By I.V. Novozhilov
This publication considers equipment of approximate research of mechanical, elec tromechanical, and different structures defined through traditional differential equa tions. smooth mathematical modeling of refined mechanical structures contains a number of levels: first, development of a mechanical version, after which writing acceptable equations and their analytical or numerical ex amination. often, this method is repeated numerous instances. no matter if an preliminary version properly displays the most homes of a phenomenon, it de scribes, quite often, many pointless info that make equations of movement too complex. As adventure and experimental facts are collected, the researcher considers less complicated types and simplifies the equations. hence a few phrases are discarded, the order of the equations is reduced, and so forth. This procedure calls for time, experimentation, and the researcher's intu ition. a great instance of this type of semi-experimental manner of simplifying is a gyroscopic precession equation. Formal mathematical proofs of its admis sibility seemed a few a number of many years after its profitable advent in engineering calculations. utilized arithmetic now has at its disposal many tools of approxi mate research of differential equations. program of those equipment may perhaps shorten and formalize the approach of simplifying the equations and, hence, of creating approximate movement versions. vast software of the tools into perform is hindered by way of the fol lowing. 1. Descriptions of assorted approximate equipment are scattered over the mathematical literature. The researcher, in general, doesn't comprehend what technique is best suited for a particular case. 2.
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Additional info for Fractional Analysis: Methods of Motion Decomposition
4. In Fig. 4( a) the graph of dependence of the frequency of free oscillations of a physical pendulum on the oscillations' amplitude is shown. Nonisochronism of oscillations is easily seen: the frequency decreases as the amplitude increases. This type of dependence of the frequency on the amplitude is obtained for oscillating systems with the so-called "soft" characteristic of returning force when the graph of the nonlinear elastic characteristic lies lower than the graph of its linear part. In the case of the nonlinear "stiff" characteristic, it lies higher than the corresponding linear part.
T •. 2) takes the form: dv dt = -x + ej, dx dt = v, V(O) X(O) v(O) = V. 4) j = -v. 4) for e « 1 does not decompose into equations in the fast and slow variables; therefore, the method of averaging cannot be used. To reduce the system to required form, a change of variables should be used. The following speculations help to obtain an appropriate form of the variable change. 5) 56 CH. III. 6) where a, tp are amplitude and phase of harmonic oscillations that satisfy the equations: da =0 dt ' dt = a(O} 1, = Jx~ + V5, vo = arctan -xo .
Let nonlinear terms begin with the cubic one. 17) changes. 14) changes also. Then dependence of the frequency on the amplitude is presented by Fig. 4(b). CHAPTER III Decomposition of motion in systems with fast phase Among the problems considered in the previous chapter special attention was paid to slowly changing motions parameters, for example, slowly increasing amplitudes in Sec. 2, and slow Magnus drifts in Sec. 3. The Magnus drifts were obtained by averaging the right-hand side of Eqs. 29) with respect to the time variable which is a part of high-frequency oscillating components.