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Spring-mass system for problem 5
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
Create a new github repository called ‘04_linear_algebra’.
Add rcc02007 and pez16103 as collaborators.
Add rcc02007 and zhs15101 as collaborators.
Clone the repository to your computer.
Complete this before 10/26 by midnight
Create the 4x4 and 5x5 Hilbert matrix as H:
What are the 2-norm, frobenius-norm, 0-norm and infinity-norm of the 4x4 and 5x5 Hilbert matrices?
What are the 2-norm, frobenius-norm, 0-norm and infinity-norm of the 4x4 and 5x5 inverse Hilbert matrices?
What are the condition numbers for the 2-norm, frobenius-norm, 0-norm and infinity-norm of the 4x4 and 5x5 Hilbert matrices?
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);
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);
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);
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);
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Spring-mass system for problem 5
K1=K3=K4=1000 N/m, K2=1000e-12 N/m
lu_tridiag.m and solve_tridiag.m to solve for the displacements of hanging masses 1-4 shown above, if all masses are 1 kg.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.
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Spring-mass system for analysis
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 \(p^{2} = \frac{P}{EI}\)
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+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)
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+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 |
@@ -66,6 +79,6 @@ due 3/28/17 by 11:59pm
| 5 | ... | ... | ... |
| 6 | ... | ... | ... |
| 10 | ... | ... | ... |If the segment length (\(\Delta x\)) approaches 0, how many eigenvalues would there be?
+If the segment length approaches 0, how many eigenvalues would there be?