
Abstract
With exponential form of the amplitude–phase–frequency characteristic (1.​16), relationship (1.​12) takes the form
Leonid Chechurin, Sergej Chechurin

Parametric Oscillations of Linear Periodically Nonstationary Systems

Frontmatter
Chapter 4. The First Parametric Resonance
Abstract
Linear periodically nonstationary systems are described by linear differential equations with either one or several periodically variable coefficients. In contrast to the coordinates, that describe the motion of the system, the periodically variable coefficients are called periodically variable parameters.
Leonid Chechurin, Sergej Chechurin
Chapter 5. Parametric Resonances of the Second and Higher Orders
Abstract
As before, let a T-periodic variable parameter be described as \(a(t) = a_{0} + a\sin\Omega (t - \tau )\) and its input signal takes the form \(x(t) = x_{0} + \tilde{x}(t) = x_{0} + A\sin\Omega t,\) here x0 and \(\tilde{x}(t)\) are the constant and variable components of the same frequency Ω, correspondingly.
Leonid Chechurin, Sergej Chechurin
Chapter 6. Higher Order Parametric Systems
Abstract
As it follows from Chap. 4, the single-frequency parametric oscillation excitation depends rather on the natural frequencies and the resonance pick values of the frequency response than on the order of the mathematical model of the system. Let us consider a system with two natural frequencies.
Leonid Chechurin, Sergej Chechurin

Parametric Resonance in Nonlinear System Oscillations

Frontmatter
Chapter 7. Nonlinear System Oscillations: Harmonic Linearization Method
Abstract
The method of describing functions is mainly used to analyze nonlinear dynamic systems in a single-frequency harmonic approximation. The method is also known as the method of harmonic linearization or harmonic balance method or the first harmonic method. The method directly arises from spectral and asymptotic methods as their first approximation. For the first time, the method was applied in 1934 by the Soviet scientist V. Kotelnikov to evaluate the performance of self-excited oscillation generators.
Leonid Chechurin, Sergej Chechurin
Chapter 8. Nonlinear System Oscillation Stability
Abstract
One of the obvious effects of motion stability loss is known as jump resonance which occurs under certain situations in nonlinear dynamic systems.
Leonid Chechurin, Sergej Chechurin
Chapter 9. Synchronization of Oscillations
Abstract
The synchronization of oscillatory systems is widely used in technology.
Leonid Chechurin, Sergej Chechurin

Parametric Oscillation Control

Frontmatter
Chapter 10. Parametric Damping and Exciting of Oscillations
Abstract
Let us consider an oscillatory system, comprising the gravitational pendulum on an elastic suspension (spring). Thus, the vertical motion of the mass happens when the spring is compressed or elongated. Since a parameter (length) of the gravitational pendulum sways periodically varies, we can sometimes observe the parametric oscillations of this angular coordinate.
Leonid Chechurin, Sergej Chechurin
Chapter 11. Parametric Regulators
Abstract
A swinging person has been known to be able to control sway oscillations by the periodic variation of system parameter that is the effective pendulum length. It follows from Sect. 10.3 that the operator can vary not only the parameter variation amplitude but also the variation phase; thereby, the excitation and damping of the parametric swing oscillations are affected. This is the manual parametric control.
Leonid Chechurin, Sergej Chechurin

Applications

Frontmatter
Chapter 12. Mechanical Problems
Abstract
The vertical oscillations of a horizontal dead-end elastic beam of rectangular but substantially non-square are considered. Assuming the beam width is much greater than its thickness, horizontal stiffness of the beam is taken to be infinitely large.
Leonid Chechurin, Sergej Chechurin
Chapter 13. Parametric Resonance in Fluid Dynamics
Abstract
In numerous complex problems of fluid dynamics, either the motion of a medium (liquid, gas, and air) in which an object is situated or the motion of an object (aircraft, a rocket, and a ship) inside the medium as well as medium-object interference are considered. As a rule, those problems cannot be analytically solved. They are addressed numerically and by physical modeling of the medium and the object.
Leonid Chechurin, Sergej Chechurin
Chapter 14. Electro and Radio Engineering Problems
Abstract
The excitation of parametric resonance oscillations are known to be feasible in linear electric circuits with periodically variable parameters. This phenomenon is widely used in radio engineering for generators and parametric amplifiers design.
Leonid Chechurin, Sergej Chechurin
Chapter 15. Economics
Abstract
The progress logic of mathematic modeling in economics was similar to any other field of engineering, e.g., mechanics. The was the evolution from simplest models to complex ones, from static relations to dynamic, from scalar problems to multidimensional, from linear problems to nonlinear, etc. However, the progress in classical mechanics was inseparably linked with the progress in mathematics and vice versa, they started applying well-developed by that time mathematical formalism to economy in the second half of the 20th century only. In many aspects, the difficulties of modeling of economics as a nature object are explained. Up to now, there is no such set of universally recognized laws which could be widely applicable, for example, Hook’s law or Coulomb’s law.
Leonid Chechurin, Sergej Chechurin

Multifrequency Oscillations, Stability and Robustness

Frontmatter
Chapter 16. Single-Frequency Approximation Correction of Periodically Nonstationary Systems
Abstract
The direct solution of the problem of multifrequency process analysis meets enormous difficulties. Indeed, the simplest calculation, for example, of a two-frequency process in a dynamic single-frequency parameter system has three frequencies, three amplitudes, and three phases as unknown quantities. Finding a solution in the nine-dimensional space of unknowns concerning a system with many state coordinates is not a simple problem even using modern computer power.
Leonid Chechurin, Sergej Chechurin
Chapter 17. Single-Frequency Approximation Correction of Nonlinear Systems
Abstract
This chapter provides the multiple-frequency analysis of the nonlinear dynamic system oscillations. We are interested in the influence of the higher harmonics on the evaluations given in Chap. 3 by the first harmonic approximation.
Leonid Chechurin, Sergej Chechurin
Chapter 18. Robust Dynamic Systems
Abstract
Problem background. There is roughly one and half century history of investigations of sensitivity with respect to different kinds of uncertainties in the description of the system or external disturbances.
Leonid Chechurin, Sergej Chechurin
Backmatter
Metadata
Title
Physical Fundamentals of Oscillations
Authors
Leonid Chechurin
Sergej Chechurin
Copyright Year
2017
Electronic ISBN
978-3-319-75154-2
Print ISBN
978-3-319-75153-5
DOI
https://doi.org/10.1007/978-3-319-75154-2