Synchronization and effect of Zommerfelda as typical resonant samples (стр. 2 из 3)

In the case when the system is far from resonance, i.e.

, eqs. (6) can easily be solved using the Poincaré perturbation method applied to the small non-resonant terms in order . However, in the resonant case, as , the first-order nonlinear approximation solution should contain the so-called secular terms appearing due to the known problems of small denominators. To overcome such a problem one usually applies the following trick. As soon as and the quantities and are changing rapidly, with approximately the same rate, it is natural to introduce a new generalized slow phase , where is a small variation of the angular velocity. Then after the averaging over the fast variable , one obtains the equations for the slow variables only, which are free of secularity. Such equations are called the evolution equations or truncated ones. In the case of set (6) the truncated equations hold true:

where

is the small frequency detuning, is the new generalized phase. Note that for the problem of averaging over the fast variable is enough to write .

Stationary oscillations in the absence of energy dissipation

Now the usual condition of a steady motion, i.e.

, is applied. We are looking now for the stationary oscillatory regimes in vacuo, i.e. . The solution corresponding to these regimes reads

This solution describes a typical resonant curve at

. The plus sign in front of the unit is selected when , otherwise .

The next stage of the study is to test the stability properties of stationary solutions. To solve this problem, one should obtain the equations in perturbations. The procedure for deriving these equations is that, firstly, one performs the following change of variables

where

is the steady-state amplitude of oscillations, then after replacing the variables the perturbation equations get the following form

To solve the stability problem evoking the Lyapunov criterion we formulate the eigenvalue problem defined by the following cubic polynomial, implicitly presented by determinant of the third order

Now we can apply one of the most widely known criteria, for example, the Hurwitz criterion, for the study the stability properties in the space of system parameters. The result is that the descending branch of the resonant curve, when

, cannot be practically observed because of the volatility associated with the fact that the driver is of limited power. This cannot maintain the given stationary oscillation of the elastic base near the resonance. This result corresponds to the well-known paradigm associated with the so-called Sommerfeld effect.

Formally, there are stable stationary regains, when

. However, this range of angular velocity is far beyond the accuracy of the first-order nonlinear approximation.

Damped stationary oscillations

A small surprise is that the response of the electromechanical system (2) has a significant change in the presence of even very small energy dissipation. Depending on the parameters of the set (2) the small damping can lead to typical hysteretic oscillatory patterns when scanning the detuning parameter

. While let the dissipation be sufficiently large, then a very simple stable steady-state motions, inherent in almost linear systems, holds true.

From the stationary condition, one looks for the stationary oscillation regimes

, and , as . The equations corresponding to these regimes are the following ones

;

;

.

For a small damping the solution of these equations describes a typical non-unique dependence between the frequency and amplitude, i.e.

, defined parametrically upon the phase . Near the resonance ( ), at some given specific parameters of the problem, say, , , and , the picture of this curve is shown in Fig. 1. Accordingly, the dependence of the angular velocity is presented in Fig. 2.

Fig. 1. The frequency-amplitude dependence

near the resonance at (arbitrary units).

Fig. 2. The angular velocity

changes (arbitrary units).

To study the stability problem of stationary solutions to the perturbed equations we should formulate the eigenvalue problem. This leads to the following characteristic cubic polynomial

with the coefficients[1]

;

;

;

.

Now one traces the stability properties by finding the areas of system parameters by applying the Routh-Hurwitz criterion, which states the necessary and sufficient conditions of positivity of the following numbers

, , , . These conditions are violated along the frequency-amplitude curve when scanning the parameter between the points Aand C. The characteristic points Aand B originate from the traditional condition that the derivative of function approaches infinity. The point C appears due to the multiple and zero valued roots of the characteristic equation , as the determinants in the Routh-Hurwitz criterion approach zero, more precisely, . At the direct scanning of the parameter together with increasing the angular velocity of the driver, one can observe a “tightening” of oscillations up to the point A. Then, the upper branch of the resonant curve becomes unstable and the stationary oscillations jump at the lower stable branch. At the reverse scan the angular velocity of the driver at the point C, in turn, there is a loss of stability of stationary oscillations at the lower branch and the jumping to stable oscillations with the greater amplitude at the upper branch of the resonance curve. The point B, apparently, is physically unrealizable mode of oscillations.