I am trying to solve a practical problem for a mechanical engineering application: finding the angular acceleration given measurements from the linear acceleration and the angular rate in two different points A and B. In a solid rigid, with points A and B fixed (no Coriolis forces) this is related by:
\begin{equation}
\vec{a_B} = \vec{a_A} + \vec{\alpha} \times \vec{r} + \vec{\omega}\times(\vec{\omega} \times \vec{r})
\end{equation}
Where $\vec{a_A}$, $\vec{a_B}$ represent the linear accelerations of $A$ and $B$ respectively, $\vec{\alpha}$ the angular acceleration, $\vec{\omega}$ the angular rate, $\vec{r}$ the vector from $A$ to $B$, and for all what matters here they are all vectors or pseudovectors on $\mathbb{R}^3$. Re-arranging:
\begin{equation}
\vec{\alpha} \times \vec{r} = \vec{a_B}- \vec{a_A} – \vec{\omega}\times(\vec{\omega} \times \vec{r})
\end{equation}
and renaming the rhs for convenience to $\vec{c}$
\begin{equation}
\vec{\alpha} \times \vec{r} = \vec{c}
\end{equation}
When trying to solve the system of 3 equations resulting from the cross product through Gauss elimination, I find that the system is linearly dependent and I can't isolate an $\alpha_i$ component without cancelling out the whole row.
What is the geometrical intuition behind this result? If the unknown variable was any of the linear accelerations $\vec{a_A}$, or $\vec{a_B}$, solving it would be trivial. Why having the unknown variable involved in a cross product makes the problem unsolvable?
Best Answer
The intuition is that adding a multiple of $\vec{r}$ to $\vec{\alpha}$ preserves $\vec{\alpha}\times\vec{r}$, so $\vec{\alpha}$ isn't unique. You can't solve for $\vec{\alpha}$ without some more information.