User:EGM6341.s11.TEAM1.WILKS/Mtg15

=EGM6321 - Principles of Engineering Analysis 1, Fall 2010=

Mtg 15: Thu, 23 Sep 10

[[media: 2010_09_23_14_52_54.djvu | Page 15-1]]
SC-L1-ODE-CC:

HW Use [[media:2010_09_23_13_57_17.djvu  | Eq.(1) and Eq.(2) p.14-3]], together with Eq.(1) to obtain [[media:2010_09_23_13_57_17.djvu | Eq.(4)p.14-1, p.14-2]] END HW

= Example: Roll control of rocket - prevent rolling by activating ailerons =

Bryson & Ho 1975 p.169; F10 Lecture plan $$ \delta\ = \ $$ aileron angle (deflection) $$ \phi\ = \ $$ roll angle $$ \omega\ = \ $$ roll angle velocity $$ Q = \ $$ aileron effectiveness $$ \tau\ = \ $$ roll time constant

[[media: 2010_09_23_14_52_54.djvu | Page 15-2]]
HW  Put Eq.(2) thru Eq.(4) p.15-1 in form of Eq(1)p.15-1 END HW

= 2nd order ODEs (cont'd from [[media:2010_09_16_14_59_27.djvu | Page 12-4]]) =

General N2-ODE: $$ F(x,y,y',y'')=0 \ $$

$$ F(.)=0 \ $$ is exact means $$ \exists \ \phi\ (x,y,y') =k \ $$ such that $$ F(x,y,y',y'') = \frac{d}{dx} \phi\ (x,y,y') \ $$

Define $$ p := y' \ $$ and thus $$ y'' = p' \ $$.

where Eq.(2) is equivalent to the first condition of exactness.

[[media: 2010_09_23_14_52_54.djvu | Page 15-3]]
2nd condition of exactness:

If both exactness conditions are satisfied, then from Eq.(4)p.15-2, we have where $$ h(x,y) \ $$ is a function of integration determined using Eq.(3) p.15-2.

Application:

Consider $$ x(y')^2+yy'+(xy)y''=0 \ $$ Where $$ x(y')^2+yy' = g(x,y,y') \ $$ and $$ (xy) = f(x,y,y') \ $$ and $$ y'=p \ $$

1st exact condition (1) is satisfied.

2nd exact condition (2) is also satisfied (F09 HW; see [[media: egm6321.f09.mtg10.djvu | F09 Mtg 10, p10-3]]).

NOTE:

For lectures on the exact conditions for N2-ODE, see: F09 Lecture plan, [[media: egm6321.f09.mtg10.djvu | Mtg 10]], [[media: egm6321.f09.mtg11.djvu | Mtg 11]], [[media: egm6321.f09.mtg12.djvu | Mtg 12]], [[media: egm6321.f09.mtg13.djvu | Mtg 13]], [[media: egm6321.f09.mtg14.djvu | Mtg 14]].

ENDNOTE

$$ f_x = y \Rightarrow \ f_{xx}=0 \ $$

[[media: 2010_09_23_14_52_54.djvu | Page 15-4]]
$$ g_y=p \Rightarrow \ g_{yp}=1 \ $$ Eq.(3)p15-3 $$ \Rightarrow \ \ $$ where $$ \int_{}^{} (xy)\, dp=xyp \ $$ using definition of g in Eq(3)p.15-2: