Two referentials ${\cal A} (A,{\bf \hat{a}} _1,{\bf \hat{a}} _2,{\bf \hat{e}} _3)$ and ${\cal B} (B,{\bf \hat{b}} _1,{\bf \hat{b}} _2,{\bf \hat{e}} _3)$
are in rotational motion about axis $(A,{\bf \hat{e}} _3)$ and $(B, {\bf \hat{e}} _3)$, respectively, relative to a referential
${\cal E} (A,{\bf \hat{e}} _1,{\bf \hat{e}} _2,{\bf \hat{e}} _3)$. Points $A$ and $B$ are located on axis $(O,{\bf \hat{e}} _1)$, with $O$ midpoint of $AB$.
${\cal A}$ rotates at constant angular velocity
$\omega$ in the counterclockwise direction. ${\cal B}$ rotates at constant angular velocity
$2\omega$ in the clockwise direction. The axes $(A,{\bf \hat{a}} _1)$ and $(B,{\bf\hat{ b}} _1)$ both coincide with axis $(O,{\bf \hat{e}} _1)$ at $t=0$. Define $2l$ the distance $|AB|$.
a. Find the kinematic screw $\{ {\cal V}_{\cal B/A} \}$ resolved at point $A$, then at point $B$.
b. Deduce that $\cal B$ is in instantaneous rotation relative to $\cal A$. Determine the corresponding instantaneous axis of rotation $(I,{\bf \hat{e}} _3)$. Describe the loci of $I$ relative to $\cal A$ and $\cal B$. Then characterize the motions of $\cal A$ and $\cal B$ relative to $\cal E$.
