User:Egm6322.s14.team2/report5

=Report 5 PEA 2 Spring 2014 Team 2=

Problem Statement
Submit term paper topics on Piazza.

Problem Statement
Find the expression for the Laplacian in the following coordinate systems

(a): Elliptical

(b): Parabolic

Solution

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On our honor, we did this assignment on our own, without looking at the solutions in previous semesters or other online solutions.
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Part(a): Elliptical Coordinates
The definition of the Laplacian in curvilinear coordinates is:

Elliptical coordinates in 2D (u3=constant and du3=0):

The derivatives of u1 and u2 are:

Using both trigonometric identities $$sin^{2}u+cos^{2}u=1$$ and $$cosh^{2}u-sinh^{2}u=1$$:

The magnitude of the tangent vector h are then identified as follows:

In two dimensions the general equation 2-1 simplifies to:

Part(b): Parabolic Coordinates
The derivatives of u1 and u2 are:

The magnitude of the tangent vector h are then identified as follows:

Problem Statement
(a): Show the following equalities:

(b): Use the Generalized Binomial Theorem to prove:

Solution

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On our honor, we did this assignment on our own, without looking at the solutions in previous semesters or other online solutions.
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Part(a): Scalar Product
By vector addition:

Using the scalar product and the magnitude of a vector |$$\vec{r}$$|=$$\sqrt{\vec{r}{\cdot}\vec{r}}$$:

Factoring $$r_{Q}^{2}$$ and substituting $${\rho}=\frac{r_{P}}{r_{Q}}$$:

Using the property of the scalar product $$\vec{a}{\cdot}\vec{b}=|a||b|cos{\theta}$$:

Or in spherical coordinates:

Using the trigonometric identity $$cos(x-y)=cosxcosy+sinxsiny$$:

Part (b): Generalized Binomial Theorem
Using the Generalized Binomial Theorem:

Where the binomial coefficient is defined:

With y=-x and r=-1/2, the (-1/2) can be factored out of the numerator leaving a product of i odd integers. I even integers are then left in the denominator because $$2^{i}i!=(2){\cdot}(4){\cdot}{\cdot}{\cdot}(2i)$$:

Problem Statement
Use the Navier Equation, Hooke's Law, and the Laplacian to derive the Equation of Elasticity in polar coordinates.

Solution

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On our honor, we did this assignment on our own, without looking at the solutions in previous semesters or other online solutions.
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The Navier Equation, Hooke's Law, and the vector Laplacian in polar coordinates are:

Where $$\boldsymbol{\epsilon}$$ is the symmetric part of $$\nabla \boldsymbol{u}$$:

Also need in this problem is the gradient and divergence in polar coordinates given by:

Applying the product rule to ($$):

Writing ($$), ($$), and ($$) in index notation:

Plugging ($$) into ($$):

Plugging ($$) into ($$):

(Note: we are assuming $$\lambda$$ and $$\mu$$ are constant within a material)

Using the Kronecker delta: $$i=j$$

Since $$k$$ is a dummy index, ($$) can be rewritten as:

Moving back to vector notation:

Plugging in ($$):

Looking at the r component using the gradient as defined in ($$) and the laplacian in ($$):

Looking similarly at the theta component:

Problem Statement
Problems taken from sec.70c Pb-60-[8-10] Course Notes

(a) Let $$f(x)$$ be a periodic function and $$ g(.)$$ be any function. Show that the following function is also periodic:

is also periodic.

Then show that $$\cos (e \sin \phi), \ \sin (e \sin \phi)$$ are also periodic

(b) Fourier series of a periodic function with period $$T$$:

Can also be written as:

Discus the pros and cons of the first representation versus the second.

