User:EGM6341.s11.TEAM1.WILKS/Mtg3

=EGM6321 - Principles of Engineering Analysis 1, Fall 2010= Mtg 3: Thur, 26 Aug 10

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NOTE: - page numbering 3-1 defined as meeting number 3, page 1 - T = torque [[media: 2010_08_24_15_22_30.djvu | Fig.p.1-1]] - HW* END NOTE [[media: 2010_08_26_13_57_13.djvu | Eq.(3)P.2-1]] : "Ordinary" Differential Equation (ODE) order = highest order of derivative Nonlinearity = What is linearity? ; use intuition for now, formal definition soon. System has 3 unknowns: $$ Y^1(t) \ $$ $$ u^1(S,t) \ $$ $$ u^2(S,t) \ $$ 1 nonlinear ODE (Ordinary Differential Equation) for $$ Y^1(t) \ $$, and 2 PDEs (Partial Differential Equations) for $$ u^1(S,t) \ $$ and $$ u^2(S,t) \ $$. 3 equations are coupled $$ \Rightarrow \ \ $$ Numerical Methods; Ref: VQ&O 1989

= Simplify for analytical solution =

2nd Order $$ \rightarrow \ \ $$ 2nd Order nonlinear $$ \rightarrow \ \ $$ linear unknown varying coefficient $$ \rightarrow \ \ $$ known varying coefficient

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NOTE: Math structure of coefficient  $$ c_i(Y^1,t) \ $$ for $$ i=0,1,...3 \ $$ is known, but not their values until $$ u^1 \ $$ and $$ u^2 \ $$ are known (solved for) END NOTE General structure of Linear 2nd order ODEs with varying coefficients (L2-ODE-VC) where $$ y''=\frac{d^2y}{dx^2} \ $$ $$ x= \ $$ independent variable $$ y(x)= \ $$ dependent variable (unknown function to solve for) Many linear PDEs in engineering applications can be solved by separation of variables. Some examples include, but are not limited to: Heat, solids, fluids, acoustics and electromagnetics. Examples of these types equations are: the Helmholz equation: $$ \Delta X+k^2X=0 \ $$ and the Laplace equation: $$ \Delta X=0 \ $$ [[media: Egm6321.f09.mtg28.djvu| Ref F09 Mtg.28]], [[media: Egm6321.f09.mtg29.djvu| Ref F09 Mtg.29]] , [[media: Egm6321.f09.mtg30.djvu| Ref F09 Mtg.30]]

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In 3-D,, a point $$\displaystyle x$$ has coordinates $$\displaystyle (x_1,x_2,x_3)$$ and is written as $$\displaystyle x=(x_1,x_2,x_3)$$.

where the lowercase $$ x \ $$ in the left-hand-side term $$ X(x) \ $$ is defined as $$ x=(x_1,x_2,x_3) \ $$ and $$ X_1(x_1)X_2(x_2)X_3(x_3) \ $$ is the separation of variables



[[media:EGM6341.s11.TEAM1.WILKS EC3.vql.svg|figure svg file]] Where $$ \xi\ \ $$ in the first term $$ X( \xi\ ) \ $$ is defined as $$ \xi\ =( \xi_1, \xi_2, \xi_3) \ $$ and $$ X_1( \xi_1)X_2( \xi_2)X_3( \xi_3) \ $$ is the separation of variables Separated equations for $$ i=1,2,3 \ $$

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Simplify: $$ \xi_i \rightarrow \ x \ $$ $$ X_i ( \xi_i) \rightarrow \ y(x) \ $$ $$ g_i ( \xi_i) \rightarrow \ g(x) \ $$ $$ f_i ( \xi_i) \rightarrow \ a_0(x) \ $$ [[media: 2010_08_26_14_53_13.djvu | Eq.(3)p.3-3]]: Where $$ \frac{g'(x)}{g(x)} = a_1(x) \ $$ Particular case of [[media: 2010_08_26_14_53_13.djvu | Eq.(1)p.3-2]]

= Definition: Linearity =

Let $$ F(.) \ $$ be an operator. $$ u \ $$ and $$ v \ $$ are 2 possible arguments (could be functions) of $$ F(.) \ $$ $$ F( \alpha\ u+ \beta\ v = \alpha\ F(u) + \beta\ F(v) \ $$ Where $$ \alpha\ \ $$ and $$ \beta\ \ $$ are any arbitrary number.

END Definition