Synchronization and Sommerfeld Effect as Typical Resonant Patterns
Kovriguine D.A.
Abstract
This paper presents results of theoretical studies inspired bythe problem of reducing the noise and vibrations by using hydraulic absorbers asdampers 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 propertiesof hydraulic absorbers to save an electric power and protect the equipment itselfdue to utilizing the synchronous modes of rotation of the rotors.
Key words: Synchronization; resonance, stability, rotorvibrations; dampers.
Introduction
The phenomenon of the phase synchronization, had being firstphysically described by Huygens, was intensively studied mathematically only sincethe mid 20-th century, in parallel with significant advances in electronics [1-4].Fundamental results on the synchronization in terms of the qualitative theory ofdifferential equations and bifurcation theory prove the resonance nature of thisphenomenon [5, 6]. Now the application of this theory is widely used to solve pressingpractical problems in a wide range of activities, from microelectronics to powersupply [7-9]. Now the research interest in advanced fields of the synchronizationtheory is concentrated, apparently due to the rapid development of new technologies,on studying complex systems with chaotic dynamics, discrete objects and systemswith time delay variables. However, in the traditional areas of human activity suchas, for instance, energy and transport, there is also noticeable growth of attentionin this phenomenon focused on the searching effective ways to save the energy andintegrity of power units.
Progressive developments in the scientific researches are constantlyimproving 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 ofa pair of oscillators or rotators reads />, where /> is a small frequency (or angular velocity)detuning, /> isthe depth of the phase modulation, /> is the time. This one being a verysimple 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 showsthat the phase mismatch must be small, or, accordingly, the parameter of modulationmust be sufficiently large, otherwise the synchronization may be destroyed.
A more detailed mathematical study of this problem, referredto a two-rotor system based on an elastic base, turns out that the reduced modelis incomplete. Namely, one draws some surprising attention to that the model lacksany description of that element of the system which provides the coupling betweenthe rotors. More detailed studies lead to the following structure of the refinedmodel:
/>, />,
where /> describes a measure of the amplitudeof oscillations of the elastic foundation. This additional equation appears as aresult of the phase modulation of the angular velocity of rotors due to the elasticvibrations of the base. So that, the perturbed rotors, in turn, cause the resonantexcitation of vibrations of the base, described by the first equation. In the studyof the refined model one can explain that the stable synchronization requires thesame condition: />. But, one more necessary conditionis required, namely, the coefficient of the resonant excitation of vibrations ofthe base /> shouldnot exceed the rate of energy dissipation />, i. e. />. The last restriction significantlyalters the stability region of the synchronization in the parameter space of thesystem that will be demonstrated by some specific computational examples below.
The equations of motion
We consider the motion of two asynchronous drivers mounted onan elastic base. A mathematical model is presented by the following system of widelycited differential equations [10, 11]
/>;
(1) />;
/>,
where /> is the mass of the base, modeled asa rigid body with one degree of freedom, characterized by a linear horizontal displacement/>, /> is the coefficientof elasticity of the platform, /> is the damping coefficient, /> are the small massesof eccentrics with the eccentricities /> (radii of inertia), /> are the momentsof inertia of rotors in the absence of imbalance, /> stands for the driving moments, /> denotes the resistancemoment of the rotor. There is installed the pair of asynchronous drivers (unbalancedrotors) on the platform, whose rotation axes are perpendicular to the directionof base oscillation. The angles of rotation of the rotor/>are measured from the directionof the axis /> counter-clockwise.Assume that the moment characteristics of each driver and torque resistance havea simplest form, i. e. />, />. Here/>are the constant parameters, respectivefor the starting points, /> and />stand for the drag coefficients ofthe 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 staticcharacteristics of the devices, the dimensionless form of eqs. (1) can be rewrittensuch as follows:
/>;
(2) />;
/>,
where /> appears in the role of the-small parameterof the problem. The parameters /> and /> are of order of unity such that /> and />, where /> and />. We introducenew notations: />, />, /> (/>). Here /> is the oscillation frequency of thebase in the absence of the devices, /> is the dimensionless damping coefficient,/> is the newdimensionless linear coordinate measured in fractions of the radius of inertia ofthe eccentrics. The set (2), in contrast to the original equations, depends nowon the dimensionless time/>.
The problem (2) admits an effective study by the method of asmall 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. Theintermediate steps of this procedure are the follows ones. Firstly, we introducethe new variables, />, />, />, associated with the initial dependentvariables by differential relations: />, />, />. Assume that /> in the set (2). Then onedefines the transform to the new dependent variables based on the method of variedconstants: />,/>, />, />, />, where />, />, /> and /> are the partialangular velocities of devices. Here />, />, />, />, />, /> are the six new variables of the problem.The sense of these new variables: />, /> are the amplitude and phase of baseoscillations, respectively, />, /> are the angular accelerations and/>, /> are the angularvelocities 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 readercan trace in detail the stages of its derivation [12].
Solution of the system in a standard form is solved as transformseries in the small parameter />:
/>;
/>;
(3) />;
/>;
/>;
/>.
Here, the kernel expansion depends upon the slow temporal scales/>, which characterizethe evolution of resonant processes. The variables with superscripts denote smallrapidly oscillating correction to the basic evolutionary solution.
