User: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 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

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$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

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

Axial displacement under moving wheel/magnet

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

(1)

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:

$\displaystyle \left. f(S,t) \right|_{S=Y^1(t)} = f(Y^1(t),t)$

(2)

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

(3)

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

$\displaystyle \begin{align} \frac{d^2f}{dt^2}=f_{,S}(Y^1,t) \ddot Y ^1 + f_{,SS}(Y^1,t) ( \dot Y ^1)^2 + 2f_{,St}(Y^1,t) \dot Y ^1 +f_{,tt}(Y^1,t) \end{align}$

(4)

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} \$.