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53 changes: 53 additions & 0 deletions HW5/README.html
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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
<html xmlns="http://www.w3.org/1999/xhtml">
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<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
<meta http-equiv="Content-Style-Type" content="text/css" />
<meta name="generator" content="pandoc" />
<title></title>
<style type="text/css">code{white-space: pre;}</style>
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<body>
<h1 id="homework-5">Homework #5</h1>
<h2 id="due-32817-by-1159pm">due 3/28/17 by 11:59pm</h2>
<p><em>Include all work as either an m-file script, m-file function, or example code included with ``` and document your code in the README.md file</em></p>
<ol style="list-style-type: decimal">
<li><p>Create a new github repository called ‘linear_algebra’.</p>
<ol style="list-style-type: lower-alpha">
<li><p>Add rcc02007 and pez16103 as collaborators.</p></li>
<li><p>Clone the repository to your computer.</p></li>
</ol></li>
<li><p>Create an LU-decomposition function called <code>lu_tridiag.m</code> that takes 3 vectors as inputs and calculates the LU-decomposition of a tridiagonal matrix. The output should be 3 vectors, the diagonal of the Upper matrix, and the two off-diagonal vectors of the Lower and Upper matrices.</p>
<p><code>[ud,uo,lo]=lu_tridiag(e,f,g);</code></p></li>
<li><p>Use the output from <code>lu_tridiag.m</code> to create a forward substitution and back-substitution function called <code>solve_tridiag.m</code> that provides the solution of Ax=b given the vectors from the output of [ud,uo,lo]=lu_tridiag(e,f,g). <em>Note: do not use the backslash solver <code>\</code>, create an algebraic solution</em></p>
<p><code>x=solve_tridiag(ud,uo,lo,b);</code></p></li>
<li><p>Test your function on the matrices A3, A4, …, A10 generated with <code>test_arrays.m</code> Solving for <code>b=ones(N,1);</code> where N is the size of A. In your <code>README.md</code> file, compare the norm of the error between your result and the result of AN.</p></li>
</ol>
<pre><code>| size of A | norm(error) |
|-----------|-------------|
| 3 | ... |
| 4 | ... |</code></pre>
<div class="figure">
<img src="spring_mass.png" alt="Spring-mass system for analysis" />
<p class="caption">Spring-mass system for analysis</p>
</div>
<ol start="5" style="list-style-type: decimal">
<li><p>In the system shown above, determine the three differential equations for the position of masses 1, 2, and 3. Solve for the vibrational modes of the spring-mass system if k1=10 N/m, k2=k3=20 N/m, and k4=10 N/m. The masses are m1=1 kg, m2=2 kg and m3=4 kg. Determine the eigenvalues and natural frequencies.</p></li>
<li><p>The curvature of a slender column subject to an axial load P (Fig. P13.10) can be modeled by</p></li>
</ol>
<p><span class="math inline">\(\frac{d^{2}y}{dx^{2}} + p^{2} y = 0\)</span></p>
<p>where <span class="math inline">\(p^{2} = \frac{P}{EI}\)</span></p>
<p>where E = the modulus of elasticity, and I = the moment of inertia of the cross section about its neutral axis.</p>
<p>This model can be converted into an eigenvalue problem by substituting a centered finite-difference approximation for the second derivative to give <span class="math inline">\(\frac{y_{i+1} − 2y_{i} + y_{i−1} }{\Delta x^{2}}+ p^{2} y_{i}\)</span></p>
<p>where i = a node located at a position along the rod’s interior, and <span class="math inline">\(\Delta x\)</span> = the spacing between nodes. This equation can be expressed as <span class="math inline">\(y_{i−1} − (2 − \Delta x^{2} p^{2} )y_{i} + y_{i+1} = 0\)</span> Writing this equation for a series of interior nodes along the axis of the column yields a homogeneous system of equations. (See 13.10 for 4 interior-node example)</p>
<p>Determine the eigenvalues for a 5-segment (4-interior nodes), 6-segment (5-interior nodes), and 10-segment (9-interior nodes). Using the modulus and moment of inertia of a pole for pole-vaulting ( <a href="http://people.bath.ac.uk/taf21/sports_whole.htm" class="uri">http://people.bath.ac.uk/taf21/sports_whole.htm</a>) E=76E9 Pa, I=4E-8 m^4, and L= 5m.</p>
<p>Include a table in the <code>README.md</code> that shows the following results: What are the largest and smallest eigenvalues for the beam? How many eigenvalues are there?</p>
<pre><code>| # of segments | largest | smallest | # of eigenvalues |
| --- | --- | --- | --- |
| 5 | ... | ... | ... |
| 6 | ... | ... | ... |
| 10 | ... | ... | ... |</code></pre>
<p>If the segment length approaches 0, how many eigenvalues would there be?</p>
</body>
</html>
83 changes: 83 additions & 0 deletions HW5/README.md
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# Homework #5
## due 3/28/17 by 11:59pm

*Include all work as either an m-file script, m-file function, or example code included
with \`\`\` and document your code in the README.md file*

1. Create a new github repository called 'linear_algebra'.

a. Add rcc02007 and pez16103 as collaborators.

b. Clone the repository to your computer.

