User:Egm6321.f10.team2.vrasul/HW1

=Problem 1: Perform Dimensional analysis on Governing Equation of Force acting along Maglev Deformed Guideway=

Given
Governing Equation of force acting along a Maglev deformed guideway
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$$C_0\left ( Y^1,t \right)=-F^1\left [ 1-\overline{R}{u^2_{,ss}}\left ( Y^1,t \right) \right] -F^2 {u^2_{,s}}\left ( Y^1,t \right) - \frac{T}{R} +M\left [ \left [ 1-\overline{R}{u^2_{,ss}}\left ( Y^1,t \right) \right ] \left [ {u^1_{,tt}}\left ( Y^1,t \right)-\overline{R}{u^2_{,stt}}\left ( Y^1,t \right) \right ]+{u^2_{,s}}\left ( Y^1,t \right){u^2_{,tt}}\left ( Y^1,t \right) \right ]
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$$     (Eq 1.1)
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Note: Argument $$\left ( Y^1, t \right)$$ will be omitted for clarity in subsequent steps.

The following are the known variables of (Eq 1.1):

$$F^{1}$$:vertical force

$$F^{2}$$:horizontal force

$$F^{1}$$:vertical force

$$\overline{R}$$:distance from deformed guide way centerline to wheel centerline

$$ u^2_{,s}$$:slope of deform guide way

$$T$$:Torque exerted on wheel

$$R$$:radius of wheel

$$M$$:mass of wheel

The following are the derived variables of (Eq 1.1):

$$ u^2_{,ss}$$

$$ u^1_{,tt}$$

$$ u^2_{,stt}$$

$$ u^2_{,tt}$$

Find
Verify validity of (Eq 1.1) via “dimensional analysis”

Solution
Dimensional analysis of each variable must be completed for each variable in the equation prior to determining the overall dimension of the equation.


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$$[F^{1}]\longrightarrow$$ $$F$$ (Eq 1.2)
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$$[F^{2}]\longrightarrow$$ $$F$$ (Eq 1.3)
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$$[\overline{R}]\longrightarrow$$ $$L$$ (Eq 1.4)
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$$ [u^2_{,s}]=\frac{\partial{u}^2}{\partial s}\longrightarrow $$ $$\frac{L}{L}$$ $$ \longrightarrow [1] $$ (Eq 1.5)
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$$[T] \longrightarrow$$ $$F\cdot L$$ (Eq 1.6)
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||  (Eq 1.7)
 * $$[R]\longrightarrow$$ $$L$$
 * $$[R]\longrightarrow$$ $$L$$
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(Eq 1.8)
 * $$[M]\longrightarrow $$ $$L$$
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(Eq 1.9)
 * $$ [u^2_{,ss}]=\frac{\partial^2{u}^2}{\partial s^2}\longrightarrow $$$$\frac{L}{{L}^2}$$ $$ \longrightarrow \frac{1}{L}$$
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(Eq 1.10)
 * $$ [u^1_{,tt}]=\frac{\partial^2{u}^1}{\partial s^2}\longrightarrow $$$$\frac{L}{T}^2$$
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(Eq 1.11)
 * $$ [u^2_{,stt}]=\frac{\partial {u}^2}{\partial s\partial^2 {t}}\longrightarrow $$$$\frac{L}{L\cdot{T}^2}$$ $$ \longrightarrow \frac{1}{T}^2$$
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(Eq 1.12)
 * $$ [u^2_{,tt}]=\frac{\partial^2 {u}^2}{\partial {t}^2} \longrightarrow $$$$\frac{L}{T^2}$$
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The dimensional analysis of the entire equation can now be applied to (Eq 1.1) by substitution of (Eq 1.2) to (Eq 1.12).


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$$ [C_0\left ( Y^1,t \right)]=F\left [ 1-L \frac{1}{L}\right] -F \cdot\frac{L}{L} -\frac{F \cdot L}{L} +M\left [ \left [ 1-L \cdot \frac{1}{L} \right] \left [ \frac{L}{T^2} - \frac{L}{T^2} \right] +1 \cdot \frac{L}{T^2} \right]
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$$     (Eq 1.13)
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In conclusion, it can be deduced by simplification of fractions that the components of equation (Eq 1.13) will reduce down to force ($$F$$) dimensions and mass x acceleration ($$M \cdot \tfrac{L}{T^2}$$), which further reduces to force ($$F$$) dimensions.

=Problem 2: Proof of Nonlinarity=

Given
Mass component of equation of motion for Maglev train.


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$$C_0\left ( Y^1,t \right) \ddot{Y}(t) = M\left [1 - \overline{R} {u^2_{,ss}}\left ( Y^1,t \right) \right]
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$$     (Eq 2.1)
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The following are the known variables of (Eq 2.1):

$$\overline{R}$$:distance from deformed guide way centerline to wheel centerline

$$ u^2_{,s}$$:slope of deform guide way

$$M$$:mass of wheel

Find
Prove that $$C_0\left ( Y^1,t \right) \ddot{Y}(t) $$ is nonlinear with respect to $$Y^1$$.

Solution
For (Eq 2.1) to be nonlinear, the following condition must exist:


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$$F\left ( \alpha u + \beta v \right) \ne \alpha F\left ( u \right) + \beta F\left ( v\right)
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$$     (Eq 2.2)
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The left hand condition of (Eq 2.2) is then applied to (Eq 2.1) and the following results are yielded:
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$$ C_0\left ( \alpha Y^1,t \right) \ddot{Y}(t)= M\left [1 - \overline{R} {u^2_{,ss}}\left ( \alpha Y^1,t \right) \right]
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$$     (Eq 2.3)
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The right hand condition of (Eq 2.2) is then applied to (Eq 2.1) and the following results are yielded:
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$$ \alpha C_0\left ( Y^1,t \right) \ddot{Y}(t)= \alpha \left [ M\left [1 - \overline{R} {u^2_{,ss}}\left ( Y^1,t \right) \right]\right]
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$$     (Eq 2.4)
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The results of (Eq 2.3) and Eq 2.4) show that the right sides of each of these equations are indeed not equal to eachother and therefore, this proves that (Eq 2.1) is nonlinear.