University of Florida/Egm6321/F10.TEAM1.WILKS/Mtg1

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1-1 defined as (Meeting Number) - (Slide Number)
German Transrapid Emsland 500 km/h , youtube, Uploaded by TransrapidSupporter on Feb 14, 2007
Vu-Quoc & Olson (1989) CMAME
The magnet (i.e vehicle) and structure (i.e. guideway) interaction is shown below.




Where:
${\displaystyle y^{1}(t)=\ }$ nominal motion
${\displaystyle u^{1}(s,t)=\ }$ axial displacement of guideway
${\displaystyle u^{2}(s,t)=\ }$ transversal displacement of guideway

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${\displaystyle f(y^{1}(t),t)\ }$
Total time derivative of f is given as:

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

Where ${\displaystyle {\dot {y}}^{1}(t)={\frac {d}{dt}}y^{1}(t)\ }$
${\displaystyle {\frac {d}{dt}}f=f_{,s}(y^{1},t){\dot {y}}^{1}+f_{,t}(y^{1},t)}$

${\displaystyle \displaystyle {\begin{aligned}{\frac {d^{2}}{dt^{2}}}f=f_{,s}(y^{1},t){\ddot {y}}^{1}+f_{,ss}(y^{1},t)({\dot {y}}^{'})+2f_{,st}(y^{1},t){\dot {y}}^{1}+f_{,tt}(y^{1},t)\end{aligned}}}$	(2)

Where ${\displaystyle {\dot {y}}^{'}={\dot {y}}^{2}}$
Coriolis forces - Dynamics

Material time derivative - Continuum Mechanics

Reynolds transport theorem - Continuum Mechanics

${\displaystyle \displaystyle {\begin{aligned}c_{3}(y',t){\ddot {y}}^{1}+c_{2}(y^{1},t)({\dot {y}}^{1})^{2}+c_{1}(y^{1},t){\dot {y}}^{1}+c_{0}(y^{1},t)=0\end{aligned}}}$	(3)

is an example of a nonlinear ordinary differential equation (ODE)

References