problem_4.m

% Problem 5

clear
clc
% Set the default text and plot size according to this script.
setdefaults

% Young's Modulus for aluminum [Pascals]:
E = 70e9;
% Width of beam [meters]:
b = 0.1;
% Height of beam [meters]:
h = 0.01;
% Second moment of inertia [meters^4]:
I = (b*h.^3)/12;
% Density of aluminum beam [kg/m^3]:
rho = 2700;
% Cross sectional area of the beam [m^2]:
A = b*h;
%--------------------------------------------------------------------------
% Define the 6 segment matrix:
% Segment length for 6 segments
h_6 = 1/6;
A_6 = zeros(5);
b_6 = zeros(5,1);
% Diagonal vectors of the A matrix:
ones_vect_6 = ones(3,1);
fours_vect_6 = -4*ones(4,1);
sixes_vect_6 = 6*ones(5,1);
% Create A matrix by summing the diagonal vectors.
A_6 = diag(ones_vect_6,-2) + diag(fours_vect_6,-1) + diag(sixes_vect_6) + diag(ones_vect_6,2) + diag(fours_vect_6,1);
A_6(1,1) = 5;
A_6(5,5) = 5;
% Solve the eigenvalue problem using Matlab's "eig" function to return
% eigenvectors and eigenvalues.
[eig_6,lamb_6] = eig(A_6);
% Find natural frequency
omeg_6 = sqrt((lamb_6 * E * I)/(rho*A*h_6^4))'/2/pi;

% Plotting the shape of the beam
x_plot_6 = 0:1/4:1;
figure;
plot(x_plot_6,eig_6(:,1:3));
title('Shape of Beam for First 3 Natural Frequencies: \newline     Beam Deflection vs. Horizontal Distance\newline                           \it 6 Segments');
xlabel('x [m]');
ylabel('Beam Deflection [m]');
legend('\omega_{1}', '\omega_{2}', '\omega_{3}','Location','southeastoutside');
%-------------------------------------------------------------------------------
% Define the 10 segment matrix:
% Segment length for 10 segments
h_10 = 1/10;
A_10 = zeros(9);
b_10 = zeros(9,1);

% Diagonal vectors of the A matrix:
ones_vect_10 = ones(7,1);
fours_vect_10 = -4*ones(8,1);
sixes_vect_10 = 6*ones(9,1);
% Create A matrix by summing the diagonal vectors.
A_10 = diag(sixes_vect_10)+diag(fours_vect_10,-1)+diag(fours_vect_10,1)+diag(ones_vect_10,-2)+diag(ones_vect_10,2);
A_10(1,1) = 5;
A_10(9,9) = 5;
% Solve the eigenvalue problem using Matlab's "eig" function to return
% eigenvectors and eigenvalues.
[eig_10,lamb_10] = eig(A_10);
% Find natural frequency
omeg_10 = sqrt((lamb_10 * E * I)/(rho*A*h_10^4))'/2/pi;

% Plotting the shape of the beam
x_plot_10 = 0:1/8:1;
figure;
plot(x_plot_10,eig_10(:,1:3));
title('Shape of Beam for First 3 Natural Frequencies: \newline     Beam Deflection vs. Horizontal Distance\newline                          \it 10 Segments');
xlabel('x [m]');
ylabel('Beam Deflection [m]');
legend('\omega_{1}', '\omega_{2}', '\omega_{3}','Location','southeastoutside');
%--------------------------------------------------------------------------
% Define the 20 segment matrix:
% Segment length for 20 segments
h_20=1/20;

A_20 = zeros(19);
b_20 = zeros(19,1);
% Diagonal vectors of the A matrix:
sixes_vect_20 = 6*ones(1,19);
fours_vect_20 = -4*ones(1,18);
ones_vect_20 = ones(1,17);
% Create A matrix by summing the diagonal vectors.
A_20 = diag(sixes_vect_20)+diag(fours_vect_20,-1)+diag(fours_vect_20,1)+diag(ones_vect_20,-2)+diag(ones_vect_20,2);
A_20(1,1) = 5;
A_20(19,19) = 5;
% Solve the eigenvalue problem using Matlab's "eig" function to return
% eigenvectors and eigenvalues.
[eig_20,lamb_20] = eig(A_20);
% Find natural frequency
omeg_20 = sqrt((lamb_20 * E * I)/(rho*A*h_20^4))'/2/pi;

% Plotting the shape of the beam
x_plot_20 = 0:1/18:1;
figure;
plot(x_plot_20,eig_20(:,1:3));
title('Shape of Beam for First 3 Natural Frequencies: \newline     Beam Deflection vs. Horizontal Distance\newline                          \it 20 Segments');
xlabel('x [m]');
ylabel('Beam Deflection [m]');
legend('\omega_{1}', '\omega_{2}', '\omega_{3}','Location','southeastoutside');