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diff --git a/Flight_Eqns/Equations/Equations.tex b/Flight_Eqns/Equations/Equations.tex new file mode 100755 index 0000000..dab0ffb --- /dev/null +++ b/Flight_Eqns/Equations/Equations.tex @@ -0,0 +1,72 @@ +\documentclass{article} + +\usepackage{amsmath} +\newcommand\inv[1]{#1\raisebox{1.15ex}{$\scriptscriptstyle-\!1$}} + + +\title{Equation of Motions already included} +\begin{document} +\maketitle + +These are the equations already included the 6DOF model. + +\[ +C_{b/n}= + \begin{bmatrix} + cos(\theta)cos(\psi) & cos(\theta)sin(\psi) & -sin(\theta) \\ + -cos(\phi)sin(\psi) + sin(\phi)sin(\theta)cos(\psi) & cos(\phi)cos(\psi) + sin(\phi)sin(\theta)sun(\psi) & sin(\phi)cos(\theta) \\ + sin(\phi)sin(\psi) + cos(\phi)sin(\theta)cos(\psi) & -sin(\phi)cos(\psi) + cos(\phi)sin(\theta)sin(\psi) & cos(\phi)cos(\theta) + \end{bmatrix} +\]\\ +Also represented as: $C_{b/n} = fn(\Theta)$ +\\ +\[ +\Omega= + \begin{bmatrix} + 0 & -R & Q \\ + R & 0 & -P\\ + -Q & P & 0 + \end{bmatrix} +\] +\\ +\[ +\left[ \begin{array}{c} + \dot{\phi} +\\ \dot{\theta} +\\ \dot{\psi} + \end{array} \right] = \begin{bmatrix} 1 & tan(\theta)sin(\phi) & tan(\theta)cos(\phi) \\ 0 & cos(\phi) & -sin(\phi)\\ +0 & sin(\phi)/cos(\theta) & cos(\phi)/cos(\theta) + \end{bmatrix} \times \left[ \begin{array}{c} P\\Q \\ R \end{array} \right] +\\ +\] +\\ Also represented as: $\dot{\Phi} = H(\Phi) \omega^b _{b/e}$ +\\ +\\ +${}^b\dot{v}^b_{CM/e} =(\frac{1}{m})F^b_{A,T} + C_{b/n} \times g + \Omega^b _{b/e}\times v^b_{CM/e} $ +\\ \\ +${}^e \dot{p}^n_{CM/T} = C_{n/b} \times v^b_{CM/e} $ +\\ \\ +${}^b\dot{\omega}^b_{b/e} = inv(J^b) \times [M^b_{A,T}- \Omega^b_{b/e}\times J^b \times \omega^b_{b/e}]$ +\\ + +These are the equations in the force-moment model. + +$\alpha = tan^-1 (w/u) $ +\\ +$Q = 0.5 \times \rho \times \lVert V \rVert ^2 $ +\\\\ +$C_{L} = C_{L0} + C_{L \alpha} \times \alpha + (\frac{C_{Lq} \times q \times c_{bar}}{2 \times \lVert V \rVert}) + C_{Lde} \times \delta_{e} $ +\\ +$C_{m} = C_{m0} + C{m \alpha} \times \alpha +(\frac{C_{mq}*q*c_{bar}}{(2 \times \lVert V \rVert})+C_{mde} \times \delta_{e}$ +\\ +$C_{D} = C_{D0} + K_{drag} \times C_{L} ^2$ +\\ \\ +$L = C_{L} \times s \times Q$ +\\ +$D = C_{D} \times s \times Q$ +\\ \\ +$Force = \left[ \begin{array}{c} -D \times cos(\alpha)+L \times sin(\alpha)+ thrust - mg \times sin(\theta)\\0 \\ -D \times sin(\alpha)-L \times cos(\alpha)+mg \times cos(\theta) \end{array} \right]$ +\\ \\ +$Moment = \left[ \begin{array}{c} 0\\C_{m} \times s\times c_{bar} \times Q \\ 0 \end{array} \right]$ + + \end{document}
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