5. Transition from local to general roads

The components of the vectors are expressed in the general axes \({e}_{1}\text{}{e}_{2}\text{}{e}_{3}\) [fig 3-a]. The matrices of the operators that connect them are therefore only valid in these axes. But the mechanics of beams are much more simply formulated in the main axes of local inertia \({t}_{1}{t}_{2}{t}_{3}\) in the current configuration. We are therefore required to change the axes from the general trihedron \({e}_{1}\text{}{e}_{2}\text{}{e}_{3}\) to the local trihedron \({t}_{1}{t}_{2}{t}_{3}\) by the product \({R}_{\text{tot}}\) of two rotations:

• the rotation \({R}_{o}\), which is invariable, which brings the general axes \(({e}_{1}\text{}{e}_{2}\text{}{e}_{3})\) onto the local axes in the reference position \(({E}_{1}\text{}{E}_{2}\text{}{E}_{3})\);

• the rotation \(R\), dependent on time, which brings the trihedron \(({E}_{1}\text{}{E}_{2}\text{}{E}_{3})\) to the local trihedron in the current deformed position \(({t}_{1}{t}_{2}{t}_{3})\), i.e.:

\({R}_{\text{tot}}={\mathrm{RR}}_{o}\) eq 5-1

Given a vector \(v\), of known components in the general trihedron, its components in the local trihedron are the components in the general trihedron of the vector:

\(V={R}_{\text{tot}}^{T}v\text{.}\) eq 5-2

We can therefore replace the calculations relating to vectors expressed in local axes in the current configuration, by the same calculations relating to the same vectors rotated by \({R}_{\text{tot}}^{T}\) and expressed in general axes. In other words, this \({R}_{\text{tot}}^{T}\) rotation makes it possible to replace the calculations in local axes of the current configuration, by the same calculations in general axes.