Frames of reference in multibody dynamics

Abstract

In this paper, a discussion is undertaken concerning the use of so-called floating frames of reference in the calculation of the kinetic and elastic energies of parts in a multibody system. The use of floating frames may simplify the calculation of the elastic energy, although sometimes at the expense of more elaborate expressions for the kinetic energy. These expressions may involve terms that couple the motion of the floating frame and the relative motion of the part. The choice of a floating frame may be arbitrary but in order to obtain as simple expressions as possible some care must be taken. When a (flexible) part is connected to a rigid part one may use a frame in which the rigid part is at rest. If so then one has, in general, to deal with coupling terms in the kinetic energy for the flexible part. There is one unique frame in which these coupling terms disappear. This frame is called the principal frame of reference. Relative to this frame the kinetic energy of the part is minimal compared to the kinetic energy relative to other frames. Two independent proofs of this property are presented. The principal frame is defined by the associated change of frame mapping. This mapping is given a full characterization. It may however be cumbersome to calculate the kinetic energy relative to the principal frame. A method for doing this is designated. A frame that has been given some attention in the literature is the principal axis frame of reference. In this paper, a full characterization of this frame and its relation to the principal frame is given. Two examples of an Euler–Bernoulli beam in rotational motion are presented and compared in the light of the theoretical findings of this paper. In conventional presentations of mechanics the Euclidean spaces associated with different frames of reference are taken to be identical. In this paper this assumption is abandoned and different frames of reference will correspond to different Euclidean spaces. From a conceptual point of view this is a natural step to take in order to increase clarity and generality. It automatically includes the dependence of the reference placement on the frame of reference. This approach has been analyzed in a previous paper by the present author. References to this paper will appear whenever needed for. Consequences of this approach are investigated in terms of transformation formulas for kinematical and dynamical quantities.