Synchronization and sommerfeld effect as typical resonant patterns

  Appleton E. V. The automatic synchronization of triode oscillator (J), Proc. Cambridge Phil. Soc., 1922, 21: 231-248. Van der Pol B.

Synchronization and sommerfeld effect as typical resonant patterns

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o 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.

 

(5)

,

where is the new slow variable (), denotes the small detuning of the partial angular velocities, . The coefficients of equations (5) are following:

 

;

;

;

.

 

Let the detuning be zero, then these equations are highly simplified up to the full their separation:

 

(6)

.

 

Equations (5) represent a generalization of the standard basic equations of the theory of phase synchronization [10], whose structure reads

 

.(7)

 

Formally, this equation follows from the generalized model (5) or (6), if we put . The equation (7) has the general solution

 

,

 

where is an arbitrary constant of integration. This solution implies the criterion of the stable phase synchronization:

 

(8) ,

 

which indicates that in the occurrence of the stable synchronization the phase detuning must be small enough, compared with the phase modulation parameter. If this condition is not satisfied, then the system can leave the zone of synchronization.

On the other hand the refined model (6) says that for the stable synchronization the performance of the above conditions (8) is not enough. It is also necessary condition that the coefficient of the resonant excitation of vibrations in the base should not exceed the rate of energy dissipation , i. e. . The last restriction significantly alters the stability zone of synchronization in the space system parameters that is demonstrated here on the specific computational examples.

 

Examples of stable and unstable regimes of synchronization

 

The table below shows the calculation of the different theoretical implementations of stable and unstable regimes of the phase synchronization. The example 1 (see the first line in the table) demonstrates a robust synchronization with a small mismatch between the angular velocities of drivers . The example 2 (see, respectively, the second line in the table, etc.) displays an unstable phase-synchronization regime at the same small difference between the angular velocities, i. e. . One can reach a stable steady-state synchronization pattern in this example by adding a damping element with the coefficient . The example number 3. This is a robust synchronization for the small differences in eccentrics () and equal angular velocities. The example number 4. This is an unstable synchronization mode with the same small differences in eccentrics () and small mismatch in angular velocities, i. e. . One can reach a stable regime in this example by adding a dissipative element with the damping coefficient. The example number 5. This is an unstable synchronization regime. One cannot reach any stable synchronization regime in this example, it is impossible, even when adding any damping element. The example number 6. This is an unstable regime of synchronization at different angular speeds. It is also impossible to achieve any sustainable sync mode in this case.

 

Table. Parameters of stable and unstable regimes of synchronization.

10.1110.50.5110.7510.75-0.244-0.20420.1110.50.5110.2510.25-0.0720.00830.1110.60.4110.250.25-0.075-0.00140.1110.60.4110.2510.25-0.0750.00950.1110.60.4111.251.250.239-0.08560.1110.50.5110.260.250.998-0.007

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

Conclusions

 

Synchronous rotations of drivers are almost idle and required no any high-powered energy set in this dynamical mode. Most responsible treatment for the drivers is their start, i. e. a transition from the rest to steady-state rotations [14]. So that, the utilizing vibration absorbers for high-powered electromechanical systems has advantageous for the two main reasons. On the one hand it provides a control tool for substantially mitigating the effects of transient shocking loads during the time of growth the acceleration of drivers. This contributes to integrities of the electromechanical system and save energy. On the other hand there is an ability to configure the appropriate damping properties of vibration absorbers to create a stable regime of synchronization when it is profitable, or even get rid of him, to destroy the synchronous movement, creating conditions for a dynamic interchange of drivers.

Acknowledgments

The work was supported in part by the RFBR grant (project 09-02-97053-р поволжье).

References

 

  1. Appleton E. V. The automatic synchronization of triode oscillator (J), Proc. Cambridge Phil. Soc., 1922, 21: 231-248.
  2. Van der Pol B. Forced Oscillations in a circuit with non-linear resistance (J), Phil. Mag., 1927, 3: 64-80.
  3. Andronov A. A, Witt A. A. By the mathematical theory of capture (J), Zhurn. Math. Physics., 1930, 7 (4): 3-20.
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  5. Arnold V.I. Geometrical methods in the theory of ordinary differential equations, Springer-Verlag, 1988: 372.
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  8. Blekhman I.I. Synchronization in Science and Technology, NY: ASME Press, 1988: 435.
  9. Blekhman I.I., Landa P. S., Rosenblum M. G. Synchronization and chaotization in interacting dynamical systems (J), Appl. Mech. Rev., 1995, 11 (1): 733-752.
  10. Samantaray A. K., Dasguptaa S. S. and R. Bhattacharyyaa. Sommerfeld effect in rotationally symmetric planar dynamical systems (J), Int. J. Eng. Sci., 2010, 48 (1): 21-36.
  11. Masayoshi Tsuchidaa, Karen de Lolo Guilhermeb and Jose Manoel Balthazarb. On chaotic vibrations of a non-ideal system with two degrees of freedom: Resonance and Sommerfeld effect (J), J. Sound and Vibration, 2005, 282 (3-5): 1201-1207.
  12. http://kovriguineda. ucoz.ru
  13. Haken H. Advanced Synergetics: Instability Hierarchies of Self-Organizing Systems and Devices. New York: Springer-Verlag: 1993: 465.
  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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