Synchronization and Sommerfeld asTypical Resonant Patterns
Part I. Single Driver Example
Kovriguine D.A.
Abstract We analyze a classical problem of oscillations arising in an elastic base caused by rotor vibrations of anasynchronous driver near the critical angular velocity. The nonlinear coupling betweenoscillations of the elastic base and rotor takes place naturally due tounbalanced masses. This provides typical frequency-amplitude patterns, even letthe elastic properties of the beam be linear one. As the measure of energydissipation increases the effect of bifurcated oscillations can disappear. Thelatter circumstance indicates the efficiency of using vibration absorbers to eliminateor stabilize the dynamics of the electromechanical system.
KeyWords Sommerfeld effect, asynchronous device; Lyapunov criterion,Routh-Hurwitz criterion, stability.
stationaryoscillation resonance synchronization
Introduction
The phenomenon of bifurcated oscillationsof an elastic base, while scanning the angular velocity of an asynchronous driver,is referred to the well-known Sommerfeld effect [1-9]. Nowadays, this plays therole of one of classical representative examples of unstable oscillations inelectromechanical systems, even being the subject of student laboratory work inmany mechanical faculties. This effect is manifested in the fact that thedescending branch of resonant curve can not be experienced in practice. A physicalinterpretation is quite simple. The driver of limited power cannot maintain givenamplitude of stationary vibrations of the elastic base. Any detailedmeasurements can reveal that the oscillation frequency of the base is always somewhathigher than that predicted by linear theory. This implies a very reasonablephysical argument. With an increase of base vibrations, for example, thegeometric nonlinearity of the elastic base should brightly manifest itself, sothat this assuredly may lead to the so-called phenomenon of “pulling”oscillations. However, a more detailed mathematical study can demonstrate that the dynamicphenomena associated with the Sommerfeld effect are of more subtle nature. If oneinterprets this effect as a typical case of resonance in nonlinear systems,then one should come to a very transparent conclusion. The appearance of the frequency-amplitude characteristic naturally encounteredin nonlinear systems, say, when regarding the Düffing-type equations, doesnot necessarily have place due to the geometric nonlinearity of the elastic base.This dependence appears as a result of nonlinear resonant coupling between oscillationsof the elastic base and rotor vibrations, even when the elastic properties beingabsolutely linear one. The latter circumstance may attract an interest in sucha remarkable phenomenon, as the effect of Sommerfeld, which is focused in thepresent paper.
The equations of motion
The equations describing a rotorrolling on an elastic base read [1-6]
/>;(1)
/>,
where /> isthe mass of a base withone degree of freedom, characterized by the linear displacement />, /> isthe elasticity coefficientof the base, /> isthe damping coefficient,/> standsfor the mass of aneccentric, /> denotesthe radius of inertia ofthis eccentric, /> isthe moment of inertia ofthe rotor in the absence of imbalance, /> isthe driving moment, /> describesthe torque resistance ofthe rotor. The single device (unbalanced rotor) set on the platform, while therotation axis is perpendicular to the direction of oscillation. The angle ofrotation of the rotor/>is measuredcounter-clockwise. Assume that the moment characteristics and the engine dragtorque are modeled by the simple functions /> and />, where/>is the starting point, /> isthe coefficientcharacterizing the angular velocity of the rotor, i.e. />, /> isthe resistance coefficient. Then the equationsof motion are rewritten as
/>
After introducing the dimensionlessvariables the basic equations hold true:
/>
where /> isthe-small parameter, />, />, />. Here/>stands for theoscillation frequency of the base, /> isthe new dimensionlesslinear coordinate measured in fractions of the radius of inertia of theeccentric, /> is the dimensionless coefficient of energydissipation, />.is the new dimensionless time.
Theset (3) is now normalizedat the linear part approaching a standard form. First, the equations can bewritten as a system of four first-order equations
/>
Then we introduce the polarcoordinates, /> and />. So that the equations take the followingform
/>
Now the set (5) experiences thetransform on the angular variable />. Then the equations obtain the form closeto a standard form
/>
Here /> denotesthe partial angularvelocity of the rotor. The system of equations (6) is completely equivalent tothe original equations. It is not a standard form, allowed for the higherderivatives [10], but such form is most suitable for the qualitative study ofstationary regimes of motion, due to the explicit presence of generalized velocitiesin the right-hand side terms.
