Mtg 13: Tue, 21 Sept10

Application (Engineering): Motion of particle (e.g.,rocket) in air

Where ${\displaystyle v:=\left\Vert \mathbf {v} \right\|\ }$ , normal = magnitude

${\displaystyle k,n\in \mathbb {R} \ }$

${\displaystyle m=\ }$ mass of the particle

${\displaystyle g=\ }$ acceleration of gravity

HW 1) Derive equation of motion

${\displaystyle \displaystyle {\begin{aligned}m{\frac {dv_{x}}{dt}}=-kv^{n}\cos \alpha \ \end{aligned}}}$

(1)

${\displaystyle \displaystyle {\begin{aligned}m{\frac {dv_{y}}{dt}}=-kv^{n}\sin \alpha \ -mg\end{aligned}}}$

(2)

${\displaystyle \displaystyle {\begin{aligned}v^{2}=(v_{x})^{2}+(v_{y})^{2}\end{aligned}}}$

(3)

${\displaystyle \displaystyle {\begin{aligned}\tan \alpha \ ={\frac {dy}{dx}}={\frac {\frac {dy}{dt}}{\frac {dx}{dt}}}={\frac {v_{y}}{v_{x}}}\end{aligned}}}$

(4)

Where Eq.(1) and Eq.(2) are a system of coupled N1_ODEs = SC_N1_ODE (numerical methods)

2) Particular case ${\displaystyle k=0\ }$ : verify ${\displaystyle y(x)\ }$ is parabola

3) Consider ${\displaystyle k\neq 0\ }$ , ${\displaystyle v_{x0}=0\ }$

Eq.(2)P.13-1 ${\displaystyle \displaystyle {\begin{aligned}\Rightarrow \ m{\frac {dv_{y}}{dt}}=-k(v_{y})^{n}-mg\end{aligned}}}$

(1)

3.1) Find ${\displaystyle v_{y}(t),y(t)\ }$ for ${\displaystyle m=constant\ }$

3.2) ${\displaystyle m=m(t)\ }$

Is Eq.(1) "exact" (= either exact or can be made exact by integrating factor method) (IFM) END HW

Application (Engineering): Control Engineering - Linear System:
1) Time invariant
2) Time variant:

${\displaystyle \displaystyle {\begin{aligned}\mathbf {\dot {x}} (t)=\mathbf {A} (t)\mathbf {x} (t)+\mathbf {B} (t)\mathbf {u} (t)\end{aligned}}}$

(2)

Where ${\displaystyle \mathbf {\dot {x}} (t)\ }$ is a nx1 matrix,
${\displaystyle \mathbf {A} (t)\ }$ is a nxn matrix,
${\displaystyle \mathbf {x} (t)\ }$ is a nx1 matrix,
${\displaystyle \mathbf {B} (t)\ }$ is a nxm matrix,
${\displaystyle \mathbf {u} (t)\ }$ is a mx1 matrix

Time invariant ${\displaystyle \mathbf {A} ,\mathbf {B} \ }$ constant

Eq.(2)P.13-2 : SC_L1_ODEs

EXAMPLE Application (Engineering)

END EXAMPLE
HW 1) Derive

${\displaystyle \displaystyle {\begin{aligned}m_{1}l^{2}{\ddot {\theta _{1}}}=-ka^{2}(\theta _{1}-\theta _{2})-m_{1}gl\theta _{1}+u_{1}l\end{aligned}}}$

(1)

${\displaystyle \displaystyle {\begin{aligned}m_{2}l^{2}{\ddot {\theta _{2}}}=-ka^{2}(\theta _{2}-\theta _{1})-m_{2}gl\theta _{1}+u_{2}l\end{aligned}}}$

(2)

2) Write Eq.(1) and Eq.(2) in form of Eq.(2)P.13-2

${\displaystyle \mathbf {x} :=\left\lfloor \theta _{1}\ {\dot {\theta _{1}}}\ \theta _{2}\ {\dot {\theta _{2}}}\right\rfloor \ ^{T}}$

Where ${\displaystyle \mathbf {x} }$ is a 4x1 matrix

Find ${\displaystyle \mathbf {A} ,\mathbf {B} ,\mathbf {u} :={\begin{Bmatrix}u_{1}l\\u_{2}l\end{Bmatrix}}\ }$

Where ${\displaystyle \mathbf {A} }$ is a 4x4 matrix
${\displaystyle \mathbf {B} }$ is a 4x2 matrix
${\displaystyle \mathbf {u} }$ is a 2x1 matrix END HW