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Synchronization and sommerfeld effect as typical resonant patterns (стр. 2 из 4)

Solution of the system in a standard form is solved as transform series in the small parameter

Synchronization and sommerfeld effect as typical resonant patterns:

Synchronization and sommerfeld effect as typical resonant patterns;

Synchronization and sommerfeld effect as typical resonant patterns;

(3)

Synchronization and sommerfeld effect as typical resonant patterns;

Synchronization and sommerfeld effect as typical resonant patterns;

Synchronization and sommerfeld effect as typical resonant patterns;

Synchronization and sommerfeld effect as typical resonant patterns.

Here, the kernel expansion depends upon the slow temporal scales

Synchronization and sommerfeld effect as typical resonant patterns, which characterize the evolution of resonant processes. The variables with superscripts denote small rapidly oscillating correction to the basic evolutionary solution.

Then it is necessary to identify the resonant conditions in the standard form. The resonance in the system (2) occurs within the first-order nonlinear approximation theory, when

Synchronization and sommerfeld effect as typical resonant patterns and when
Synchronization and sommerfeld effect as typical resonant patterns or if the both parameters are close to unity,
Synchronization and sommerfeld effect as typical resonant patterns. All these cases require a separate study. Now we are interested in the phenomenon of the phase synchronization in the system (2). This case, in particular, is realized when
Synchronization and sommerfeld effect as typical resonant patterns, though the both partial angular velocities should be sufficiently far and less than unity, in order to overcome the instability predicted by the Sommefeld effect, since the first-order approximation resonance is absent in the system (2) in this case. Such a kind of resonance is manifested in the second approximation only.

In addition to the resonance associated with the standard phase synchronization in the system (2) there is one more resonance, when

Synchronization and sommerfeld effect as typical resonant patterns, which apparently has no practical significance, since its angular velocities fall in the zone of instability.

Note that other resonances in the system (2) are absent within the second-order nonlinear approximation theory. The next section investigates these cases are in detail.

Synchronization

After the substitution the expressions (3) into the standard form of equations and the separation between fast and slow motions within the first order approximation theory in the small parameter

Synchronization and sommerfeld effect as typical resonant patterns one obtains the following information on the solution of the system. In the first approximation theory, the slow steady-state motions (when
Synchronization and sommerfeld effect as typical resonant patterns) are the same as in the linearised set, i. e.
Synchronization and sommerfeld effect as typical resonant patterns,
Synchronization and sommerfeld effect as typical resonant patterns;
Synchronization and sommerfeld effect as typical resonant patterns,
Synchronization and sommerfeld effect as typical resonant patterns;
Synchronization and sommerfeld effect as typical resonant patterns;
Synchronization and sommerfeld effect as typical resonant patterns. This means that the slowly varying generalized coordinates
Synchronization and sommerfeld effect as typical resonant patterns,
Synchronization and sommerfeld effect as typical resonant patterns,
Synchronization and sommerfeld effect as typical resonant patterns and
Synchronization and sommerfeld effect as typical resonant patterns,
Synchronization and sommerfeld effect as typical resonant patterns и
Synchronization and sommerfeld effect as typical resonant patterns do not depend within the first approximation analysis upon the physical time
Synchronization and sommerfeld effect as typical resonant patterns nor the slow time
Synchronization and sommerfeld effect as typical resonant patterns. Solutions to the small non-resonant corrections appear as it follows:

Synchronization and sommerfeld effect as typical resonant patterns

Synchronization and sommerfeld effect as typical resonant patterns(4)

Synchronization and sommerfeld effect as typical resonant patterns

Synchronization and sommerfeld effect as typical resonant patterns

Synchronization and sommerfeld effect as typical resonant patterns

Synchronization and sommerfeld effect as typical resonant patterns.

This solution describes a slightly perturbed motion of the base with the same frequencies as the angular velocities of rotors, that is manifested in the appearance of combination frequencies in the expression for the corrections to the amplitude

Synchronization and sommerfeld effect as typical resonant patterns and the phase
Synchronization and sommerfeld effect as typical resonant patterns. Amendments to the angular accelerations
Synchronization and sommerfeld effect as typical resonant patterns,
Synchronization and sommerfeld effect as typical resonant patterns and the velocities
Synchronization and sommerfeld effect as typical resonant patterns,
Synchronization and sommerfeld effect as typical resonant patterns also contain the similar small-amplitude combination harmonics at the difference and sum.

Now the solution of the first-order approximation is ready. This one has not suitable for describing the synchronization effect and call to continue further manipulations with the equations along the small-parameter method. Using the solution (4), after the substitution into eqs. (3), one obtains the desired equation of the second-order nonlinear approximation, describing the synchronization phenomenon of a pair of drivers on the elastic foundation. So that, after the second substitution of the modified representation (3) in the standard form and the separation of motions into slow and fast ones, we obtain the following evolution equations.