linear_algebra

lu_tridiag is a function that does L U decomposition on a 3x3 matrix and can be found in the files folder. solve_tridiag is a function that solves for x given A*x = b and can be found in the files folder.

lu_tridiag can only be conducted on 3x3 matrix. To evaluate solve_tridiag with matrices lager than 3x3 a script had to be made to get the diagonals of matrices for use in solve_tridiag. This can be found in large_mat_eval.m

###Table of errors made from backslash operator

    Size    norm_error
    ____    __________

     3      1.1547    
     4         1.5    
     5         1.6    
     6      2.0412    
     7      2.2678    
     8       2.315    
     9      2.6667    
    10       2.846    

###Finding Eigenvalues and Natural Frequencies

The following code solves the 3-DOF mass spring system's eigenvalues and natural frequencies

% Assigns mass values kg
m1 = 1;
m2 = 2;
m3 = 4;

% Assigns spring constants N/m
k1 = 10;
k2 = 20;
k3 = 20;
k4 = 10;

% Creates mass matrix
m = [m1 0 0; 0 m2 0; 0 0 m3]; 

% Creates spring constant matrix
k = [k1+k2, -k2, 0; -k2, k2+k3, -k3; 0, -k3, k3+k4];

lam = eig(k,m) % outputs eigenvalues

omega = lam.^.5 % outputs natural frequency rad/s

OUTPUT

lam =

    2.5690
   14.4090
   40.5220


omega =

    1.6028
    3.7959
    6.3657

Pole eigenvalues

Code

% Inputs
L = 5; % meters
segments = 5;

%constants
E = 76*10^9; % Pa
I = 4*10^-8; % m^4

nodes = segments-1;
k = zeros(nodes,nodes);
dx = (L/segments)^2;
scale = 1/dx;

matrix = scale*(2*eye(nodes) + diag(-1*ones(1, nodes-1),1) + diag(-1*ones(1, nodes-1),-1));

lambda = eig(matrix)

Table

    Num_Segments    Largest    Smallest    Num_Eigen
    ____________    _______    ________    _________

     5               3.618      0.382       4       
     6              3.7321     0.2679       5       
    10              3.9021     0.0979       9       

As dx approaches zero the eigenvalues increase in the opposite magnitude (become 10^x times larger)