Dynamics of the conductivity bodies of pendulum types in alternating magnetic field

Abstract

The investigations fulfilled in this article are founded on two results. The first is experiments of M. Beteno, Y. Duboshinsky. The description of these experiments is adduced in [1]. In these experiments the low frequency oscillations of iron ball suspended on the thin string [1] were obtained (frequency of these oscillations is order to eigen frequency of pendulum). And the second is theoretical conclusions about possible stabilizations of early (without alternating magnetic field) unstable equilibrium positions. This theoretical result was obtained by the asymptotical solution of Lagrange-Maxwell equations of dynamic of electromechanical systems suspended in alternating magnetic field [2].The first step of this study is consideration of the motions of pendulum system the role of solid body of which is carried out by the closed loop of current. This solid loop is stiffed connected with point of suspension by weightless rod (Fig. 1). In first case it's supposed that the magnetic field, in which pendulum system is situated, is uniform and its frequency much more than frequency of small pendulum oscillations. We designate the angle of deviation of the pendulum from vertical axis by θ and assume that angle θ = 0 corresponds to the lower position of the pendulum. The expressions for the kinetic energy is T , magnetic field energy -W and potential energy -Π have the formsThe Lagrange-Maxwell equations for the considered electromechanical system are ) L(i). + B 0 S sin νt cos θθ + B 0 Sν cos νt sin θ + Ri = 0where θ -angle of deviation of pendulum, i -the current in loop, B 0 and ν -amplitude and frequency of external magnetic field, m, S, I, and, R -mass, square, inertia moment, and resistance of conductive counter accordingly, l -length of rod. The solution of this type of equations will be conducted by method of division of movements [3].

Numerical and experimental study of dynamics of two pendulums under a magnetic field

“…Skubov and Vavilov 21 investigated theoretically and numerically the dynamics of conductive bodies of pendulum types in an alternating magnetic field. The problem of possible stabilization of early unstable equilibrium configurations and the asymptotical solution to the Lagrange–Maxwell equations of the dynamics of the electromechanical system embedded into an alternating magnetic field has been addressed.…”