diff --git a/mr/ubC/mrC.pdf b/mr/ubC/mrC.pdf index 754052d..81b63cb 100644 --- a/mr/ubC/mrC.pdf +++ b/mr/ubC/mrC.pdf Binary files differ diff --git a/mr/ubC/mrC.tex b/mr/ubC/mrC.tex index c1170d0..dc2a82b 100644 --- a/mr/ubC/mrC.tex +++ b/mr/ubC/mrC.tex @@ -134,34 +134,32 @@ 0\\0\\-\omega_1^2+\omega_2^2-\omega_3^2+\omega_4^2-\omega_5^2+\omega_6^2 \end{pmatrix}$ \item $l\begin{pmatrix} - 0 \\ -k_T\omega_1^2\\ 0 - \end{pmatrix} \\ - \tau_{T2} = l\begin{pmatrix} - \frac{\sqrt{3}}{2}k_T\omega_2^2 \\ -\frac{1}{2}k_T\omega_2^2\\ 0 - \end{pmatrix} \\ - \tau_{T3} = l\begin{pmatrix} - \frac{\sqrt{3}}{2}k_T\omega_3^2 \\ \frac{1}{2}k_T\omega_3^2\\ 0 - \end{pmatrix} \\ + 0 \\ -k_T\omega_1^2\\ 0 + \end{pmatrix},\ \tau_{T2} = l\begin{pmatrix} + \frac{\sqrt{3}}{2}k_T\omega_2^2 \\ -\frac{1}{2}k_T\omega_2^2\\ 0 + \end{pmatrix},\ \tau_{T3} = l\begin{pmatrix} + \frac{\sqrt{3}}{2}k_T\omega_3^2 \\ \frac{1}{2}k_T\omega_3^2\\ 0 + \end{pmatrix}, \\ \tau_{T4} = l\begin{pmatrix} - 0\\ k_T\omega_4^2\\ 0 - \end{pmatrix} \\ + 0\\ k_T\omega_4^2\\ 0 + \end{pmatrix}, \ \tau_{T5} = l\begin{pmatrix} - -\frac{\sqrt{3}}{2}k_T\omega_5^2 \\ \frac{1}{2}k_T\omega_5^2\\ 0 - \end{pmatrix} \\ + -\frac{\sqrt{3}}{2}k_T\omega_5^2 \\ \frac{1}{2}k_T\omega_5^2\\ 0 + \end{pmatrix}, \ \tau_{T6} = l\begin{pmatrix} - -\frac{\sqrt{3}}{2}k_T\omega_6^2 \\ -\frac{1}{2}k_T\omega_6^2\\ 0 + -\frac{\sqrt{3}}{2}k_T\omega_6^2 \\ -\frac{1}{2}k_T\omega_6^2\\ 0 \end{pmatrix} \\ \tau= \begin{pmatrix} - l \frac{\sqrt{3}}{2}k_T(\omega_2^2\cdot \omega_3^2\cdot-\omega_5^2\cdot -\omega_6^2)\\ - l\frac{1}{2}k_T(-2\omega_1^2-\omega_2^2\cdot \omega_3^2 \cdot 2\omega_4^2 \cdot \omega_5 \cdot-\omega_6)\\ + l \frac{\sqrt{3}}{2}k_T(\omega_2^2+ \omega_3^2-\omega_5^2 -\omega_6^2)\\ + l\frac{1}{2}k_T(-2\omega_1^2-\omega_2^2+\omega_3^2+2\omega_4^2 +\omega_5^2 -\omega_6^2)\\ k_Q (-\omega_1^2+\omega_2^2-\omega_3^2+\omega_4^2-\omega_5^2+\omega_6^2) \end{pmatrix} $ \item $ \Gamma = \begin{pmatrix} k_T&k_T&k_T&k_T&k_T&k_T \\ - 0 & l \frac{\sqrt{3}}{2}k_T & l \frac{\sqrt{3}}{2}k_T & 0& -l\frac{\sqrt{3}}{2}k_T& -l\frac{\sqrt{3}}{2}k_T \\ - -lk_T &l\frac{1}{2}k_T &-l\frac{1}{2}k_T & lk_T&l\frac{1}{2}k_T &-l\frac{1}{2}k_T \\ + 0 & \frac{\sqrt{3}}{2}lk_T & \frac{\sqrt{3}}{2}lk_T & 0& -\frac{\sqrt{3}}{2}lk_T& -\frac{\sqrt{3}}{2}lk_T \\ + -lk_T &\frac{1}{2}lk_T &-\frac{1}{2}lk_T & lk_T&\frac{1}{2}lk_T &-\frac{1}{2}lk_T \\ -k_Q &k_Q & -k_Q &k_Q &-k_Q&k_Q \\ \end{pmatrix} $ @@ -177,10 +175,37 @@ 0\\0\\ 9.87 \frac{m}{s^2} \end{pmatrix} =\begin{pmatrix} 0\\0\\ 5.836 \frac{m}{s^2} - \end{pmatrix} $ - - \item b TODO - \item c TODO + \end{pmatrix} $ + \item $\tau=\begin{pmatrix} + lk_T\omega_2^2-lk_T\omega_4^2\\ + -lk_T\omega_1^2+lk_T\omega_3^2\\ + k_Q(-\omega_1^2+\omega_2^2-\omega_3^2+\omega_4^2) + \end{pmatrix}$, $\alpha=I^{-1}\tau\approx \begin{pmatrix} + 142.857&0&0\\ + 0&142.857&0\\ + 0&0&83.333 + \end{pmatrix}\tau$\\ + maximal acceleration around x and y in both directions is the same:\\ + $\alpha_{max}=\begin{pmatrix} + 142.857(lk_T\omega_2^2-lk_T\omega_4^2)\\ + 142.857(-lk_T\omega_1^2+lk_T\omega_3^2)\\ + 83.333\cdot k_Q(-\omega_1^2+\omega_2^2-\omega_3^2+\omega_4^2) + \end{pmatrix}$\\ + $\alpha_{x,max}\approx 142.857\frac{1}{kgm^2}(lk_T\omega_2^2)\approx 558.785l\frac{1}{s^2}$\\ + $\alpha_{z,max}\approx 83.333\frac{1}{kgm^2}\cdot k_Q(\omega_2^2+\omega_4^2)\approx 15.917\frac{1}{s^2}$ + \item \begin{itemize} + \item $\alpha_{x,max}\approx 558.785l\frac{1}{s^2}$\\ + where: + $\omega_2=838\frac{rad}{s},\omega_4=0\frac{rad}{s}, \omega_1=\omega_3=\sqrt{\frac{\frac{9.81N}{k_T}-\omega_2^2}{2}}\approx 727.660\frac{rad}{s} < 838\frac{rad}{s}$\checkmark\\ + accordingly for $-x,y,-y$ + \item $\text{max: }\alpha_{z}=k_Q(-\omega_1^2+\omega_2^2-\omega_3^2+\omega_4^2)$ s.t.\\ + $\omega_2=\omega_4,\ \omega_1=\omega_3,\ \omega_1^2+\omega_2^2+\omega_3^2+\omega_4^2=\frac{9.81N}{k_T}$\\ + $\Rightarrow \omega_2=\omega_4=838\frac{rad}{s}\\ + \Rightarrow \omega_1^2+\omega_3^2=\frac{9.81N}{k_T}-2\cdot \left(838\frac{rad}{s}\right)^2\\ + \Rightarrow \omega_1=\omega_3=\sqrt{\frac{9.81N}{2k_T}-838^2\frac{rad^2}{s^2}}\approx 422.334\frac{rad}{s}$\vspace{5mm}\\ + $\Rightarrow\alpha_{z,max}=83.333\frac{1}{kgm^2}\cdot k_Q(-422.334^2+838^2-422.334^2+838^2)\frac{rad^2}{s^2}\approx 11.875\frac{1}{s^2}$ + \end{itemize} + \end{enumerate} \end{document}