1. Introduction

1.1. General principle

A linear kinematic relationship of type RBE3 involves a so-called master node and several so-called slave nodes. The relationship has the effect of distributing the torsor of effort seen by the master node over the slave nodes. The hypotheses for constructing linear constraints require, for each slave node:

• a distribution of torsors weighted by the distance between the master node and the current slave node,

• an additional relative weighting using coefficients entered by the user.

By entering only one coefficient (not zero) for all the nodes, all the slave nodes are considered with the same weight, and only the weighting with the distance will be active.

The distribution is done in such a way that the torsors on the nodes involved are in equilibrium.

1.2. Ratings

\({X}_{M}\)	Master Node Coordinates
\({X}_{i}\)	Coordinates of the \(i\) -th slave node (\(1\mathrm{\le }i\mathrm{\le }n\))
\({\mathrm{\xi }}_{i}={X}_{i}-{X}_{M}=\left[\begin{array}{c}{\mathrm{\xi }}_{\mathit{ix}}\\ {\mathrm{\xi }}_{\mathit{iy}}\\ {\mathrm{\xi }}_{\mathit{iz}}\end{array}\right]\)	Relative coordinates of the \(i\) -th slave node
\({\omega }_{i}\)	Weighting coefficient of the \(i\) -th slave node
\({F}_{i}=\left[\begin{array}{}{F}_{\mathrm{ix}}\\ {F}_{\mathrm{iy}}\\ {F}_{\mathrm{iz}}\end{array}\right]\)	Efforts applied to node \(i\)
\({M}_{i}=\left[\begin{array}{}{M}_{\mathrm{ix}}\\ {M}_{\mathrm{iy}}\\ {M}_{\mathrm{iz}}\end{array}\right]\)	Moments applied to node \(i\)
\({u}_{i}=\left[\begin{array}{c}{u}_{\mathit{ix}}\\ {u}_{\mathit{iy}}\\ {u}_{\mathit{iz}}\end{array}\right]\)	Node moves \(i\)
\({\mathrm{\theta }}_{i}=\left[\begin{array}{c}{\mathrm{\theta }}_{\mathit{ix}}\\ {\mathrm{\theta }}_{\mathit{iy}}\\ {\mathrm{\theta }}_{\mathit{iz}}\end{array}\right]\)	Node rotations
\({d}_{i}=\left[\begin{array}{c}{u}_{i}\\ {\mathrm{\theta }}_{i}\end{array}\right]\)	Degrees of freedom of movement, potentially carried by the \(i\) node
\({T}_{i}=\left[\begin{array}{}{F}_{i}\\ {M}_{i}\end{array}\right]\)	Force twister at the \(i\) node

The transpose of the \(M\) matrix will be noted \({M}^{T}\).

Moreover, it is considered that the nodes carry the displacement and rotation components by default, i.e. six degrees of freedom per node.