User:Eml5526.s11.team01.roark/Mtg13

[[media: Fe1.s11.mtg13.djvu| Mtg 13]]: Fri, 28 Jan 11 [[media: Fe1.s11.mtg13.djvu| Page 13-1]] Note:

HW 1.1 [[media: Fe1.s11.mtg5.djvu| p. 5-6]] FBD
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$$\displaystyle \mathop{\sum }^{}forces=\underbrace{N\left( x+dx \right)}_{\sigma (x+dx)A(x+dx)\cdot \underbrace{1}_{n(x+dx)}+\left[ \sigma (x)A(x)\cdot \underbrace{1}_{n(x)} \right]}+N(x)$$


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$$\displaystyle ={{\left. \underbrace{\left( \sigma A \right)}_{T} \right|}_{x+dx}}-{{\underbrace{\left. \left( \sigma A \right) \right|}_{T}}_{x}}$$ $$\displaystyle ~=T\left( x+dx \right)-T(x)$$ $$\displaystyle \frac{dT(x)}{dx}dx+\underbrace{hot}_{d{{x}^{2}},d{{x}^{3}},...}$$

Inertia Force [[media: Fe1.s11.mtg13.djvu| Page 13-2]]
 * 1)  $$\displaystyle {{F}_{I}}=p\left( x \right)A\left( x \right)dx\frac{{{\partial }^{2}}u}{\partial {{t}^{2}}}$$ [[media: Fe1.s11.mtg6.djvu| p. 6-1]]
 * 2)  $$\displaystyle {{F}_{I}}=\underbrace{p\left( x+dx \right)}_{\left( 1 \right)}\frac{1}{2}\left[ h\left( x \right)+h(x+dx) \right]bdx\frac{{{\partial }^{2}}u}{\partial {{t}^{2}}}$$

TS (Taylor series)

$$\displaystyle \left( 1 \right)=\rho \left( x+\frac{dx}{2} \right)=\rho (x)+\frac{d\rho (x)}{dx}\frac{dx}{2}+hot$$

$$\displaystyle \left( 2 \right)=\frac{1}{2}\left[ h(x)+\underbrace{h(x+dx)}_{h(x)+\frac{dh(x)}{dx}dx+hot} \right]=h(x)+\frac{1}{2}\frac{dh(x)}{dx}dx+hot$$

$$\displaystyle \left( 1 \right)\times \left( 2 \right)=\rho (x)h(x)+\underbrace{\left[ \rho \frac{dh}{dx}+h\frac{d\rho }{dx} \right]}_{\frac{d}{dx}\left( \rho h \right)}\frac{dx}{2}+hot=\left( \rho h \right)+\frac{d}{dx}\left( \rho h \right)\frac{dx}{2}+hot$$

$$\displaystyle {{F}_{I}}=\left[ \rho hbdx+hot \right]\frac{{{\partial }^{2}}u}{\partial {{t}^{2}}}$$

Neglect higher order terms -> back to case 1

End Note [[media: Fe1.s11.mtg13.djvu| Page 13-3]]

HW 2.9

[[media: Fe1.s11.mtg12.djvu| p. 12-3]] Continued 8) Compute $$\displaystyle u_{n}^{h}(x=0.5)$$ for n=2,4,6. Error $$\displaystyle {{e}_{n}}(0.5)=u(0.5)-{{u}^{h}}\left( 0.5 \right)$$ Plot $$\displaystyle {{e}_{n}}(0.5)$$ vs. n (convergence)

End HW 2.9

HW 3.1:

Repeat HW 2.9 w/ polynomial basis functions $$\displaystyle \left\{ {{\left( x+k \right)}^{j}},j=0,1,2,...,n \right\}$$ Choose k=1 to avoid $$\displaystyle {{{b}'}_{j}}(0)=0$$.

End HW 3.1

Remember:

Actually no need to shift basis functions $$\displaystyle \left\{ {{x}^{i}} \right\}$$ (unlike $$\displaystyle \left\{ \cos jx \right\}$$) since there is no “singularity” at x=0 i.e., $$\displaystyle {{{b}'}_{j}}(0)\ne 0$$ for some j’s, since

$$\displaystyle \left\{ {{b}_{j}}(x) \right\}=\left\{ {{x}^{j}} \right\}=\left\{ 1,x,{{x}^{2}},... \right\}$$

$$\displaystyle \left\{ {{{{b}'}}_{j}}(0) \right\}=\left\{ 0,\underbrace{1,}_{{{{{b}'}}_{1}}(0)}0,0... \right\}$$

See HW in [[media: Fe1.s11.mtg17.djvu| Mtg 17]].

End Section