Integral criterion op stability for systems with quasicyclic coordinates and energy relations for oscillations of current-carrying conductors: PMM vol. 33, no. 1, 1969, pp. 85–100

Abstract

We consider systems with quasicyclic coordinates and analyze the motions in which velocities, impulses and position (but not quasicyclic) coordinates are periodic functions of time. We assume that the generalized forces corresponding to quasicyclic coordinates either depend on time only, or are proportional to quasicyclic generalized coordinates and that the latter are small. We show that, when certain requirements are imposed on the nonpotential forces with reference to the position coordinates in stable motions, then the quasicyclic impulses assume (up to the small order terms) mean values yielding the minimum of some function Λ of these mean values. This function can be expressed in terms of the Routh's kinetic potential of the system, by the virial describing the forces acting upon the position sybsystem by the quasicyclic subsystem, etc. This in turn yields various versions of the integral criterion of stability. Applying this criterion to the case of the oscillations of linear current-carrying conductors, we can relate mean periodic values of the magnetic fluxes to the extermal conditions of the combination of the averaged values of the magnetic field energy, magnetization energy and of the mechanical kinetic potential (or the virial of the ponderomotive forces). The case when the Routh's equations are linear with respect to the position coordinates is considered separately, and we refer back to our previous papers on the problems on excitation of oscillations [ 1,2] to analyze the possibility of representing the conditions of excistence and stability in terms of the harmonic coefficients of action of an oscillating system, and to give specific expressions for Λ. We thus find that the systems in question constitute a second class of systems admitting the integral criterion. Earlier, Blekhman e.a. [ 3–5] studied the systems of synchronizable objects with weak constraints. Differences occurring between these two classes are related to the form of the Lagrangian and of the generalized forces as well as to the assumptions on smallness. This leads to different formulations of the criterion. In particular, systems with quasicyclic coordinates, unlike the synchronizable systems, can admit the integral criterion also when considerable dissipation occurs over the position coordinates.