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

:

;

;

(3)

;

;

;

.

Here, the kernel expansion depends upon the slow temporal scales

, 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

and when or if the both parameters are close to unity, . 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 , 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

, 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

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

(4)

.

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

and the phase . Amendments to the angular accelerations , and the velocities , 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.