Resonance
We study the resonance phenomenon inthe dynamical system (6). Let/>, then eqs. (6) are reduced to the followingset: />, />, />, />, which has a simple solution
/>
where />, />, />, /> are the integration constants. Now thesolution (7) is substituted into the right-hand terms of eqs. (6). Then one discardsall the terms in order/>and higher, as well, toperform the averaging over the period of fast rotating phases. In the problem(6) the fast variables are the angles/>and />, accordingly, the slow variables are /> and />. The average of an arbitraryfunction/>is calculated as
/>.
Now the average /> isexamined for thepresence of jumps along a smooth change of system parameters. One of which representsthe partial angular velocity />. It is easy to see that the jump of theaverage takes place at the value />.
The equations of slow motions
In the case when the system is farfrom resonance, i.e. />, eqs. (6) can easily be solved using thePoincaré perturbation method applied to the small non-resonant terms inorder/>. However, in the resonant case, as/>, the first-order nonlinearapproximation solution should contain the so-called secular terms appearing dueto the known problems of small denominators. To overcome such a problem one usuallyapplies the following trick. As soon as /> and the quantities /> and /> arechanging rapidly, withapproximately the same rate, it is natural to introduce a new generalized slowphase />, where /> isa small variation of theangular velocity. Then after the averaging over the fast variable/>, one obtains the equations for theslow variables only, which are free of secularity. Such equations are called theevolution equations or truncated ones. In the case of set (6) the truncated equationshold true:
/>
where /> is the small frequencydetuning, /> isthe new generalizedphase. Note that for the problem of averaging over the fast variable is enoughto write />.
Stationary oscillations in theabsence of energy dissipation
Now the usual condition of a steadymotion, i.e. />, is applied. We are looking now for the stationaryoscillatory regimes in vacuo, i.e. />. The solution corresponding to theseregimes reads
/>
This solution describes a typicalresonant curve at />. The plus sign in front of the unitis selected when />, otherwise />.
The next stage of the study is to testthe stability properties of stationary solutions. To solve this problem, oneshould obtain the equations in perturbations. The procedure for deriving theseequations is that, firstly, one performs the following change of variables
/>
where /> isthe steady-stateamplitude of oscillations, then after replacing the variables the perturbation equationsget the following form
/>
To solve the stability problem evokingthe Lyapunov criterion we formulate the eigenvalue problem defined by thefollowing cubic polynomial, implicitly presented by determinant of the thirdorder
/>
Now we can apply one of the mostwidely known criteria, for example, the Hurwitz criterion, for the study thestability properties in the space of system parameters. The result is that thedescending branch of the resonant curve, when/>, cannot be practically observed becauseof the volatility associated with the fact that the driver is of limited power.This cannot maintain the given stationary oscillation of the elastic base nearthe resonance. This result corresponds to the well-known paradigm associatedwith the so-called Sommerfeld effect.
Formally, there are stable stationaryregains, when />. However, this range of angularvelocity is far beyond the accuracy of the first-order nonlinear approximation.
Damped stationary oscillations
A small surprise is that the responseof the electromechanical system (2) has a significant change in the presence ofeven very small energy dissipation. Depending on the parameters of the set (2) thesmall damping can lead to typical hysteretic oscillatory patterns when scanningthe detuning parameter/>. While let the dissipation besufficiently large, then a very simple stable steady-state motions, inherent inalmost linear systems, holds true.
From the stationary condition, onelooks for the stationary oscillation regimes />, /> and />, as />. The equations corresponding tothese regimes are the following ones
/>;
/>;
/>.
For a small damping the solution ofthese equations describes a typical non-unique dependence between the frequencyand amplitude, i.e. />, defined parametrically upon thephase />. Near the resonance/>(/>), at some given specific parametersof the problem, say, />, />, /> and />, the picture of this curve is shownin Fig. 1. Accordingly, the dependence of the angular velocity is presented inFig. 2.
