University of Florida/Egm6341/s11.TEAM1.WILKS/Mtg3

EGM6321 - Principles of Engineering Analysis 1, Fall 2010

Mtg 3: Thur, 26 Aug 10

Page 3-1

NOTE: - page numbering 3-1 defined as meeting number 3, page 1

- T = torque Fig.p.1-1

- HW*

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)\$

$\equiv \$ Partial Differential Equations (PDE)

3 equations are coupled $\Rightarrow \ \$ Numerical Methods

Simplify for analytical solution Ref:VQ&O 1989

2nd Order $\rightarrow \ \$ 2nd Order

nonlinear $\rightarrow \ \$ linear

unknown varying coefficient $\rightarrow \ \$ known varying coefficient

Page 3-2

Note: Math structure of coefficient $c_{i}(Y',t)\$ for $i=0,1,...3\$ is known, but not their values until $u^{1}\$ and $u^{2}\$ are known (solved for)

General structure of Linear 2nd order ODEs with varying coefficients (L2_ODE_VC)

$\displaystyle {\begin{aligned}P(x)y''+Q(x)y'+R(x)y=F(x)\end{aligned}}$

(1)

where $y''={\frac {d^{2}y}{dx^{2}}}\$

$x=\$ independant variable

$y(x)=\$ dependant variable (unknown function to solve for)

Many applications in engineering are a result of solving PDEs by separation of variables. Some examples include, but are not limited to: Heat, Solids, Fluids, Acoustics and electrmagnetics.

Examples of these types equations are:

the Helmholz equation: $\Delta \ X+k^{2}X=0\$

and the Laplace Equation: $\Delta \ X=0\$

Ref F09 Mtg.28, Ref F09 Mtg.29 , Ref F09 Mtg.30

Page 3-3

In 3_D, $x=(x_{1},x_{2},x_{3})\$

$\displaystyle {\begin{aligned}X(x)=X_{1}(x_{1})X_{2}(x_{2})X_{3}(x_{3})\end{aligned}}$

(1)

Where the lowercase $x\$ in the first 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

$\displaystyle {\begin{aligned}X(\xi \ )=X_{1}(\xi \ _{1})X_{2}(\xi \ _{2})X_{3}(\xi \ _{3})\end{aligned}}$

(2)

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\$

$\displaystyle {\begin{aligned}{\frac {1}{g_{i}(\xi \ _{i})}}{\frac {d}{d\xi \ _{i}}}\left[g_{i}(\xi \ _{i}){\frac {dX_{i}(\xi \ _{i})}{d\xi \ _{i}}}\right]+f_{i}(\xi \ _{i})X_{i}(\xi \ _{i})=0\end{aligned}}$

(3)

Page 3-4

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)\$

Eq.(3)p.3-3:

$\displaystyle {\begin{aligned}y''+{\frac {g'(x)}{g(x)}}y'+a_{0}(x)y=0\end{aligned}}$

(1)

Where ${\frac {g'(x)}{g(x)}}=a_{1}(x)\$

Particular case of Eq.(1)p.3-2

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.