User:EGM6321.f11.team1.Zhichao Gong/Mtg2

Mtg 2: Thu, 25 Aug 11

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Motivation: High-speed trains (con't) German Transrapid (electromagnetic attraction) Japanese Maglev (electrodynamicrepulsion) French TGV (wheel on rail) Vu-Quoc &amp; Olsson 1989 CMAME Vehicle/structure in interaction, where vehicle is the high-speed maglev and the structure is  <P>the flexible guideway</P> <P> </P> <P> </P> page2-2

$${Y}^{'}(t)={\color{blue}nominal \ position} \ of \ wheel \ ({\color{blue}without} \ guideway \ deformation)$$

$$S={x}^{1}={\color{blue}horizontal} \ coordinate$$

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

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

$$ {u}^{2}_{,s} \underset{\underbrace{:=}} \  \frac{ \partial{u}^{2}(S,t)}{ \partial S}=guideway \ {\color{blue}slope} \ (small \ deformation)$$

<P><U>Note:</U> latex code</P> <P>Use \displaystyle to prevent the fraction from </P> <P>being squished down, i.e., preserve the font size in</P> <P>the numerator and in the denominator.</P>

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<P><U>Note:</U> Equal-by-definition symbols</P> <P> </P>

$$:=$$ <P>non-symmetric symbol, goodto use,flexible.</P> <P>The quantity on the side of the colon is defined</P> <P>by the quantity on the side of the equal sign.</P> <P>e.g.,</P> <P>A:=B    A is defined by B</P> <P>A=:B    B is defined by A</P>

$$ \overset{def}{=}$$

symmetricsymbol, not clear

$$A\overset{def}{=}B$$

<P>A is defined by B? OR</P> <P>B is defined by A?</P>

<P>Need a convention: left-hand side defined</P> <P>by right-hand side (read "def" from left to</P> <P>right)</P>

$$ \overset{ \triangle}{=} \ similar to \ \overset{def}{=},$$

<P>but worse, since at least "def"</P> <P>has an implicit reading direction from left to</P> <P>right, whereas</P>$$ \triangle$$ <P>iscompletely symmetric</P>

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<P>Axial displacement of guideway under moving </P> <P>wheel/magnet</P>

$${u}^{1}(S,t) {|}_{{S}=Y^{1}(t)}={u}^{1}({Y}^{1},t) \ {\color{red}(1)}$$

$${u}^{1}(S,t) {|}_{{S}=Y^{1}} \ is \ {u}^{1}(S,t) \ evaluated \ at \ S={Y}^{1}(t)$$

<P>Note: latex code</P> <P>"\left." is used to balance "\right|" to get nothing</P> <P>on the left and the vertical bar | on the right.</P> <P>Note the dot "." after "left"</P>

General setting

$$f(S,t) {|}_{S=Y^{1}(t)}=f(Y^{1}(t),t) \ {\color{red}(2)}$$

<P>First total time derivative:</P> <P> </P>

$$ \frac{d}{dt}f({Y}^{1}(t),t)= \frac{ \partial f({Y}^{1}(t),t)}{ \partial S}{ \overset{.}{Y}}^{1}+ \frac{ \partial f({Y}^{1}(t),t)}{ \partial t} \ {\color{red}(3)}$$

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$$with \ \overset{.}{{Y}^{1}}:= \frac{d{Y}^{1}(t)}{dt}$$

<U>R</U><SUP>*</SUP><U>1.1</U>: Second total time derivative

Show that

$$ \frac{{d}^{2}f}{d{t}^{2}}=f,s({Y}^{1},t) \overset{..}{{Y}^{1}}+f,ss({Y}^{1},t)({{ \overset{.}{Y}}^{1}})^{2}+2f,st({Y}^{1},t){ \overset{.}{Y}}^{1}+f,tt({Y}^{1},t) \ {\color{red}(1)}$$

$$f,s({Y}^{1},t):= \frac{ \partial f({Y}^{1},t)}{ \partial S } \ {\color{red}(2)}$$

$$f,st({Y}^{1},t):= \frac{{ \partial}^{2}f({Y}^{1},t)}{ \partial  S \partial t} \ {\color{red}(3)}$$

Similarly for the other partial derivatives.