README.md

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 | ... |

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

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

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 $\frac{y_{i+1}-2y_i+y_{i-1}}{\mathrm{\Delta }{x}^2}+p^2{y}_i$

where i = a node located at a position along the rod’s interior, and $\mathrm{\Delta }x$ = the spacing between nodes. This equation can be expressed as $y_{i-1}-(2-\mathrm{\Delta }{x}^2{p}^2)y_i{y}_{i+1}=0$ 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) 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 ($\mathrm{\Delta }x$) approaches 0, how many eigenvalues would there be?