User:EGM6321.Fall2012.Team 5.Enze Zhu

= R1.3 = Author: Enze Zhu I am not quite sure that the dimension of $$\displaystyle [u^2_{,Stt}] $$ (1/time^2)is the acceleration of the wheel/magnet over the spacial displacement or not. Sorry about that.

Problem Statement
Analyze the dimension of all terms in following function,and provide the physical meaning.

Solution

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 The problem was solved independently without referring to any reports from previous classes.
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First term
in which

The dimension of $$\displaystyle [F] $$ is the axial force of the wheel/magnet $$\displaystyle [F^1]=F $$

The dimension of $$\displaystyle 1 $$ is non-dimensional $$\displaystyle [1]=1 $$

The dimension of $$\displaystyle [\bar R] $$ is length. Distance between the guideway and center of the wheel/magnet $$\displaystyle[\bar R]=L $$

The dimension of $$\displaystyle [u^2] $$ is length. It represents the transversal displacement of the guideway $$\displaystyle[u^2]=L $$

The dimension of $$\displaystyle [u^2_{,SS}] $$ is the curvature of the guideway deformed by the force of the wheel/magnet $$\displaystyle[u^2_{,SS}]=\frac{1}{R}=\frac{1}{L} $$

So

Second Term
in which

The dimension of $$\displaystyle [F^2] $$ is force of the wheel/magnet in the vertical direction $$\displaystyle[F^2]=F$$

The dimension of $$\displaystyle u^2_{,S}(Y^1,t) $$ is dimensionless. It is the guideway slope $$\displaystyle [1]=1 $$

So

Third Term
in which

The dimension of $$\displaystyle T (torque) $$ is Force-length of the wheel/magnet $$\displaystyle [T]= F\cdot L $$

The dimension of $$\displaystyle R $$(radius of the wheel) is length $$\displaystyle [R]= L $$

so

Fourth Term
in which

The dimension of $$\displaystyle M $$ is mass of wheel/magnet $$\displaystyle [M]=M $$

The dimension of $$\displaystyle [u^1_{,tt}] $$ is Length / time^2. This is the acceleration of the wheel/magnet $$\displaystyle [u^1_{,tt}]=\frac{L}{S^2} $$

The dimension of $$\displaystyle [u^2_{,Stt}] $$ is 1/time^2, This is the acceleration of the wheel/magnet over the spacial displacement $$\displaystyle [u^2_{,Stt}]=\frac{1}{S^2} $$

so

Conclusion
All in all, the dimension of $$\displaystyle C_0 $$ should be Force.