# Lund University Publications

## LUND UNIVERSITY LIBRARIES

### Frames of reference in multibody dynamics

(2019) In Mathematics and Mechanics of Solids 24(1). p.98-151
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,... (More)

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.

(Less)
author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
elastic energy, Euler–Bernoulli beam, frame of reference, kinetic energy, Multibody dynamics, principal axis frame, principal frame
in
Mathematics and Mechanics of Solids
volume
24
issue
1
pages
98 - 151
publisher
SAGE Publications Inc.
external identifiers
• scopus:85044732300
ISSN
1081-2865
DOI
10.1177/1081286517731485
language
English
LU publication?
yes
id
f2258599-a868-41df-bd66-cfd934f3266a
2018-04-12 16:12:11
date last changed
2019-02-20 11:14:03
```@article{f2258599-a868-41df-bd66-cfd934f3266a,
abstract     = {<p>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.</p>},
author       = {Lidström, Per},
issn         = {1081-2865},
keyword      = {elastic energy,Euler–Bernoulli beam,frame of reference,kinetic energy,Multibody dynamics,principal axis frame,principal frame},
language     = {eng},
number       = {1},
pages        = {98--151},
publisher    = {SAGE Publications Inc.},
series       = {Mathematics and Mechanics of Solids},
title        = {Frames of reference in multibody dynamics},
url          = {http://dx.doi.org/10.1177/1081286517731485},
volume       = {24},
year         = {2019},
}

```