User:Egm6322.s12.team2.steele.m2/Mtg2

=EGM6321 - Principles of Engineering Analysis 1, Fall 2011=

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Motivation: High-speed trains (cont’d) German Transrapid (electromagnetic attraction) Japanese Maglev (electrodynamic repulsion) French TGV (wheel on rail) Vu-Quoc & Olsson 1989 CMAME: Vehicle/structure interaction, where vehicle is the high-speed maglev and the structure is the flexible guideway.

=Equations of Motion (EOM) of Wheel/Magnet=

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$$Y^1(t) =$$ nominal position of wheel (without 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 $$\displaystyle u^2_(,s) := \frac{\partial u^2(S, t)}{\partial S} =$$ guideway slope (small deformation)
 * = -> “equal by definition (non symmetric”	Note latex code for reuse and for training. Demo with online latex equation editor.

Note: latex code Use \display style to prevent fraction from being squished down, i.e., preserve the font size in the numerator and in the denominator.

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Note: Equal-by-definition symbols
 * = 	non-symmetric symbol, good to use, flexible.

The quantity on the side of the colon is defined by the quantity on the side of the equal sign. e.g.,		A := B		A is defined by B 			A =: B		B is defined by A $$\overset{=}$$ symmetric symbol, not clear $$A \overset{=} B$$	A is defined by B? OR B is defined by A? Need a convention: left-hand side defined by right-hand side (read “def” from left to right) $$\overset{\Delta}{=}$$ similar to $$\overset{=}$$, but worse, since at least “def” has an implicit reading direction from left to right, whereas  is completely symmetric. ///

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Axial displacement of guideway under moving wheel/magnet

$$\displaystyle \left. u^1(S,t) \right|_{S=Y^1(t)}$$ is  evaluated at $$S = Y^1(t)$$ Note: latex code “\left.” Is used to balance “\right|” to get nothing on the left and the vertical bar | on the right. Note the dot “.” (after “left”).

General setting: First total time derivative:

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With $$\displaystyle \dot Y^1 := \frac{dY^1(t)}{dt}$$ R*1.1: Second total time derivative Show that Similarly for the other partial derivatives.