2. Definitions

The construction of kinematic relationships requires the introduction of matrices that make it possible to explain the formulas for changing the reduction point for the various torsors. Recall that the force twister of a slave node \(i\) carried to the master node \(M\) is written:

(2.1)\[ {T} _ {M} =\ left [\ begin {array} {c} {c} {F} {F} _ {M} _ {M}\ end {array}\ right] =\ left [\ begin {array} {c} {c} {c} {F} {F} _ {f} _ {i} _ {i} + {\ mathrm {\ xi}}} _ {i} _ {i} _ {i} _ {i} _ {i} _ {i} _ {i} _ {i} _ {i}}\ end {array}\ right]\]

The matrix \({S}_{i}\) is thus introduced:

(2.2)\[\begin{split} {S} _ {i} =\ left [\ begin {array} {cccccc} 1& 0& 0& 0& {\ mathrm {\ xi}} _ {\ mathit {iz}}} & - {\ mathrm {\ xi}}} _ {\ mathrm {\ xi}}} _ {\ mathit {ix}}} _ {\ mathit {iy}}} _ {\ mathit {iy}}} _ {\ mathit {iy}}} _ {\ mathit {iy}}} iz}} & 0& {\ mathrm {\ xi}}} _ {\ mathit {ix}}\\ 0& 0& 1& {\ mathrm {\ xi}} _ {\ mathit {iy}}} & - {\ mathrm {\ xi}} _ {\ mathit {ix}} _ {\ mathit {ix}} & 0\\ 0& 0& 0& 0& 1& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 1\ end {array}\ right]\end{split}\]

So that we can write:

(2.3)\[ {T} _ {M} = {S} _ {i} ^ {T} {T} _ {i}\]

Calculating relationships requires scaling the rotation components so that the relationships created are not changed when the scale of the problem changes. For this, the following characteristic length is defined:

\[\]

: label: eq-4

{L} _ {c} =frac {{sum} _ {i=1} _ {i=1} ^ {n}parallel {xi} _ {i}parallel} {n}

and the matrices (diagonals), associated with the weighting coefficient for the \(i\) -th slave node:

\[\]

: label: eq-5

{W} _ {i} = {omega} _ {i}left [begin {array} {cccccc} 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\ 0\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& {L} _ {c} ^ {2} & 0\ 0& 0& 0& 0& 0& 0& {L} _ {c} ^ {2}end {array} {2}end {array}right]