ME3220Honors
This code is intended to serve as a 1D finite element analysis for a beam that deflects in the transverse direction. The code takes an input the number of segments the beam is divided into (n), the density (rho), the cross-sectional area (A), Young's Modulus (E), and the length of the beam (L) and returns the response of the beam in the transverse direction over time for a specified time range.
This first function sets up the mass matrix for the beam using an ODE.
function M = mass(n,rho,A,L)
C = rho*A*L/2;
matrix_m = 2/3*diag(ones(n+1,1))-1/6*diag(ones(n,1),1)-1/6*diag(ones(n,1),-1);
matrix_m(1,1) = 1/3;
matrix_m(n+1,n+1) = 1/3;
M = C.*matrix_m;
endThe second function sets up a matrix for the stiffness of the beam
function K = stiffness(n,A,E,L)
C = 2*A*E/L
matrix_k = 2*diag(ones(n+1,1))-1*diag(ones(n,1),1)-1*diag(ones(n,1),-1);
matrix_k(1,1) = 1;
matrix_k(n+1,n+1) = 1;
K = C.*matrix_k;
endThe third function sets up an ODE that will solve for the natural frequencies and eigenvectors of the beam.
function [w,u]=KM_solve(K,M)
[u,w] = eig(K,M);
w = sqrt(w);
w = diag(w);
endThis final code takes all the previously defined ODE solvers and gives them inputs for a standard beam. Responses for a beam divided into 4, 20, and 50 segments are plotted on a graph for comparison.
A = 0.2;
E = 200*10^9;
rho = 8050;
L = 10;
for n = [4, 20, 50]
K = stiffness(n,A,E,L);
m = mass(n,rho,A,L);
[w,u]=KM_solve(K,m);
t_end = 2*pi/min(w);
t = 0:0.0001:0.025;
init = zeros(2*n,1);
init(2*n) = -0.1;
matrix_1 = zeros(n+1,2*(n+1));
matrix_2 = zeros(n+1,2*(n+1));
matrix_1(:,2:2:end) = u(:,1:1:end);
matrix_2(:,1:2:end) = u(:,1:1:end);
matrix_3 = [matrix_1;matrix_2];
C = matrix_3\init;
x_t = u(:,1)*C(1)*sin(w(1)*t);
for i = 2:numel(w)
x_t = x_t + u(:,i)*C(i)*sin(w(i)*t);
end
f = figure(1);
plot(t*1000,x_t(n,:)*1000)
hold on;
end
title('Continuous Rod Deflection Over Time');
xlabel('Time (ms)')
ylabel('Position (mm)')
legend('n=4','n=20','n=50');
saveas(f,'HonorsProject.png');
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