Zwangskräfte bei einem Mehrkörpersystem mit drei Freiheitsgraden

Abstract

A mechanical system with three degrees of freedom in a gravitational field is analyzed for chaotic versus regular behavior, and for moments of constraint that act on its axes. The system consists of three connected rigid bodies each of which rotating about one axes. The vertical axis of the first body is fixed to the immobile base; the second body rotates about a horizontal axis which moves with the first, and the third body rotates about an axis in the second. Its moments of inertia A, B, C are assumed to obey C=A B. The motion is regular if the body is symmetric, A=B, otherwise the asymmetry parameter µ=(B-A)/B determines the degree of chaoticity. The different types of motion were identified with the help of Poincaré sections while the study of forces of constraints as functions of time was carried out using methods of time-frequency analysis: Fourier analysis in the case of regular, and wavelet analysis in the case of chaotic motion. It was noticed that for large values of µ the motion of the system shows an effect that can be described as intermittency, that is, there exist of long phases of motion during which the system behaves regularly. The reason for the occurence of such an effect is explained. It is observed that the forces of constraints obtain their maximal values in connection with the transition from one regular phase to another. In spite of the presence of chaos in the Poincaré-sections, for large intervals of µ, the wavelet transformation diagrams reflect a relatively regular behavior of the forces of constraints. Using classical statistical methods it would be possible, on the basis of analysis, to evaluate the long-term effects of chaotic motion on the wearing of materials in the bearings of the system.