2. Create an LU-decomposition function called `lu_tridiag.m` that takes 3 vectors as inputs
and calculates the LU-decomposition of a tridiagonal matrix. The output should be 3
vectors, the diagonal of the Upper matrix, and the two off-diagonal vectors of the Lower
and Upper matrices.

```[ud,uo,lo]=lu_tridiag(e,f,g);```

3. Use the output from `lu_tridiag.m` to create a forward substitution and
back-substitution function called `solve_tridiag.m` that provides the solution of
Ax=b given the vectors from the output of [ud,uo,lo]=lu_tridiag(e,f,g). *Note: do not use
the backslash solver `\`, create an algebraic solution*

```x=solve_tridiag(ud,uo,lo,b);```

4. Test your function on the matrices A3, A4, ..., A10 generated with `test_arrays.m`
Solving for `b=ones(N,1);` where N is the size of A. In your `README.md` file, compare
the norm of the error between your result and the result of AN\b.

```
| size of A | norm(error) |
|-----------|-------------|
| 3 | ... |
| 4 | ... |
```

![Spring-mass system for analysis](spring_mass.png)

5. In the system shown above, determine the three differential equations for the position
of masses 1, 2, and 3. Solve for the vibrational modes of the spring-mass system if k1=10
N/m, k2=k3=20 N/m, and k4=10 N/m. The masses are m1=1 kg, m2=2 kg and m3=4 kg. Determine
the eigenvalues and natural frequencies.

6. The curvature of a slender column subject to an axial load P (Fig. P13.10) can be
modeled by

$\frac{d^{2}y}{dx^{2}} + p^{2} y = 0$

where $p^{2} = \frac{P}{EI}$

where E = the modulus of elasticity, and I = the moment of inertia of the cross section
about its neutral axis.

This model can be converted into an eigenvalue problem by
substituting a centered finite-difference approximation for the second derivative to give
$\frac{y_{i+1} -2y_{i} + y_{i-1} }{\Delta x^{2}}+ p^{2} y_{i}$

where i = a node located at a position along the rod’s interior, and $\Delta x$ = the
spacing between nodes. This equation can be expressed as $y_{i-1} - (2 - \Delta x^{2}
p^{2} )y_{i} y_{i+1} = 0$ Writing this equation for a series of interior nodes along the
axis of the column yields a homogeneous system of equations. (See 13.10 for 4
interior-node example)

Determine the eigenvalues for a 5-segment (4-interior nodes), 6-segment (5-interior
nodes), and 10-segment (9-interior nodes). Using the modulus and moment of inertia of a
pole for pole-vaulting (
[http://people.bath.ac.uk/taf21/sports_whole.htm](http://people.bath.ac.uk/taf21/sports_whole.htm))
E=76E9 Pa, I=4E-8 m^4, and L= 5m.

Include a table in the `README.md` that shows the following results:
What are the largest and smallest eigenvalues for the beam? How many eigenvalues are
there?

```
| # of segments | largest | smallest | # of eigenvalues |
| --- | --- | --- | --- |
| 5 | ... | ... | ... |
| 6 | ... | ... | ... |
| 10 | ... | ... | ... |
```

If the segment length ($\Delta x$) approaches 0, how many eigenvalues would there be?
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12 changes: 12 additions & 0 deletions HW5/lu_tridiag.m
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function [ud,uo,lo]=lu_tridiag(e,f,g);```
% lu_tridiag calculates the components for LU-decomposition of a tridiagonal matrix
% given its off-diagonal vectors, e and g
% and diagonal vector f
% the output is
% the diagonal of the Upper matrix, ud
% the off-diagonal of the Upper matrix, uo
% and the off-diagonal of the Lower matrix, lo
% note: the diagonal of the Lower matrix is all ones

end

11 changes: 11 additions & 0 deletions HW5/solve_tridiag.m
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function x=solve_tridiag(ud,uo,lo,b);```
% solve_tridiag solves Ax=b for x
% given
% the diagonal of the Upper matrix, ud
% the off-diagonal of the Upper matrix, uo
% the off-diagonal of the Lower matrix, lo
% the vector b
% note: the diagonal of the Lower matrix is all ones

end

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