(c) Consider a product of two functions:

(i) Find whether $$h(x)$$ is half-period odd/even if $$f(x)$$ or $$g(x)$$ is half-period odd/even

(ii) Show that $$\cos{(n \omega x)},\sin{(n \omega x)}$$ are:

half-period odd when n is odd

half-period even when n is even

(iii) Consider a periodic function $$f(x)$$ with period $$T$$ that is also half-period odd. Show that $$f(x)$$ can be expressed as a Fourier sine series of the form:

(iv) (iii)  Consider a periodic function $$f(x)$$ with period $$T$$ that is also half-period even. Show that $$f(x)$$ can be expressed as a Fourier cosine series of the form:

(v) Show that:

Is a half-period odd function.

So that the the Fourier series can be written as:

(vi) Show that:

Is a half-period even function.

So that the the Fourier series can be written as:

Solution

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On our honor, we did this assignment on our own, without looking at the solutions in previous semesters or other online solutions.
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Part a
If $$f(x)$$ is periodic with period $$T$$

Then:

By the definition of a periodic function.

Likewise a composite function:

Is also periodic with period T since:

Using the results above it can be seen that $$\cos (e \sin \phi), \ \sin (e \sin \phi)$$ are both periodic since $$e \sin \phi$$ is periodic.

Part b
The advantage of the second formulation of the Fourier series is that it collapses the number of expressions for the coefficients to just two integral equations. This expression is more compact compared to the first expression which requires a third coefficient equation.

(i)
The definition of a half-period odd function is:

Where $$L$$ is the half period $$\frac{T}{2}$$

The definition of a half-period even function is:

A product of half-period even/odd functions has 3 cases of interest (the fourth is a repeated case.)

Case 1: $$f(x)$$ half-period odd, $$g(x)$$ half-period odd:

Product is even.

Case 2: $$f(x)$$ half-period even, $$g(x)$$ half-period even:

Product is even.

Case 3: $$f(x)$$ half-period even, $$g(x)$$ half-period odd:

Product is odd.

(ii)
The functions $$\cos{(n \omega x)}$$ and $$\sin{(n \omega x)}$$ are half-periodic odd or even depending on n.

Here $$\omega=\frac{2 \pi}{T}=\frac{\pi}{L}$$

Using summation trig identities:

When n is even, $$\cos{(n \pi)}=1$$

(Note: $$\sin{(n \pi)}= 0$$ for all n.)

$$\cos{(n \omega x)}$$ and $$\sin{(n \omega x)}$$ are half-periodic even when n is even.

When n is odd, $$\cos{(n \pi)}=-1$$

$$\cos{(n \omega x)}$$ and $$\sin{(n \omega x)}$$ are half-periodic odd when n is odd.

(iii)
Consider a function $$f(\theta)$$ that is a half-period odd function.

Since $$ f(\theta)$$ is odd, $$c_n=0$$ and the Fourier series collapses into a sine series:

When n is odd, $$sin{(\frac{n \pi \theta}{L})}$$ is half-periodic odd (shown in part (ii)), and the product of $$f(\theta)sin{(\frac{n \pi \theta}{L})}$$ is half-period even (part (i)).

Therefore, for odd values of n, the integral in ($$) equal to:

since $$f(\theta+L) \, \sin{(\frac{n \pi (\theta+L)}{L})}=f(\theta) \, \sin{(\frac{n \pi \theta}{L})}$$

Similarly, when n is even, $$sin{(\frac{n \pi \theta}{L})}$$ is half-periodic even (shown in part (ii)), and the product of $$f(\theta)sin{(\frac{n \pi \theta}{L})}$$ is half-period odd(part (i)).

The integral of in ($$) is then equal to zero $$f(\theta+L) \, \sin{(\frac{n \pi (\theta+L)}{L})}=-f(\theta) \, \sin{(\frac{n \pi \theta}{L})}$$

The Fourier sine series then only has odd coefficients and can be written as:

(iv)
Consider a function $$f(\theta)$$ that is a half-period even function.

Recall ($$), $$, and ($$).

Since $$ f(\theta)$$ is even, $$c_n=0$$ and the Fourier series collapses into a cosine series:

When n is odd, $$sin{(\frac{n \pi \theta}{L})}$$ is half-periodic odd (shown in part (ii)), and the product of $$f(\theta)sin{(\frac{n \pi \theta}{L})}$$ is half-period odd (part (i)).

