Capabilities of the system maple in studying dynamic systems of magnetically interacting free bodies
- PDF / 204,096 Bytes
- 5 Pages / 595 x 842 pts (A4) Page_size
- 2 Downloads / 167 Views
BRIEF COMMUNICATIONS CAPABILITIES OF THE SYSTEM MAPLE IN STUDYING DYNAMIC SYSTEMS OF MAGNETICALLY INTERACTING FREE BODIES L. V. Grigor’eva,a V. V. Kozorez,b and S. I. Lyashkoa
UDC 531.381:531.53:537.61:537.2
The problem of n magnetically interacting free bodies is considered. Its dynamic model of order 12n is obtained based on the Lagrangian formalism. The Maple software is used to numerically solve the Cauchy problem and to plot phase portraits. The model is relevant to magnetic levitation. Keywords: free bodies, magnetic interaction, dynamics. Modern systems of computer mathematics create new possibilities of investigation of complicated objects. In particular, distinctive features of the system Maple [1] whose kernel and libraries absorbed the most part of mathematical knowledge are as follows: the output of results in symbolic (analytical) form, automation of routine computations, and unique graphics of representation of solutions of problems, including solutions that could not be obtained earlier. One of concrete examples of such problems is the “problem of n magnets.” The traditional “problem of n bodies” that called forth the classical mechanics, analytical dynamics [2], and stability theory [3] basically deals with material points each of which has three degrees of freedom. A magnetic analogue of this problem must take into account not only the translational coordinates but also the coordinates of rotary motion around the centers of masses since magnetic interaction is noncentral and translational motions essentially depend on orientational motions. Thus, the problem of n magnets is the problem of analysis of a system of usual differential equations of order 12n. If the nonlinearity of these equations is taken into account, then the complexity of this problem becomes obvious. For example, the motion of only three free magnets is described by a nonlinear thirty-sixth order system. The subject of this article is the approbation of the Maple system with the help of a model of a special case of the problem of n magnets with an arbitrary n. Here, a mathematical model is constructed that describes the dynamics of free bodies with magnetic interaction of superconducting rings placed one under another and gravity directed along the Oz axis of the inertial reference frame Oxyz (Fig. 1). A superconducting ring is fixed on a platform that vibrates in the direction of the same axis with a given frequency W. The center of mass of each body is located on the common axis of the two superconducting rings at end faces of the body, and the mentioned axis is one of axes of the dynamic symmetry of the body. The spatial position of a body is described by three Cartesian coordinates of its center of mass with respect to the inertial system and three orientation angles. Such angles can be the Euler angles (the nutation q, precession y, and pure rotation j (Fig. 2)) or the Euler–Krylov angles (the roll a, tangage b, and yaw g (Fig. 3)). The main problem in constructing the model is the determination of the magnetic component of
Data Loading...
