User:EGM6341.S11.team5.cavalcanti/Mtg35

NM1 Mtg 35: Fri, 25 Mar 11 [[media: Nm1.s11.mtg35.djvu | Page 35-1]] Euler equations: Rate of change of linear momentum p = Sum of forces $$\sum_{\color{red}{i}}\underline{F}_{\color{red}{i}} \Rightarrow$$(6)p. 34-4 Use $$(\overline{x}, \overline{y})$$ coordinate sustem to simplify composition of $$\underline{\color{blue}{v}}$$ Reminder: In (x, y) coordinates, momentum p involves both $$v(t) \ and \ \gamma(t)$$  "End Reminder"  Linear momentum composition: $$(p_{\overline{x}}, p_{\overline{y}})$$ $$p_{\overline{x}} = mv$$ $$ \underbrace{\frac{dp_{\overline{x}}}{dt}}_{\color{blue}{\overline{\dot{mv}}}-\dot{m}v+m\dot{v}} = \sum F_{\overline{x}}=(T-D)cos\gamma - Lsin\gamma - \underbrace{mg}_{W}sin\gamma$$ D & D 1962, S & Z 2007: neglect $$\dot{m}v$$ in front of $$ m\dot{v}$$. [[media:Nm1.s11.mtg35.djvu | Page 25-2]] Note: At $$t+dt$$, consider $$v+dv$$. Can show $$dp_{\overline{y}}=mvd\gamma$$ by neglecting hot. (S10 HW) [[media:Nm1.s11.mtg35.djvu | Page 35-3]] Summary: Equations of motion (EOM) Kinematics: (4) & (5)p. 32-3 Kinetics: (1) & (4)p. 35-2 Initial conditions: At $$t=t_0$$ (usually $$t_0 = 0$$) Abstract formulation: $$ \underline{z}=\underbrace{\left \lfloor x,y,V,\gamma \right \rfloor ^{\color{red}T \ (transpose)}}_{\color{red}row \ matrix \ 1 \times 4} $$ EOM = set of nonlinear 1st order ODEs. $$\underline{\dot{z}}_{\color{red}{4 \times 1}} = \underline{f}_{\color{red}{4 \times 1}} \left( \underline{z}_{\color{red}{4 \times 1}}, \underline{u}_{\color{red}{2 \times 1}}, t \right) $$ underbar: = matrix quantity boldface \mathbf $$\underline{u}_{\color{red}{4 \times 1}} = \left \lfloor T, \alpha \right \rfloor^{\color{red}{T}} \ \color{blue}controls $$ [[media:Nm1.s11.mtg35.djvu | Page 35-4]] Kinematics: (4) & (5)p.32-3 Kinetics: (1) & (4)p. 35-2 $$ \dot{x} = Vcos\gamma$$ $$ \dot{y} = Vsin\gamma$$ $$ \dot{V} = \frac{T-D}{m}cos\alpha - \frac{L}{m}cos\alpha-\frac{q}{v}cos\gamma$$ $$ \underbrace{\dot{\gamma}}_{\color{blue}{\underline{\dot{z}}}} = \underbrace{\frac{T-D}{mv}sin\alpha + \frac{L}{mv}cos\alpha-\frac{q}{v}cos\gamma}_{\color{blue}{\underline{f}(\underline{z}, \underline{u}, t)}}$$