User:Egm6321.f11.team4.YuChen/HW1

= Problem 1.1 - First and Second Derivative of the Equation of Maglev Trains =

Given
The equation of the meglev trains:$$\displaystyle f(Y^1 (t),t) $$

Find
Derive the first and second time derivatives for the given equation.

Derivation of Multivariable Functions
Suppose $$\displaystyle u(x) $$ and $$\displaystyle v(x) $$ are functions of $$\displaystyle x $$. Then the first partial derivative with respect to $$\displaystyle x $$ of the multivariable function $$\displaystyle f(u,v)=f(u(x),v(x)) $$ is:

First Derivative
Now let $$\displaystyle s=Y^1(t) $$ for ease,so the first time derivative of the given function should be:

Use $$\displaystyle \dot Y^1(t) = \frac{dY^1}{dt} $$ for ease:

Second Derivative
Derive $$\displaystyle (Eq. 1) $$ with respect to $$\displaystyle t $$ again:

For ease, use $$\displaystyle f_{,x} := \frac{\partial f}{\partial x} $$ and $$\displaystyle f_{,xy} := \frac{\partial^2 f}{\partial x \partial y} $$, so (1.3) is actually:

=Problem 2.1-Show the Relationship of the Derivatives in Problem 1 and the Reynolds Transport Theorem & the Derivation of Coriolis Force=

Given
$$\displaystyle (Eq.1) $$ and $$\displaystyle (Eq.2) $$ in problem 1.

Find
Show the relationship of $$\displaystyle (Eq.1) $$ and the Reynolds Transport Theorem in Continuum Mechanics. Show the similarity of $$\displaystyle (Eq.2) $$ and the Derivation of Coriolis Force.

Reynolds Transport Theorem
Consider the function $$\displaystyle f=f(x(t),t) $$ over the time-dependent region $$\displaystyle \Omega(t) $$ that has boundary $$\displaystyle \partial \Omega(t) $$, then take the derivative with respect to time $$\displaystyle t $$:

Now we wish to move the derivative under the integral sign.

In continuum mechanics this theorem is often used for material elements, which are parcels of fluids of solids which no material enters or leaves. Now suppose $$\displaystyle \Omega(t) $$ is a 3-dimensional region (higher dimension also applies). $$\displaystyle \mathbf{v} $$ is the velocity field of such material element, $$\displaystyle \Sigma $$ is the

NOT FINISHED

=Problem 1.6 - Proof of Nonlinearity of Certain Function= From [[media:Nm1.s11.mtg23.djvu|Mtg 23-1]], [[media:Nm1.s11.mtg26.djvu|Mtg 26-1]]

Given
The first term in EOM:

where

Find
Show function (6.1) is nonlinear with respect to $$\displaystyle Y^1 $$.

Definition of Nonlinearity
If a function $$\displaystyle F(x) $$ does notsatisfy:

then it is nonlinear.

Proof of Nonlinerarity
Let $$\displaystyle F(x)=M[1-\bar{R} u_{,ss}^2 (x,t)] \ddot x $$, then:

If $$\displaystyle F(\alpha x + \beta y) = \alpha F(x) +\beta F(y) $$ is true, then:

is true for any real number $$\displaystyle \alpha, \beta $$ and any $$\displaystyle x, y $$ in the domain. That means:

is true for any real number $$\displaystyle \alpha, \beta $$ and any $$\displaystyle x, y $$ in the domain, which is absolutely not true. So equation (6.3) is not true for $$\displaystyle c_3(Y^1,t) \ddot Y^1 $$, and $$\displaystyle c_3(Y^1,t) \ddot Y^1 $$ is nonlinear.