User:Eml5526.s11.team5.JA/Mtg6

 Mtg 6: Sat, 15 Jan 11

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1) Do balance of forces $$ \rightarrow \ $$ PDE

2) Discuss the case in which the bar has a rectangular cross section as shown below




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$$ \underbrace{ \rho \left ( x + \frac{dx}{2} \right ) \frac{1}{2} \left [ h(x) + h(x+dx) \right ] b }_{\color{Blue} m} \ $$

$$
 *  $$ \displaystyle
 * }

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Use the above mass per unit length in the inertia force and perform balance of forces again to derive PDE




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$$ q(x) \ $$ = heat flux at x

$$ Q(x) \ $$ = heat flow (outflow)  though $$ A(x) \ $$ with $$ \underbrace{ n(x)}_{\color{Blue} outward \ normal } = {\color{Red}-1}  $$ (outward normal)

$$ Q(x) = q(x)A(x)\underbrace{n(x)}_{\color{Blue} -1} = - q(x)A(x) \ $$

$$
 *  $$ \displaystyle
 * }

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$$ q(x+dx) \ $$ = heat flux at $$ x + dx \ $$

$$ Q(x+dx) \ $$ = heat outflow  through $$ A(x + dx) \ $$ with outward normal $$ n(x+dx) = {\color{Red}+1} \ $$

$$ Q(x+dx) = q(x+dx) \ A(x + dx) \ \underbrace{n(x + dx)}_{\color{Blue} +1} = q(x+dx) \ A(x + dx) \ $$

$$ r(x,t) \ $$ = heat source (per volume)

$$ r(x, {\color{Red}t})A(x) = f(x,{\color{Red}t}) \ $$ = heat source per length

$$
 *  $$ \displaystyle
 * }

Balance of heat:

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$$ {\color{Blue}H_1} \ $$ = heat flow  into  $$ \omega \ $$ (control volume) = $$ \underbrace{\color{Red}-}_{\color{Blue} inflow} \left [ \underbrace{Q(x)}_{\color{Blue} outflow} + \underbrace{Q(x+dx)}_{\color{Blue}outflow} \right ] \ $$

$$
 *  $$ \displaystyle
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$$ {\color{Blue}H_2} \ $$ = heat generated by $$ r(x,t) \ $$

$$
 *  $$ \displaystyle
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$$ {\color{Blue}H_2} = f(x,t) \ dx \ $$

$$ {\color{Blue}H_3} \ $$ = heat due to change in temperature = $$ \underbrace{ \rho(x) \ A(x)}_{\color{Blue}m(x)} c \  \frac{\partial u}{\partial t} \ $$

$$
 *  $$ \displaystyle
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Balance of heat:


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$$
 *  $$ \displaystyle {\color{Red}(1)}
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Generalized 1-D PDE ([[media: Fe1.s11.mtg4.djvu| page 4-4 ]]) continued


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$$ P(u):= \frac{\partial}{\partial x} \left [ a_2(x) \frac{\partial u}{\partial x} \right ] + f(x,t) - \bar{m} \frac{\partial^{\color{Red}s} u}{\partial t^{\color{Red}s}} \ $$

$$
 * <p style="text-align:right"> $$ \displaystyle {\color{Red}(2)}
 * }

<span id="(1)">
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$$

\bar{m} := \begin{cases} m, & for \ {\color{Red} s = 2} \ {\color{Blue}(elastic \ bar) }\\ mc, & for \ {\color{Red}s = 1} {\color{Blue}(1-D \ heat) } \end{cases}

$$

$$
 * <p style="text-align:right"> $$ \displaystyle {\color{Red}(3)}
 * }

Weighted Residual Form (WRF)
<span id="(1)">
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$$
 * <p style="text-align:right"> $$ \displaystyle {\color{Red}(4)}
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$$ \int_{\Omega} w(x) P(u) dx = 0 \Leftrightarrow \underbrace{P(u) = 0}_{\color{Blue} PDE} \ $$ (cf. F&B, p.47 [3.13a])

$$
 * <p style="text-align:right"> $$ \displaystyle {\color{Red}(5)}
 * }