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 $\to$ 2nd Order

nonlinear $\to$ linear

unknown varying coefficient $\to$ known varying coefficient

Page 3-2

Note: Math structure of coefficient $c_i(Y^{\prime },t)$ 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)

$\begin{array}{c}P(x)y^{″}+Q(x)y^{\prime }+R(x)y=F(x)\end{array}$

where $y^{″}=\frac{d^{2}y}{d{x}^2}$

$x=$ 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: $\mathrm{\Delta }X+k^{2}X=0$

and the Laplace Equation: $\mathrm{\Delta }X=0$

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

Page 3-3

In 3_D, $x=(x_1,x_2,x_3)$

$\begin{array}{c}X(x)=X_1(x_1)X_2(x_2)X_3(x_3)\end{array}$

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

$\begin{array}{c}X(\xi )=X_1(\xi {}_1)X_2(\xi {}_2)X_3(\xi {}_3)\end{array}$

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 $i=1,2,3$

$\begin{array}{c}\frac{1}{g_i(\xi {}_i)}\frac{d}{d\xi {}_i}[g_i(\xi {}_i)\frac{d{X}_i(\xi {}_i)}{d\xi {}_i}]+f_i(\xi {}_i)X_i(\xi {}_i)=0\end{array}$

Page 3-4

Simplify:

$\xi {}_i\to x$

$X_i(\xi {}_i)\to y(x)$

$g_i(\xi {}_i)\to g(x)$

$f_i(\xi {}_i)\to a_0(x)$

Eq.(3)p.3-3:

$\begin{array}{c}{y}^{″}+\frac{g^{\prime }(x)}{g(x)}{y}^{\prime }+a_0(x)y=0\end{array}$

Where $\frac{g^{\prime }(x)}{g(x)}=a_1(x)$

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.

References

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