/>
Fig. 1. The frequency-amplitudedependence/>near the resonance at /> (arbitraryunits).
/>
Fig. 2. The angular velocity/>changes (arbitraryunits).
To study the stability problem ofstationary solutions to the perturbed equations we should formulate theeigenvalue problem. This leads to the following characteristic cubic polynomial
/>
with the coefficients[1]
/>;
/>;
/>;
/>.
Now one traces the stability propertiesby finding the areas of system parameters by applying the Routh-Hurwitzcriterion, which states the necessary and sufficient conditions of positivityof the following numbers />, />, />, />. These conditions are violated along thefrequency-amplitude curve when scanning the parameter/>between the points Aand C. The characteristicpoints A and B originatefrom the traditional condition that the derivative of function /> approaches infinity. The point Cappears due to the multiple and zero valued roots of the characteristicequation />, as the determinants in the Routh-Hurwitz criterionapproach zero, more precisely, />. At the direct scanning of theparameter/>together with increasing the angular velocityof the driver, one can observe a “tightening” of oscillations up to the pointA. Then, the upper branch of the resonant curve becomes unstable and thestationary oscillations jump at the lower stable branch. At the reverse scanthe angular velocity of the driver at the point C, in turn, there is a loss ofstability of stationary oscillations at the lower branch and the jumping tostable oscillations with the greater amplitude at the upper branch of theresonance curve. The point B, apparently, is physically unrealizable mode of oscillations.
However, with the growth of thedissipation the instability zone shrinks. Then the frequency-amplitude curvebecomes unambiguous, and the instability zone is completely degenerated. Inthis case the Sommerfeld effect also disappears.
Conclusions
Near the resonance the rotor issubstantially influenced by the pair of forces acting from the vibrating base.The average value of this moment is a definite value proportional to quadrateof the amplitude of vibrations of the base. Therefore, near the resonance someincrease in the angular velocity of the engine is experienced. This leads tothe phenomenon of ‘pulling’ hesitation, despite the fact that the elastic propertiesof the base are linear. Together with the growth of dissipation the zone of theSommerfeld instability narrows down to its complete disappearance. This leadsto the idea of efficiency of utilizing vibration absorbers to stabilize themotion of electromechanical systems.
Acknowledgments
Thework was supported in part by the RFBR grant (project 09-02-97053-рповолжье).
References
[1] Vibrationsin Engineering, v. 2, Moscow. Mechanical Engineering, 1979: 351 (in Russian).
[2] FrolovK.V. Vibrations of machines with limited capacity power source and the variableparameters (Proc. K.V. Frolov ed.), Nonlinear oscillations and transientprocesses in machines, Moscow: Nauka, 1972: 5-16 (in Russian).
[3] KononenkoV.O. Nonlinear vibrations of mechanical systems. Kiev: Nauk. Dumka, 1980: 382 (inRussian).
[4] NagaevR.F. Quasiconservative systems. St. Petersburg: Nauka, 1996: 252 (in Russian).
[5] BlekhmanI.I. Synchronization in Nature and Technology. Moscow: Nauka, 1977: 345 (inRussian).
[6] BlekhmanI.I., Landa P.S., Rosenblum M.G. Synchronization and chaotization ininteracting dynamical systems (J) Appl. Mech. Rev., 1995, 11(1): 733-752.
[7] SamantarayA.K., Dasguptaa S.S.and R. Bhattacharyyaa. Sommerfeld effect in rotationallysymmetric planar dynamical systems (J), Int. J. Eng. Sci., 2010, 48(1): 21-36.
[8] MasayoshiTsuchidaa, Karen de Lolo Guilhermeb and Jose Manoel Balthazarb. On chaotic vibrationsof a non-ideal system with two degrees of freedom: Resonance and Sommerfeldeffect (J), J. Sound and Vibration, 2005, 282(3-5): 1201-1207.
[9] LeonovG.A., Ponomarenko D.V. and Smirnova V.B. Frequency-domain methods for nonlinearanalysis. Theory and applications. Singapore: World Sci., 1996: 498.
[10] Zhuravlev V.F., Klimov D.M. Appliedmethods in oscillation theory, Moscow: Nauka, 1988: 328 (in Russian).