User:Eml5526.s11.team5.JA/Mtg40

Mtg 40: Mon, 11 Apr 11

[[media: Fe1.s11.mtg40.djvu| Page 40-1 ]]

Example 2



Boundary Conditions [[media: Fe1.s11.mtg35.djvu|  page 35-4 ]]

Mesh [[media: Fe1.s11.mtg29.djvu|  page 29-2 ]]


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$$ \eta = \left \{ 1, 2, 3, .., 16 \right \} \ $$

$$
 *  $$ \displaystyle
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$$ \eta_g = \left \{ 1, 2, 3, 4, {\color{Red}5}, 6, 7, 8, {\color{Red}9}, 10, 11, 12, {\color{Red}13}, {\color{Red}14}, {\color{Red}15}, 16 \right \} \ $$

$$
 *  $$ \displaystyle
 * }

[[media: Fe1.s11.mtg39.djvu| (4) page 39-3 ]] $$ \eta_{\color{Red}F} := \eta \setminus  \eta_g = \left \{ 6, 7, 10, 11 \right \} \ $$


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$$ \underbrace{\boldsymbol{ \Gamma}_{\color{Red}FF}}_{\color{Red}4x4} = \left [ < \underbrace{ b_{\color{Blue}i} }_{\color{Blue}row},\underbrace{ b_{\color{Red}j} }_{\color{Red}column}>, \ {\color{Blue}i}, {\color{Red}j} \ \in \ \eta_{\color{Red}F} \right ] \ $$

$$
 *  $$ \displaystyle
 * }

End Example 2

Note: Recall the capacitance operator $$ \tilde{m} \ $$ [[media: Fe1.s11.mtg34.djvu| (4)  page 34-3 ]] and the capacitance matrix $$ \mathbf{M} = [ M_{ij} ] [ \tilde{m} (b_i, b_j) \ $$ [[media: Fe1.s11.mtg22.djvu| (2)  page 22-3: ]]


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If $$ \rho c = {\color{Red}constant} \Rightarrow \mathbf{M} = \rho \ c \ \boldsymbol{ \Gamma}\Rightarrow \ $$ only need to compute $$ \mathbf{M} \ $$, and $$ \boldsymbol{ \Gamma} = \frac{1}{\rho c} \mathbf{M} \ $$.

$$
 *  $$ \displaystyle
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[[media: Fe1.s11.mtg40.djvu| Page 40-2 ]]

If $$ \rho \ c \ne {\color{Red}constant} \Rightarrow \mathbf{M} \ne \rho \ c \ \boldsymbol{\Gamma} \Rightarrow \ $$ better use capacitance operator $$ \tilde{m} \ $$ for projection, example:

, where subscript $$ {\color{Red}M} \ $$ represents "mass/capacitance"

Instead of [[media: Fe1.s11.mtg39.djvu| (2) page 39-2 ]], use


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$$ < w, u (\mathbf{x}, t=0) >_{\color{Red}M} = _{\color{Red}M} \ \forall \ w \ $$

$$
 *  $$ \displaystyle {\color{Red}(2)}
 * }

$$
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[[media: Fe1.s11.mtg40.djvu| Page 40-3 ]]

Summary

Use matlab ode 45 to integrate nonlinear ODE's

[[media: Fe1.s11.mtg40.djvu| Page 40-4 ]]

Vibrating membrane, Transient analytical: continued [[media: Fe1.s11.mtg38.djvu| Page 38-4 ]]

PDE:

$$ u(\mathbf{x}, t) = \ $$ Transverse displacement

$$ T = \ $$ tension (force/length) = constant

$$ f(\mathbf{x}, t) = \ $$ distributed transverse force

$$ \rho(\mathbf{x}) = \ $$ mass density (mass/area)

[[media: Fe1.s11.mtg40.djvu| Page 40-5 ]]

Find static solution $$ u^h_{\color{Red}s} \ $$ to $$ 10^{-6} \ $$ accuracy at center.

1) Dynamic (Transient): $$ \rho = 3 \ $$

2a) Free vibration: Solve general eigenvalue problem


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$$ \boldsymbol{\Kappa} \underbrace{\boldsymbol{\Phi}}_{ {\color{Blue}eigenvector} } = \underbrace{\lambda}_{ {\color{Blue}eigenvalue} } \mathbf{M} \boldsymbol{\Phi} \ $$

$$
 *  $$ \displaystyle {\color{Red}(1)}
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Reference: EML 4500 F08, page 41-2, found in http://en.wikiversity.org/wiki/User:Eml4500.f08

for first 3 eigenpairs $$ (\lambda_i, \boldsymbol{\Phi}_i ), \ i = 1, 2, 3 \ $$

Fundamental frequency: $$ \omega_1 = (\lambda_1)^{\color{Red}\frac{1}{2}}, \ f_1 = \frac{\omega_1}{2 \pi} \ $$

 Fundamental period:  $$ T_1 = \frac{1}{f_1} \ $$
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