# Synchronization and sommerfeld effect as typical resonant patterns

## Reducing the noise and vibrations by using hydraulic absorbers as dampers to dissipate the energy of oscillations in railway electric equipments. The phenomenon of the phase synchronization. Examples of stable and unstable regimes of synchronization.

 Рубрика Физика и энергетика Вид статья Язык английский Дата добавления 25.03.2011 Размер файла 153,4 K

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Synchronization and Sommerfeld Effect as Typical Resonant Patterns

Kovriguine D.A.

# Table. Parameters of stable and unstable regimes of synchronization.

1

0.1

1

1

0.5

0.5

1

1

0.751

0.75

-0.244

-0.204

2

0.1

1

1

0.5

0.5

1

1

0.251

0.25

-0.072

0.008

3

0.1

1

1

0.6

0.4

1

1

0.25

0.25

-0.075

-0.001

4

0.1

1

1

0.6

0.4

1

1

0.251

0.25

-0.075

0.009

5

0.1

1

1

0.6

0.4

1

1

1.25

1.25

0.239

-0.085

6

0.1

1

1

0.5

0.5

1

1

0.26

0.25

0.998

-0.007

The matching condition .

After substitution from the expressions (3) into the standard form of equations (2), separation of fast and slow motions within the first-order approximation in the small parameter , under the assumption that , one obtains the following evolutionary equations

; (9)

,

where

is the new slow variable (), is the small detuning. The coefficients of eqs. (9) are as it follows:

;

;

;

.

The resonance of this type, as already mentioned, has no practical significance. Let the detuning be zero, then these equations (9) are highly simplified up to the full their separation:

;

(10)

.

The formal criterion of stability is extremely simple. Namely, the coefficient of the resonant excitation of vibrations in the base exceeds no the rate of energy dissipation , i. e. , but the synchronization is awfully destroyed at any positive values of other parameters.

synchronization phase resonant pattern

# [14] Rumyantsev S. A., Azarov E. B. Study of transient dynamics vibrating and transporting machines using a mathematical model (J) Transport of Ural, 2005, 4 (7): 45-51 (in Russian).

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