User:Eml5526.s11.team01.roark/Mtg23

Mtg 23: Wed, 23 Feb 11

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

Paraglider

Disk

Sphere-volume

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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 p. 22-4

Use (5)-(6) p. 22-1, (1)-(7) p. 22-3, (1)-(6) p. 22-3, (1)-(3) p. 22-4 =>

$\displaystyle \underbrace{{{c}_{E}}}_{=0}\cdot \underbrace{\left[ \left( {{\mathbf M}_{EE}}{{g}^{\left( s \right)}}+{{ \mathbf K}_{EE}}g \right)+{{ \mathbf M}_{Ef}}{{\mathbf d}_{F}}^{\left( s \right)}+{{ \mathbf K}_{EF}}{{\mathbf d}_{F}}-{{\mathbf F}_{E}} \right]}_{\ne 0}=0$

(1)

DWF-U (unconstrained):

$\displaystyle {{\mathbf c}_{F}}\cdot \left[ \underbrace{\left( {{\mathbf M}_{FE}}{{g}^{\left( s \right)}}+{{ \mathbf K}_{FE}}g \right)}_{Known}+{{\mathbf M}_{FF}}{{\mathbf d}_{F}}^{\left( s \right)}+{{ \mathbf K}_{FF}}{{\mathbf d}_{F}}-{{\mathbf F}_{F}} \right]=0\,\,\,\forall {{\mathbf c}_{F}}$

(2)

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since $\displaystyle {{c}_{F}}$ is unconstrained, => select arbitrarily

$\displaystyle {{\mathbf c}_{F}}=\underbrace{\underset{1,2,...,n}{\mathop{\left\{ 1,0,...,0 \right\}}}\,,\underset{1,2,3,...,n}{\mathop{\left\{ 0,1,...,0 \right\}}}\,,\underset{1,...,n-1,n}{\mathop{\left\{ 0,...,0,1 \right\}}}\,,}_{n\,different\,choices\,for\,{{\mathbf c}_{F}}}$

(1)

$\displaystyle \underbrace{{{\mathbf{M}}_{FF}}}_{\begin{smallmatrix} \mathbf{M} \\ nxn \end{smallmatrix}}\underbrace{{{\mathbf{d}}_{F}}^{\left( s \right)}}_{\begin{smallmatrix} {{\mathbf{d}}^{\left( s \right)}} \\ nx1 \end{smallmatrix}}+\underbrace{{{\mathbf{K}}_{FF}}}_{\begin{smallmatrix} \mathbf{K} \\ nxn \end{smallmatrix}}\underbrace{{{\mathbf{d}}_{F}}}_{\begin{smallmatrix} \mathbf{d} \\ nx1 \end{smallmatrix}}=\underbrace{{{\mathbf{F}}_{F}}-\left( {{\mathbf{M}}_{FE}}{{g}^{\left( s \right)}}+{{\mathbf{K}}_{FE}}g \right)}_{\begin{smallmatrix} \mathbf{F} \\ nx1 \end{smallmatrix}}$

(2)

$\displaystyle \mathbf{M}{{\mathbf{d}}^{\left( s \right)}}+\mathbf{Kd}=\mathbf{F}$ (3)

Static: $\displaystyle {{\mathbf{d}}^{\left( s \right)}}=\underset{nx1}{\mathop{\mathbf 0}}\,,{{g}^{s}}=0$ (since g=constant)

$\displaystyle \mathbf{Kd}=\mathbf{F}$, $\displaystyle \mathbf{F}={{\mathbf{F}}_{F}}-{{\mathbf{K}}_{FE}}g$ (4)

Application: FB, p.72, Pb.3.4. $\displaystyle \left\{ {{b}_{j}} \right\}=\left\{ {{\left( x-3 \right)}^{j}},j=0,1,...,n \right\}$ $\displaystyle \mathbf{K}={{\left[ {{K}_{ij}};i,j-1,n \right]}_{nxn}}$

$\displaystyle {{K}_{ij}}=\int_{\alpha }^{\beta }{{{b}_{i}}^{\prime }{{a}_{2}}{{b}_{j}}^{\prime }dx}$

(5)

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(7) p. 22-3:

$\displaystyle {{K}_{ij}}={{\int_{\alpha }^{\beta }{\underbrace{i\left( x-3 \right)}_{{{b}_{i}}^{\prime }}}}^{i-1}}{{a}_{2}}\underbrace{j{{\left( x-3 \right)}^{j-1}}}_{{{b}_{j}}^{\prime }}dx$

(1)

(5)-(6) p. 22-3: $\displaystyle \underset{nx1}{\mathop {{\mathbf{K}}_{FE}}}\,=\left[ {{K}_{\underbrace{i}_{row}0}},i=1,...,n \right]$ , $\displaystyle {{K}_{i0}}=\int_{\alpha }^{\beta }{{{b}_{i}}^{\prime }{{a}_{2}}\underbrace{{{b}_{0}}^{\prime }}_{=0}dx}=0$

Also (5)-(6) p. 22-3: $\displaystyle {{\mathbf{K}}_{FE}}=0=\mathbf{K}_{EF}^{T}$ and from (4) p. 23-3: $\displaystyle \mathbf{F}={{\mathbf{F}}_{F}}$

Structure of $\displaystyle {{\mathbf{\tilde{K}}}_{\left( n+1 \right)\times \left( n+1 \right)}}$ (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