By A. A. Andronov, A. A. Vitt and S. E. Khaikin (Auth.)
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Extra info for Theory of Oscillators
Example text
The conditional period a n d the logarithmic decrement of the oscillations are determined by the properties of the system; the phase a n d amplitude, however, r e m a i n arbitrary a n d are determined by the initial conditions. 2. e. t o constructing its " p o r t r a i t " , representing the totality of all possible m o t i o n s . 6), we can find the equation of the family of phase p a t h s . 26) We shall show t h a t this is a family of spirals having an asymptotic point at the origin of t h e coordinates.
The region III of Fig. 25 corresponds to initial values leading to such a type of m o t i o n . The relation between the character of the m o t i o n a n d the initial conditions can be represented graphically in another form also, by showing the dependence of the m o t i o n u p o n time for all three cases /*, 77 and III. This has been done in Fig. 26 where it is assumed t h a t in all cases the initial displacement x 0 > 0 . 5. Representation of an aperiodic process on the phase plane Let us pass n o w t o the investigation of the phase p a t h s o n the phase plane x, y (y=x).
The region III of Fig. 25 corresponds to initial values leading to such a type of m o t i o n . The relation between the character of the m o t i o n a n d the initial conditions can be represented graphically in another form also, by showing the dependence of the m o t i o n u p o n time for all three cases /*, 77 and III. This has been done in Fig. 26 where it is assumed t h a t in all cases the initial displacement x 0 > 0 . 5. Representation of an aperiodic process on the phase plane Let us pass n o w t o the investigation of the phase p a t h s o n the phase plane x, y (y=x).