Mechanics of Continua and Structures
The rotation transformation is, as shown previously,
\[\begin{align} \label{eq:masterrotation} \bs{R}&=\uv{e}_i\otimes \uv{E}_i \end{align}\]This rotation is achieved in three steps: a rotation, $ \bs{R}_1$ that maps $\uv{E}_i$ to $\uv{e}’_i$, a $\bs{R}_2$ that maps $\uv{e}’_i$ to \({\uv{e}''}_i\), and finally a rotation $\bs{R}_3$ that maps \(\uv{e}''_i\) to $\uv{e}_i$, i.e.,
\[\begin{align} \uv{e}'_i&=\bs{R}_1 \uv{E}_i\\ \uv{e}''_i&=\bs{R}_2 \uv{e}'_i\\ \uv{e}_i&=\bs{R}_3 \uv{e}''_i\\ \end{align}\]If the unit vector $\uv{a}_i$ is mapped to the vectors $\uv{b}_i$ through the rotation $\bs{R}$, then, as shown previously, \(\begin{align} \bs{R}=\bs{b}_i\otimes \bs{a}_i, \end{align}\) Therefore, we have:
\[\begin{align} \bs{R}_1&=\uv{e}'_i\otimes \uv{E}_i,\\ \bs{R}_2&=\uv{e}''_i\otimes \uv{e}'_i,\\ \bs{R}_3&=\uv{e}_i\otimes \uv{e}''_i,\\ \end{align}\] \[\begin{align} \bs{R}_i&=\bs{R}(\gamma_i,\uv{g}_i) \end{align}\]Going back to the master rotation, $\eqref{eq:masterrotation}$, we have that:
Preliminary result \(\begin{align} \uv{E}_i\otimes \uv{E}_i&=\bs{I} \end{align}\)
\[\begin{align} \bs{R}&=\bs{e}_i\otimes \uv{E}_i\\ &=(\bs{R}_3\uv{e}''_i)\otimes \uv{E}_i\\ &=\uv{R}_3(\uv{e}''_i\otimes \uv{E}_i)\\ &=\uv{R}_3((\uv{R}_2\uv{e}'_i)\otimes \uv{E}_i)\\ &=\uv{R}_3(\uv{R}_2(\uv{e}'_i\otimes \uv{E}_i))\\ &=\uv{R}_3\uv{R}_2(\uv{e}'_i\otimes \uv{E}_i)\\ &=\uv{R}_3\uv{R}_2((\uv{R}_1\uv{E}_i)\otimes \uv{E}_i)\\ &=\uv{R}_3\uv{R}_2(\uv{R}_1(\uv{E}_i\otimes \uv{E}_i))\\ &=\uv{R}_3\uv{R}_2\uv{R}_1 (\uv{E}_i\otimes \uv{E}_i)\\ &=\uv{R}_3\uv{R}_2\uv{R}_1 \bs{I}\\ &=\uv{R}_3\uv{R}_2\uv{R}_1\\ \end{align}\]Each of the individual rotations are written as,
\[\begin{align} \bs{R}_p&=\bs{R}(\gamma_i, \uv{g}_i) \end{align}\]Rotation angles
Typically, $(\gamma_1,\gamma_2, \gamma_3)$ are denoted as $(\psi,\theta,\phi)$. However, this is not standard. All types of symbols are used to denote $\gamma_i$.
The rotation axis
Let $(a,b,c)$ be a permutation of $(1,2,3)$. The permutation $(3,2,3)$ is what is traditionally referred to as Euler angles.
\[\begin{align} \uv{g}_1&=\uv{E}_a\\ \uv{g}_2&=\bs{R}_1\uv{E}_b\\ \uv{g}_3&=\bs{R}_2 \bs{R}_1 \uv{E}_c\\ \end{align}\]