EGM6341.s11.TEAM1.WILKS/Mtg1

EGM6321 - Principles of Engineering Analysis 1, Fall 2010

Mtg 1: Tue, 24 Aug 10

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- course website, wiki

- high-speed trains

German transrapid (electromagentic attraction)

Japanese MAglev (electrodynamic repulsion)

Grench TGV (wheel on rail)

Vu Quoc and Olsson 1989 CMAME

vehicle/structure interaction, where vehicle is the high speed maglev and the structure is the the flexible guideway

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$Y^{1}(t)=\$ nominal position of wheel (w/o guideway definition)

$S=x^{1}\$ , horizontal coordinate

$u^{1}(S,t)=\$ axial deformation (displacement) of guideway, where $t\$ is the time parameter

$u^{2}(S,t)=\$ transverse deformation (displacement) of guideway

$u_{,s}^{2}:={\frac {\partial u^{2}(s,t)}{\partial S}}\$ , where := means equal by definition (non symmetric)

NOTE: $\Delta \$ and def are symbols (no direction)

$A:=B\$ means A is defined by B

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$A=:B\$ means B is defined by A

Axial displacement under moving wheel/magnet $=u^{1}(Y^{1}(t),t)\$ \displaystyle {\begin{aligned}u^{1}(S,t)\end{aligned}} (1)

Where $s=y^{1}(t)\$ General setting: $f(S,t)\$ , where $S=Y^{1}(t)\$ \displaystyle {\begin{aligned}=f(Y^{1}(t),t)\end{aligned}} (2)
 \displaystyle {\begin{aligned}{\frac {d}{dt}}f(Y^{1}(t),t)={\frac {\partial f(Y^{1}(t),t){\dot {y}}^{1}}{\partial S}}+{\frac {\partial f(Y^{1}(t),t)}{\partial t}}\end{aligned}} (3)

where ${\dot {y}}^{1}={\frac {dY^{1}}{dt}}$ \displaystyle {\begin{aligned}{\frac {d^{2}f}{dt^{2}}}=f_{,s}(Y^{1},t){\ddot {y}}^{1}+f_{,ss}({\dot {y}}^{1})^{2}+2f_{,st}{\dot {y}}^{1}+f_{,tt}\end{aligned}} (4)

Where $f_{,s}(Y^{1},t)={\frac {\partial f}{\partial S}}\$ and $f_{,ss}(Y^{1},t)={\frac {\partial ^{2}f}{\partial S^{2}}}\$ 