User:Egm6341.s10.team3.sa/Mtg33

Mtg 33: Thu, 23 Mar 10

 Typeset of transparencies, not lecture transcript. Subramanian Annamalai 14:07, 12 August 2010 (UTC)

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Euler equations: (Linear) Rate of change of momentum = Sum of forces

Use $$ \displaystyle (\bar{x},\bar{y})$$coordinate system to simplify the components of $$ \mathbf v $$ Note: In the $$ \displaystyle (x,y) $$ coordinate system momentum involves both $$ \mathbf v(t) $$ and $$ \gamma(t)$$

Linear momentum components: $$ \displaystyle (p_\bar{x}, p_\bar{y}) $$ $$ \displaystyle p_\bar{x} = mV $$  $$ \displaystyle \underbrace{\frac{d}{dt}(p_\bar{x})}_{\bar{m}\bar{V}} = \sum F_\bar{x} = (T-D)cos \alpha - L sin \alpha - mg sin \gamma $$ $$\displaystyle (Eq. 1) $$ Note : $$ \displaystyle {\bar{m}\bar{V}} = \dot{m}V + m\dot{V} $$ We neglect $$ \displaystyle \dot{m} V \; in\; front\; of\;  m \dot{V} $$

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$$ \displaystyle \therefore \dot{V} = \frac{(T-D)}{m}cos\alpha - \frac{L}{m}sin\alpha -gsin\gamma$$ $$\displaystyle (Eq. 1) $$

$$ \displaystyle dp_\bar{y} = mVd\gamma $$ $$\displaystyle (Eq. 2) $$

$$ \displaystyle \underbrace{\frac{d}{dt}(p_\bar{y})}_{mV \frac{d\gamma}{dt}} = \sum F_\bar{y} = (T-D)sin \alpha - L cos \alpha - mg cos \gamma $$ $$\displaystyle (Eq. 3) $$

$$ \displaystyle \therefore \frac{d\gamma}{dt} = \frac{(T-D)}{mV}sin\alpha + \frac{L}{mV}cos\alpha -\frac{g}{V}cos\gamma$$ $$\displaystyle (Eq. 4) $$


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HW:
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At t+dt, consider V+dV Show that $$\displaystyle \color{blue} dp_{\overline{y}} = mVdr$$


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[[media: Egm6341.s10.mtg33.djvu | Page 33-3]]  Summary: Equations of motion(EOM) :

Kinematics: (Eq. 4) and (Eq. 5) on [[media: Egm6341.s10.mtg32.djvu | Page 32-3]] Kinetics:  (Eq. 1) and (Eq. 4) on [[media: Egm6341.s10.mtg33.djvu | Page 33-2]]

Initial conditions: $$ \displaystyle At\; t=t_0 \; (Generally\; t_0 = 0) $$

$$ \displaystyle x(t_0)=x_0\;,\; y(t_0)=y_0\;,\; v(t_0)=v_0\;,\; \gamma(t_0)=\gamma_0$$ $$\displaystyle (Eq. 1) $$

Abstract formulation: $$ \displaystyle \underline{z} = \bigg[ x,y,v,\gamma \bigg] ^ T $$ $$\displaystyle (Eq. 2) $$

EOM: Set of non-linear first-order ODEs: $$ \displaystyle \dot{\underline{z}}_{(4 X 1)} = \underline{f}_{(4 X 1)}(\underline{z}, \underline{u}, t) $$ $$\displaystyle (Eq. 3) $$

$$ \displaystyle \underline{u} = \bigg[T,\alpha \bigg] ^ T$$ $$\displaystyle (Eq. 4) $$