University of Florida/Egm6341/s11.TEAM1.WILKS/Mtg1

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

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

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

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

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

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

${\displaystyle A:=B\ }$ means A is defined by B

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${\displaystyle A=:B\ }$ means B is defined by A

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

${\displaystyle \displaystyle {\begin{aligned}u^{1}(S,t)\end{aligned}}}$	(1)

Where ${\displaystyle s=y^{1}(t)\ }$

General setting: ${\displaystyle f(S,t)\ }$, where ${\displaystyle S=Y^{1}(t)\ }$

${\displaystyle \displaystyle {\begin{aligned}=f(Y^{1}(t),t)\end{aligned}}}$	(2)

${\displaystyle \displaystyle {\begin{aligned}{\frac {d}{dt}}f(Y^{1}(t),t)={\frac {\partial f(Y^{1}(t),t){\dot {y}}^{1}}{\partial S}}+{\frac {\partial f(Y^{1}(t),t)}{\partial t}}\end{aligned}}}$	(3)

where ${\displaystyle {\dot {y}}^{1}={\frac {dY^{1}}{dt}}}$

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

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

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

References