Статья: Synchronization and sommerfeld effect as typical resonant patterns
Synchronization and Sommerfeld Effect as Typical Resonant Patterns
Kovriguine D.A.
Abstract
This paper presents results of theoretical studies inspired by the problem of reducing the noise and vibrations by using hydraulic absorbers as dampers to dissipate the energy of oscillations in railway electric equipments. The results of experimental trials over these problem and some theoretical calculations, discussed in the text, are demonstrated the ability to customize the damping properties of hydraulic absorbers to save an electric power and protect the equipment itself due to utilizing the synchronous modes of rotation of the rotors.
Key words: Synchronization; resonance, stability, rotor vibrations; dampers.
Introduction
The phenomenon of the phase synchronization, had being first physically described by Huygens, was intensively studied mathematically only since the mid 20-th century, in parallel with significant advances in electronics [1-4]. Fundamental results on the synchronization in terms of the qualitative theory of differential equations and bifurcation theory prove the resonance nature of this phenomenon [5, 6]. Now the application of this theory is widely used to solve pressing practical problems in a wide range of activities, from microelectronics to power supply [7-9]. Now the research interest in advanced fields of the synchronization theory is concentrated, apparently due to the rapid development of new technologies, on studying complex systems with chaotic dynamics, discrete objects and systems with time delay variables. However, in the traditional areas of human activity such as, for instance, energy and transport, there is also noticeable growth of attention in this phenomenon focused on the searching effective ways to save the energy and integrity of power units.
Progressive developments in the scientific researches are constantly improving and expanding in our understanding over the synchronization phenomenon, as a consistent coherent dynamic process. This one occurs usually due to very small, almost imperceptible bonds between the individual elements of the system, which, nevertheless, cause a qualitative change in the dynamical behavior of the object.
The basic equation of the theory of phase synchronization of
a pair of oscillators or rotators reads , where
is a small frequency (or angular velocity)
detuning,
is
the depth of the phase modulation,
is the time. This one being a very
simple equation has the general solution in the following form
,
where is an arbitrary constant of integration.
From this solution follows a simple stability criterion for the stable phase synchronization:
. It shows
that the phase mismatch must be small, or, accordingly, the parameter of modulation
must be sufficiently large, otherwise the synchronization may be destroyed.
A more detailed mathematical study of this problem, referred to a two-rotor system based on an elastic base, turns out that the reduced model is incomplete. Namely, one draws some surprising attention to that the model lacks any description of that element of the system which provides the coupling between the rotors. More detailed studies lead to the following structure of the refined model:
,
,
where describes a measure of the amplitude
of oscillations of the elastic foundation. This additional equation appears as a
result of the phase modulation of the angular velocity of rotors due to the elastic
vibrations of the base. So that, the perturbed rotors, in turn, cause the resonant
excitation of vibrations of the base, described by the first equation. In the study
of the refined model one can explain that the stable synchronization requires the
same condition:
. But, one more necessary condition
is required, namely, the coefficient of the resonant excitation of vibrations of
the base
should
not exceed the rate of energy dissipation
, i. e.
. The last restriction significantly
alters the stability region of the synchronization in the parameter space of the
system that will be demonstrated by some specific computational examples below.
The equations of motion
We consider the motion of two asynchronous drivers mounted on an elastic base. A mathematical model is presented by the following system of widely cited differential equations [10, 11]
;
(1) ;
,
where is the mass of the base, modeled as
a rigid body with one degree of freedom, characterized by a linear horizontal displacement
,
is the coefficient
of elasticity of the platform,
is the damping coefficient,
are the small masses
of eccentrics with the eccentricities
(radii of inertia),
are the moments
of inertia of rotors in the absence of imbalance,
stands for the driving moments,
denotes the resistance
moment of the rotor. There is installed the pair of asynchronous drivers (unbalanced
rotors) on the platform, whose rotation axes are perpendicular to the direction
of base oscillation. The angles of rotation of the rotor
are measured from the direction
of the axis
counter-clockwise.
Assume that the moment characteristics of each driver and torque resistance have
a simplest form, i. e.
,
. Here
are the constant parameters, respective
for the starting points,
and
stand for the drag coefficients of
the rotors. Respectively, the subscript “1” refers to the first driver, while “2” to the second one. If we assume this simple linear model of the moment of static
characteristics of the devices, the dimensionless form of eqs. (1) can be rewritten
such as follows:
;
(2) ;
,
where appears in the role of the-small parameter
of the problem. The parameters
and
are of order of unity such that
and
, where
and
. We introduce
new notations:
,
,
(
). Here
is the oscillation frequency of the
base in the absence of the devices,
is the dimensionless damping coefficient,
is the new
dimensionless linear coordinate measured in fractions of the radius of inertia of
the eccentrics. The set (2), in contrast to the original equations, depends now
on the dimensionless time
.
The problem (2) admits an effective study by the method of a
small parameter. In order to explore this one, we should transform the system (2)
to a standard form of the six equations resolved for the first derivatives. The
intermediate steps of this procedure are the follows ones. Firstly, we introduce
the new variables, ,
,
, associated with the initial dependent
variables by differential relations:
,
,
. Assume that
in the set (2). Then one
defines the transform to the new dependent variables based on the method of varied
constants:
,
,
,
,
, where
,
,
and
are the partial
angular velocities of devices. Here
,
,
,
,
,
are the six new variables of the problem.
The sense of these new variables:
,
are the amplitude and phase of base
oscillations, respectively,
,
are the angular accelerations and
,
are the angular
velocities of the rotors. The standard form suitable for further analysis is ready.
Because of large records this standard form is not given, but the interested reader
can trace in detail the stages of its derivation [12].
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.
(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.
|
|
|
|
|
|
|
|
|
|
|
|
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
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.
[4] Andronov A. A, Witt A. A. Collected Works. Moscow: USSR Academy of Sciences, 1930: 70-84.
[5] Arnold V.I. Geometrical methods in the theory of ordinary differential equations, Springer-Verlag, 1988: 372.
[6] Leonov G. A., Ponomarenko D. V., Smirnova V. B. Frequency-domain methods for nonlinear analysis (Proc.). Theory and applications. Singapore: World Sci., 1996: 498.
[7] Blekhman I.I. Vibrational Mechanics. Singapore: World Sci., 2000: 509.
[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).