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Математика Introduction in Auto-oscillation systems. Basic Terminology
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Auto-oscillation isa non-damped oscillation in a non-linear dynamical system, whose amplitude and frequency can remain constant during a long period of time, are largely independent of the initial conditions, and are determined by the properties of the system itself. The term "auto-oscillation" was introduced by A.A. Andronov.

Dynamical systems capable of performing oscillations are known as auto-oscillating systems. They include clocks, generators of electric vibrations, electric bells, blow and bow musical instruments, etc. Under certain conditions auto-oscillations can also be produced in dynamical systems which normally function without auto-oscillations (e.g. the flutter of an aircraft wing, or auto-oscillations in automatic control and monitoring systems). The simplest auto-oscillating system can be represented as consisting of a constant source of energy, a mechanism regulating the supply of energy to the oscillating system, and the oscillating system itself. An essential feature of such a system is its feedback nature: on the one hand, the regulating mechanism controls the motion of the oscillating system but, on the other, it is the motion of the oscillating system that influences the operation of the regulating mechanism. From the mathematical point of view, autonomous auto-oscillating systems with one degree of freedom may be defined as systems whose equations of motion have one or more limit cycles in the phase plane.

An important typical property of auto-oscillations is that their amplitude is largely independent of the initial conditions, i.e. the existence of one or more ranges of initial conditions such that to any initial set of conditions within any one of these ranges there corresponds the same auto-oscillation amplitude. This means that oscillating systems may contain several stationary processes with different amplitudes; each one of these becomes realized in the system, depending on a choice of a region of initial conditions. This property constitutes the basic difference between periodic motion in an auto-oscillating system and periodic motion in a conservative system. Another typical property of auto-oscillating systems is the fact that the periods of the auto-oscillations are determined by the properties of the system itself and are not imposed from outside. This is the basic difference between auto-oscillations and forced oscillations.

There are many types of oscillators, and many different circuit configurations that

produce oscillations. Some oscillators produce sinusoidal signals, others produce nonsinusoidal signals. Nonsinusoidal oscillators, such as pulse and ramp (or saw-tooth) oscillators, find use in timing and control applications. Pulse oscillators are commonly found in digital-systems clocks, and ramp oscillators are found in the horizontal sweep circuit of oscilloscopes and television sets. Sinusoidal oscillators are used in many applications, for example, in consumer electronic equipment (such as radios, TVs, and VCRs), in test equipment (such as network analyzers and signal generators), and in wireless systems.

At studying of the most different areas of physics – from mechanics to a subatomic physics and astrophysics, anyhow it is necessary to face oscillatory processes. Though their nature and developing process forms are specific to each area of the physics, many regularity, used concepts, methods of the description of such processes appear the general. In it the unity of a physical picture of the world is very distinctly manifested. Such combination of unity and variety does physics of oscillations by the universal tool in studying of an extensive circle of the physical phenomena.

Oscillations are therestricted repeating traffics about some average position.

The theory of oscillations-represents the general consideration of oscillatory processes in various systems by the nature. Initial step to studying of any problem is trying to classify the investigated phenomena. Oscillations can be classified to various signs. For example, classifying oscillations to the kinematic signs,we compare them under the form and the period.

Classifying traffics on periodicity, we see that the range of periodic motions in the nature is rather wide: it can be begun with Т=1016 s (a cycle time of planets of the Sun round the Galactic centre), and to finish values Т~10-10-20 s from periods of oscillation X-ray and gamma radiations. In the interval between these conditional boundary lines values of the periods of other oscillations of the various physical nature known to us lie: acoustical, a microwave range, thermo radiation, a visible part of a spectrum of light oscillations, etc. However, classifying oscillations on the basis of periodicity, we notice that in the nature there is no strict periodicity; oscillations fluctuate and fade. Thus, injected representation about strict periodicity of movements is enough crude approximation to that is actually. In this connection induct concept about almost periodic motions.

Classifying oscillatory systems to dynamic signs, we discriminate:

Passive systems – in these systems of oscillation are made under the influence of superposed forces;

Active systems - in these systems of oscillation are made for the account of internal energy sources

Lecture 2