User:EGM6341.s11.TEAM1.WILKS/Mtg13

=EGM6321 - Principles of Engineering Analysis 1, Fall 2010= Mtg 13: Tue, 21 Sep 10

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

where $$ v := \left \Vert \boldsymbol v \right \| \ $$, norm = magnitude $$ k,n \in \mathbb R \ $$ $$ m= \ $$ mass of the particle $$ g= \ $$ acceleration of gravity HW  1) Derive equation of motion Where Eq.(1) and Eq.(2) are a system of coupled N1_ODEs = SC_N1_ODE (numerical methods) 2) Particular case $$ k=0 \ $$ : verify $$ y(x) \ $$ is parabola

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3) Consider $$ k \ne 0 \ $$, $$ v_{x0}=0 \ $$ 3.1) Find $$ v_y(t), y(t) \ $$ for $$ m=constant \ $$ 3.2) $$ m=m(t) \ $$

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

= Application: Control Engineering - Linear Systems =

1) Time invariant 2) Time variant: Where $$ \mathbf{\dot{x}}(t) \ $$ is a nx1 matrix, $$ \mathbf{A} (t) \ $$ is a nxn matrix, $$ \mathbf{x}(t) \ $$ is a nx1 matrix, $$ \mathbf{B}(t) \ $$ is a nxm matrix, $$ \mathbf{u} (t) \ $$ is a mx1 matrix

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Time invariant $$ \mathbf{A}, \mathbf{B} \ $$ constant [[media: 2010_09_21_14_56_48.djvu | Eq.(2)P.13-2]] : SC-L1-ODEs EXAMPLE Application (Engineering)

END EXAMPLE HW 1) Derive 2) Write Eq.(1) and Eq.(2) in form of [[media: 2010_09_21_14_56_48.djvu | Eq.(2)P.13-2]] $$ \mathbf{x}:= \left \lfloor \theta_1 \ \dot{ \theta_1 } \ \theta_2 \ \dot{ \theta_2 }    \right \rfloor \ ^T $$ Where $$ \mathbf{x} $$ is a 4x1 matrix Find $$ \mathbf{A}, \mathbf{B} , \mathbf{u}:= \begin{Bmatrix} u_1l \\ u_2l \end{Bmatrix} \ $$ Where $$ \mathbf{A} $$ is a 4x4 matrix $$ \mathbf{B} $$ is a 4x2 matrix $$ \mathbf{u} $$ is a 2x1 matrix END HW