Then it is necessary to identify the resonant conditions in thestandard form. The resonance in the system (2) occurs within the first-order nonlinearapproximation theory, when/>and when /> or if the both parameters are closeto unity, />.All these cases require a separate study. Now we are interested in the phenomenonof the phase synchronization in the system (2). This case, in particular, is realizedwhen/>, thoughthe both partial angular velocities should be sufficiently far and less than unity,in order to overcome the instability predicted by the Sommefeld effect, since thefirst-order approximation resonance is absent in the system (2) in this case. Sucha kind of resonance is manifested in the second approximation only.
In addition to the resonance associated with the standard phasesynchronization in the system (2) there is one more resonance, when />, which apparentlyhas no practical significance, since its angular velocities fall in the zone ofinstability.
Note that other resonances in the system (2) are absent withinthe second-order nonlinear approximation theory. The next section investigates thesecases are in detail.
Synchronization
After the substitution the expressions (3) into the standardform of equations and the separation between fast and slow motions within the firstorder approximation theory in the small parameter /> one obtains the following informationon the solution of the system. In the first approximation theory, the slow steady-statemotions (when/>) are the same as in the linearisedset, i. e. />,/>; />, />; />; />. This means thatthe slowly varying generalized coordinates />, />, /> and />, /> и /> do not depend within the first approximationanalysis upon the physical time /> nor the slow time />. Solutions to the smallnon-resonant corrections appear as it follows:
/>
/>(4)
/>
/>
/>
/>.
This solution describes a slightly perturbed motion of the basewith the same frequencies as the angular velocities of rotors, that is manifestedin the appearance of combination frequencies in the expression for the correctionsto the amplitude /> and the phase />. Amendments to the angularaccelerations />,/>and the velocities/>,/>also contain the similarsmall-amplitude combination harmonics at the difference and sum.
Now the solution of the first-order approximation is ready. Thisone has not suitable for describing the synchronization effect and call to continuefurther manipulations with the equations along the small-parameter method. Usingthe solution (4), after the substitution into eqs. (3), one obtains the desiredequation of the second-order nonlinear approximation, describing the synchronizationphenomenon of a pair of drivers on the elastic foundation. So that, after the secondsubstitution of the modified representation (3) in the standard form and the separationof motions into slow and fast ones, we obtain the following evolution equations.
/>
(5)
/>,
where /> is the new slow variable (/>), /> denotes the smalldetuning of the partial angular velocities, />. The coefficients of equations (5)are following:
/>;
/>;
/>;
/>.
Let the detuning be zero, then these equations are highly simplifiedup to the full their separation:
/>
(6)
/>.
Equations (5) represent a generalization of the standard basicequations 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 generalsolution
/>,
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 synchronizationthe 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 stablesynchronization the performance of the above conditions (8) is not enough. It isalso necessary condition that the coefficient of the resonant excitation of vibrationsin the base /> shouldnot exceed the rate of energy dissipation />, i. e. />. The last restriction significantlyalters the stability zone of synchronization in the space system parameters thatis demonstrated here on the specific computational examples.
Examples of stable and unstable regimes of synchronization
The table below shows the calculation of the different theoreticalimplementations of stable and unstable regimes of the phase synchronization. Theexample 1 (see the first line in the table) demonstrates a robust synchronizationwith 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 regimeat the same small difference between the angular velocities, i. e. />. One can reacha stable steady-state synchronization pattern in this example by adding a dampingelement with the coefficient />. The example number 3. This is a robustsynchronization for the small differences in eccentrics (/>) and equal angular velocities.The example number 4. This is an unstable synchronization mode with the same smalldifferences in eccentrics (/>) and small mismatch in angular velocities,i. e. />. Onecan reach a stable regime in this example by adding a dissipative element with thedamping coefficient/>. The example number 5. This is anunstable synchronization regime. One cannot reach any stable synchronization regimein this example, it is impossible, even when adding any damping element. The examplenumber 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 standardform of equations (2), separation of fast and slow motions within the first-orderapproximation in the small parameter />, under the assumption that />, one obtains thefollowing 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 practicalsignificance. Let the detuning be zero, then these equations (9) are highly simplifiedup 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 therate of energy dissipation />, i. e. />, but the synchronization is awfullydestroyed at any positive values of other parameters.
synchronization phase resonant pattern
Conclusions
Synchronous rotations of drivers are almost idle and requiredno any high-powered energy set in this dynamical mode. Most responsible treatmentfor the drivers is their start, i. e. a transition from the rest to steady-staterotations [14]. So that, the utilizing vibration absorbers for high-powered electromechanicalsystems has advantageous for the two main reasons. On the one hand it provides acontrol tool for substantially mitigating the effects of transient shocking loadsduring the time of growth the acceleration of drivers. This contributes to integritiesof the electromechanical system and save energy. On the other hand there is an abilityto configure the appropriate damping properties of vibration absorbers to createa stable regime of synchronization when it is profitable, or even get rid of him,to destroy the synchronous movement, creating conditions for a dynamic interchangeof drivers.
Acknowledgments
The work was supported in part by the RFBR grant (project 09-02-97053-рповолжье).
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[12] http://kovriguineda. ucoz.ru
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