README.md

Homework #4

final commit due 11/2/17 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

P1-2 Due 10/26

1. Create a new github repository called '04_linear_algebra'.

a. Add rcc02007 and zhs15101 as collaborators.

b. Clone the repository to your computer.

c. Submit clone repo link to https://goo.gl/forms/gFNxhNM4qJJKj8hE3

2. Create the 4x4 and 5x5 Hilbert matrix as H. Include the following results in your README before 10/26 by midnight:

a. What are the 2-norm, frobenius-norm, 1-norm and infinity-norm of the 4x4 and 5x5 Hilbert matrices?

b. What are the 2-norm, frobenius-norm, 1-norm and infinity-norm of the 4x4 and 5x5 inverse Hilbert matrices?

c. What are the condition numbers for the 2-norm, frobenius-norm, 1-norm and infinity-norm of the 4x4 and 5x5 Hilbert matrices?

P3-4 Due 10/30

3. Create a Cholesky factorization function called chol_tridiag.m that takes 2 vectors as inputs and calculates the Cholseky factorization of a tridiagonal matrix. The output should be 2 vectors, the diagonal and the off-diagonal vector of the Cholesky matrix.

[d,u]=chol_tridiag(e,f);

4. Use the output from chol_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 [d,u]=lu_tridiag(e,f). Note: do not use the backslash solver \, create an algebraic solution

x=solve_tridiag(d,u,b);

5. Create the stiffness matrix for the 4-mass system shown above for cases a-c. Calculate the condition of the stiffness matrices. What is the expected error when calculating the displacements of the 4 masses? Include the analysis and results in your README.

a. K1=K2=K3=K4=1000 N/m

b. K1=K3=K4=1000 N/m, K2=1000e12 N/m

c. K1=K3=K4=1000 N/m, K2=1000e-12 N/m

6. Use chol_tridiag.m and solve_tridiag.m to solve for the displacements of hanging masses 1-4 shown above in 5a-c, if all masses are 1 kg.

7. 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. Create a function, mass_spring_vibrate.m that outputs the vibration modes and natural frequencies based upon the parameters, k1, k2, k3, and k4.

8. The curvature of a slender column subject to an axial load P 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) 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 loads in the beam based upon the different shapes? How many eigenvalues are there?

| # of segments | largest load (N) | smallest load (N) | # of eigenvalues |
| --- | --- | --- | --- |
| 5 | ... | ... | ... |
| 6 | ... | ... | ... |
| 10 | ... | ... | ... |

If the segment length approaches 0, how many eigenvalues would there be?