User:Eml5526.s11.team01.roark/Mtg23

[[media: Fe1.s11.mtg23.djvu| Mtg 23]]: Wed, 23 Feb 11 [[media: Fe1.s11.mtg23.djvu| Page 23-1]]

HW 4.6:

FB, P. 74, Problem 3.9

End HW 4.6

HW 4.7:

Calculix (nonlinear FE code, open-source, ABAQUS-like input) : http://dhondt.de/

End HW 4.6

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Disk

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[[media: Fe1.s11.mtg23.djvu| Page 23-2]]
 * 1) install cgx (calculix graphics module)
 * 2) Read manual, sign up with user group to ask questions if any. Also access archive.
 * 3) Reproduce basic examples: Disk, cylinder, sphere, sphere-volume, airfoil,
 * 4) Write report for “dummies”: Explain to novices how to install and run CGX (all CGX commands in basic examples, screenshots, …)

DWF [[media: Fe1.s11.mtg22.djvu| p. 22-4]]

Use [[media: Fe1.s11.mtg22.djvu|(5)-(6) p. 22-1]], [[media: Fe1.s11.mtg22.djvu|(1)-(7) p. 22-3]], [[media: Fe1.s11.mtg22.djvu|(1)-(6) p. 22-3]], [[media: Fe1.s11.mtg22.djvu|(1)-(3) p. 22-4]] =>
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$$\displaystyle \underbrace_{=0}\cdot \underbrace{\left[ \left( {{\mathbf M}_{EE}}{{g}^{\left( s \right)}}+{{ \mathbf K}_{EE}}g \right)+{{ \mathbf M}_{Ef}}{{\mathbf d}_{F}}^{\left( s \right)}+{{ \mathbf K}_{EF}}{{\mathbf d}_{F}}-{{\mathbf F}_{E}} \right]}_{\ne 0}=0$$ DWF-U (unconstrained):
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$$\displaystyle {{\mathbf c}_{F}}\cdot \left[ \underbrace{\left( {{\mathbf M}_{FE}}{{g}^{\left( s \right)}}+{{ \mathbf K}_{FE}}g \right)}_{Known}+{{\mathbf M}_{FF}}{{\mathbf d}_{F}}^{\left( s \right)}+{{ \mathbf K}_{FF}}{{\mathbf d}_{F}}-{{\mathbf F}_{F}} \right]=0\,\,\,\forall {{\mathbf c}_{F}}$$ [[media: Fe1.s11.mtg23.djvu| Page 23-3]] since $$\displaystyle {{c}_{F}}$$ is unconstrained, => select arbitrarily
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 * $$\displaystyle {{\mathbf c}_{F}}=\underbrace{\underset{1,2,...,n}{\mathop{\left\{ 1,0,...,0 \right\}}}\,,\underset{1,2,3,...,n}{\mathop{\left\{ 0,1,...,0 \right\}}}\,,\underset{1,...,n-1,n}{\mathop{\left\{ 0,...,0,1 \right\}}}\,,}_{n\,different\,choices\,for\,{{\mathbf c}_{F}}}$$
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$$\displaystyle \underbrace_{\begin{smallmatrix} \mathbf{M} \\ nxn \end{smallmatrix}}\underbrace{{{\mathbf{d}}_{F}}^{\left( s \right)}}_{\begin{smallmatrix} {{\mathbf{d}}^{\left( s \right)}} \\ nx1 \end{smallmatrix}}+\underbrace_{\begin{smallmatrix} \mathbf{K} \\ nxn \end{smallmatrix}}\underbrace_{\begin{smallmatrix} \mathbf{d} \\ nx1 \end{smallmatrix}}=\underbrace{{{\mathbf{F}}_{F}}-\left( {{\mathbf{M}}_{FE}}{{g}^{\left( s \right)}}+{{\mathbf{K}}_{FE}}g \right)}_{\begin{smallmatrix} \mathbf{F} \\ nx1 \end{smallmatrix}}$$
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$$\displaystyle \mathbf{M}{{\mathbf{d}}^{\left( s \right)}}+\mathbf{Kd}=\mathbf{F}$$
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Static: $$\displaystyle {{\mathbf{d}}^{\left( s \right)}}=\underset{nx1}{\mathop{\mathbf 0}}\,,{{g}^{s}}=0$$ (since g=constant)


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$$\displaystyle \mathbf{Kd}=\mathbf{F}$$, $$\displaystyle \mathbf{F}={{\mathbf{F}}_{F}}-{{\mathbf{K}}_{FE}}g$$
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Application: FB, p.72, Pb.3.4. $$\displaystyle \left\{ {{b}_{j}} \right\}=\left\{ {{\left( x-3 \right)}^{j}},j=0,1,...,n \right\}$$ $$\displaystyle \mathbf{K}={{\left[ {{K}_{ij}};i,j-1,n \right]}_{nxn}}$$
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$$\displaystyle {{K}_{ij}}=\int_{\alpha }^{\beta }{{{b}_{i}}^{\prime }{{a}_{2}}{{b}_{j}}^{\prime }dx}$$ [[media: Fe1.s11.mtg23.djvu| Page 23-4]] [[media: Fe1.s11.mtg22.djvu|(7) p. 22-3]]:
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$$\displaystyle {{K}_{ij}}={{\int_{\alpha }^{\beta }{\underbrace{i\left( x-3 \right)}_{{{b}_{i}}^{\prime }}}}^{i-1}}{{a}_{2}}\underbrace{j{{\left( x-3 \right)}^{j-1}}}_{{{b}_{j}}^{\prime }}dx$$ [[media: Fe1.s11.mtg22.djvu|(5)-(6) p. 22-3]]: $$\displaystyle \underset{nx1}{\mathop {{\mathbf{K}}_{FE}}}\,=\left[ {{K}_{\underbrace{i}_{row}0}},i=1,...,n \right]$$, $$\displaystyle {{K}_{i0}}=\int_{\alpha }^{\beta }{{{b}_{i}}^{\prime }{{a}_{2}}\underbrace{{{b}_{0}}^{\prime }}_{=0}dx}=0$$
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Also [[media: Fe1.s11.mtg22.djvu|(5)-(6) p. 22-3]]: $$\displaystyle {{\mathbf{K}}_{FE}}=0=\mathbf{K}_{EF}^{T}$$ and from [[media: Fe1.s11.mtg23.djvu|(4) p. 23-3]]: $$\displaystyle \mathbf{F}={{\mathbf{F}}_{F}}$$
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Structure of $$\displaystyle {{\mathbf{\tilde{K}}}_{\left( n+1 \right)\times \left( n+1 \right)}}$$ [[media: Fe1.s11.mtg22.djvu|(3) p. 22-3]]

$$\displaystyle {{\mathbf{\tilde{K}}}_{\left( n+1 \right)\times \left( n+1 \right)}}=\left[ \begin{matrix} {{K}_{\infty }} & \underbrace{0...0}_{1\times n} \\ \underbrace{\begin{matrix} 0 \\   :  \\   0  \\ \end{matrix}}_{n\times 1} & \mathbf{K}  \\ \end{matrix} \right]$$

HW 4.8:

Find $$\displaystyle \tilde{\mathbf M}$$ for FB, p.72, pb.3.4, assuming A=1, E=2, $$\displaystyle \mathbf m=3$$ do for n=3. For dynamics, with $$\displaystyle {{u}^{h}}(\beta ,t)=g(t)=\sin 2t$$, find $$\displaystyle \mathbf F(t)$$

End HW 4.8