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
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
where i = a node located at a position along the rod’s interior, and
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?