Synchronization and sommerfeld effect as typical resonant patterns
Synchronization and Sommerfeld Effect as Typical Resonant Patterns
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
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
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
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]
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:
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 .
Solution of the system in a standard form is solved as transform
series in the small parameter :
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.
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:
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
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.
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:
Equations (5) represent a generalization of the standard basic
equations of the theory of phase synchronization , whose structure reads
Formally, this equation follows from the generalized model (5)
or (6), if we put . The equation
(7) has the general solution
an arbitrary constant of integration. This solution implies the criterion of the
stable phase synchronization:
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.
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
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:
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
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 . 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
The work was supported in part by the RFBR grant (project 09-02-97053-р
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