Therefore, for odd values of n, the integral in ($$) equal to zero:

Similarly, when n is even, $$sin{(\frac{n \pi \theta}{L})}$$ is half-periodic even (shown in part (ii)), and the product of $$f(\theta)sin{(\frac{n \pi \theta}{L})}$$ is half-period even(part (i)).

The integral of in ($$) is then equal to:

The Fourier cosine series then only has even coefficients and can be written as:

(See part b for the constant term differing by one half)

(v)
We want to show that $$\sin{(z \sin{(x)})}$$ is half-period odd.

To do this, it is sufficient to show that:

Therefore, using the results of part iv:

(vi)
We want to show that $$\cos{(z \cos{(x)})}$$ is half-period even.

To do this, it is sufficient to show that:

Therefore, using the results of part iv:

Part A
Use the recurrence relationship in ($$) to generate {$$P_{2}, P_{3},...,P_{6}$$} with $$P_{0} = 1$$ and $$P_{0} = \mu$$:

Part B
Continue the series expansion in ($$) using the Binomial Theorem to find {$$ P_{3}, ..., P_{6}  $$} and compare the results to and to earlier results obtained in part A and in the literature.

Solution

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On our honor, we did this assignment on our own, without looking at the solutions in previous semesters or other online solutions.
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Worked out by hand by Kevin Frost
Setting n = 1:

Solving:

Setting n = 2:

Solving:

Simplifying:

Setting n = 3:

Simplifying:

Setting n = 4:

Solving:

Setting n = 5:

Solving:

Programmed by Chris Neal
For this section a Matlab script was written to compute the polynomials given by the recursion relation ($$). Below are the results of the program.

Source:

Part B
In this part we look at the series expansion of ($$) to obtain results for the Legendre polynomials and then compare the results from the series expansion to the polynomials in Part A and in the literature. The series expansion of ($$) is given by the following.

A Matlab script was used to expand the terms that are in powers of i in ($$).

Source:

The result of the program gives the following.

The Legendre polynomials are defined as follows.

From observation and comparison of ($$) and ($$) we find the following.

From comparison of the above results for the results in Part A and from Wolfram Alpha's Page on Legendre Polynomials we see that there is agreement between all 3.

Problem Statement
Redo any problems in report R4.

Solution

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On our honor, we did this assignment on our own, without looking at the solutions in previous semesters or other online solutions.
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Problem Statement
This problem is motivated by notes provided by Dr. Vu-Quoc. The problem refers to Pb.70.[5,6,7] in Sec. 70b.

(a): Show that Equation 2-1 is an odd, periodic function of μ with period 2π.

(b): Show

(c): Plot φ(t) and μ(t) for τ=0, e-1/2, ω=1/2, and t [0,5T]. Find the true anomaly, $$ \theta (t) $$, and plot it on the same plot with $$\phi (t)$$ and  $$\mu (t)$$.

Solution

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On our honor, we did this assignment on our own, without looking at the solutions in previous semesters or other online solutions.
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Part c
Here we are asked to plot $$ \phi \text{ and } \mu $$ as functions of time. The results are presented below. Please note that the function $$ \mu $$ is linear with respect to time, so it is the line on the plots below.

The following relation was used to compute the values of the mean, eccentric, and true anomaly.

The analytic relation given by ($$) was found on a Wikipedia page about the True Anomaly. I do not know how that expression was derived.



Some observations about the figure above. As the angular frequency is increased, the lines tilt closer to the vertical axis. Notice how the plots for $$ \theta (t) $$ stay long the bottom of the plots. This is most likely due to the fact that an inverse cosine relation had to be used to compute the value, and as such the output of such a function is limited between 0 and $$ \pi$$.

=Contributing Team Members=


 * Cameron Stewart Solved problems 5.1, 5.4, and 5.5
 * Elizabeth Bartlett Solved problems 5.1, 5.2 and 5.3
 * Kevin Frost Solved problem 5.1, 5.6
 * Christopher Neal Solved problems 5.1, 5.6, and 5.7