User:EGM6341.s11.TEAM1.WILKS/Mtg1

=EGM6321 - Principles of Engineering Analysis 1, Fall 2010= Mtg 1: Tue, 24 Aug 10

[[media: 2010_08_24_15_22_30.djvu | Page 1-1]]
- course website, wiki - high-speed trains German Transrapid Emsland 500 km/h, youtube, Uploaded by TransrapidSupporter on Feb 14, 2007 German transrapid (electromagentic attraction) Japanese Maglev (electrodynamic repulsion) French 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

[[media: 2010_08_24_15_22_30.djvu | Page 1-2]]
$$ Y^1(t)= \ $$ nominal position of wheel (w/o guideway deformation) $$ S=x^1 \ $$, 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 $$u^2_{,s}:=\frac{\partial u^2(S,t)}{\partial S} \ $$, where := means equal by definition (non symmetric) NOTE: $$ \Delta\ $$ and def are symbols (no direction) $$ A:=B \ $$ means A is defined by B

[[media: 2010_08_24_15_22_30.djvu | Page 1-3]]
$$ A=:B \ $$ means B is defined by A

Axial displacement under moving wheel/magnet


 * {| style="width:100%" border="0"

$$  \displaystyle \left. u^1(S,t) \right|_{S=Y^1(t)} =  u^1(Y^1(t),t) $$     (1)
 * style="width:95%" |
 * style="width:95%" |
 * 
 * }

where $$\displaystyle \left. u^1(S,t) \right|_{S=Y^1(t)}$$ is $$\displaystyle u^1(S,t)$$ evaluated at $$\displaystyle S=Y^1(t)$$.

General setting:


 * {| style="width:100%" border="0"

$$  \displaystyle \left. f(S,t) \right|_{S=Y^1(t)} =  f(Y^1(t),t) $$     (2)
 * style="width:95%" |
 * style="width:95%" |
 * 
 * }


 * {| style="width:100%" border="0"

$$  \displaystyle \frac{d}{dt}f(Y^1(t),t)= \frac{\partial f(Y^1(t),t)}{\partial S} \dot Y ^1 + \frac{\partial f(Y^1(t),t)}{\partial t} $$ (3)
 * style="width:95%" |
 * style="width:95%" |
 * 
 * }

where $$ \dot Y ^1 = \frac{dY^1 (t)}{dt} $$

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