Free oscillations in conservative and dissipative systems with one degree of freedom. Representation of oscillations on a phase plane

The lecture purpose is qualitative analysis of the elementary nonlinear oscillatory systems with use of a method of a phase plane and revealing of the basic distinctive signs of nonlinear systems.

Free oscillations – the oscillationsmade only for the account of a stock of energy in power-intensive elements of oscillatory system.

As number of a degree of freedom - is called the quantity of independent variables necessary for a complete description all movements in system.

Conservative system - idealised system in which we neglect all possible aspects of an energy loss from system. In such system the full stock of energy remains to constants at any moment. Observed idealisation is useful, as allows to analyse important properties of oscillatory systems simply enough. The differential equation presenting oscillations in nonlinear conservative system:

= f (x),

Where x - displacement in mechanical system, in electric system it can be, for example, a charge or a current.

In nonlinear system period of oscillations depends on an amplitude of oscillation (an energy stock) and from system parametres. Such oscillations are called anisochronous.

Period of oscillations in linear conservative system is defined only by parametres of system and does not depend on an amplitude of oscillation (an energy store in system). Linear conservative system - system isochronous.

The dynamic condition of system is defined, if to us two magnitudes defining energy of system are known. For example, displacement х and speed for mechanical system, a charge q and a current for electric system. Through х and y (q and ) it is possible to express power relationships in oscillatory system.

The general equation of oscillations in nonlinear system with one degree of freedom in which there is a friction:

.

Leading-out of an analytical form for a traffic phase portrait in conservative and dissipative systems in neighbourhoods of singular points, the analysis of stability of system near to singular points and their classification are the purpose of the given lecture.

The equation of oscillations in nonlinear dissipative system with one degree of freedom looks like:

. (1)

To it there matches the equation presenting images on a phase plane:

. (2)

Let's observe it on purpose to find out properties of singular points and to install features of behavior of phase paths in their neighborhood.

Singular points we find, assuming at y = 0 and solving the equation f (x, 0) =0.

  Fig. 1. Phase portraits of an oscillatory damped system in a singular point neighborhood at : q <0,) q> 0.

a

  Fig. 2. Phase portraits of systems with a restoring force with positive (a) and negative (b) a friction.

a

b

Fig. 3 Aperiodic processes in a system with positive a friction (a). Phase portrait of system with repulsive force (b).

Lecture 4

Дата добавления: 2014-01-11