Completed the computation of the impulse parameter

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Karma Riuk 2023-06-22 18:19:44 +02:00
parent 95fa9cf725
commit 75c272bc89

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@ -66,6 +66,34 @@ $$ \vec r = \alpha \vv{CA} + \beta \alpha \vv{AB} $$
\newpage \newpage
\subsection{Solving for impulse parameter} \subsection{Solving for impulse parameter}
\label{app:impulse_long} \label{app:impulse_long}
[to be done :)] We start with equation \ref{eq:vp2n}:
\begin{equation*}
\begin{split}
\vec v_{p2} \cdot \vec n &= - e \vec v_{p1} \cdot \vec n\\
\left(\vec v_{ap2} - \vec v_{bp2}\right)\cdot \vec n &= - e \vec v_{p1} \cdot \vec n\\
\left( \vec v_{a2} + \omega_{a2} \times \vec r_{ap} - \vec v_{b2} - \omega_{b2} \times \vec r_{bp}
\right)\cdot \vec n &= - e \vec v_{p1} \cdot \vec n\\
\end{split}
\end{equation*}
We now expand the bracket on the left-hand side using the equations \ref{eq:va2}
- \ref{eq:omega_b2} and then simplify with equation \ref{eq:vp1}.
\[
\begin{split}
\left( \vec v_{a1} + \frac{j\vec n}{m_a} + \omega_{a1} + \frac{\vec r_{ap} \times j\vec n}{I_a} \times \vec r_{ap} - \vec v_{b1} + \frac{j\vec n}{m_b} - \omega_{b1} + \frac{\vec r_{bp} \times j\vec n}{I_b} \times \vec r_{bp} \right)\cdot \vec n &= - e \vec v_{p1} \cdot \vec n\\
\left( \frac{j\vec n}{m_a} + \frac{\vec r_{ap} \times j\vec n}{I_a} \times \vec r_{ap} + \frac{j\vec n}{m_b} + \frac{\vec r_{bp} \times j\vec n}{I_b} \times \vec r_{bp} \right)\cdot \vec n &= - (1+e) \vec v_{p1} \cdot \vec n\\
\end{split}
\]
Using the triple scalar product rule, we can derive that
\[
j \left( 1/m_a + 1/m_b + \frac{\left(( \vec r_{ap} \times \vec n
)^2\right)}{I_a} + \frac{\left( \vec r_{bp} \times \vec n \right)^2}{I_b}
\right) = -(1+e) \vec v_{p1} \cdot \vec n
\]
and therefore
\begin{equation} \begin{equation}
j = \frac{ - (1+e) \cdot \vec v_{ap1} \cdot \vec n }{\frac{1}{m_a} + \frac{1}{m_b} +
\frac{\left( \vec r_{ap} \times \vec n \right)^2}{I_a} + \frac{\left( \vec
r_{bp} \times \vec n \right)^2}{I_b}}
\end{equation} \end{equation}