Question

25 cm k-150 N/m 2m m= 1.25 kg 50 cm x2(t) Calculate the natural frequencies and the natural mode shapes of the system given in the figure. a) b) Caleulate the free motion of the system for the initial conditions, x1(t = 0) = 0 , x2(t = 0) = 15 cm e) Determine the distance between the two masses at time t-2s Verify your results for the above question with MATLAB. Provide the MATLAB script and MATLAB Command Window that shows your results.

Answer 1

clc;
clear all;
close all;
M=input('Enter the mass matrix: ');
[n,o]=size(M);
if n~=o then
error('M matrix must be square');
end
K=input('Enter the sti§ness matrix: ');
[n,o]=size(K);
if n~=o then
error('K matrix must be square');
end
qu=0;
[u,l]=eig(K,M);
% Using ìeigîin this way allows us to subtract M*w^2
% from K, instead of I*w^2 (where I is the n by n identity
% matrix).
% The output from ìeigîgives unit-length eigenvectors.
% We need to scale them with respect to M.
%
for s=1:n
alfa=sqrt(u(:,s)'*M*u(:,s));
u(:,s)=u(:,s)/alfa
end
x0=input('Enter the initial displacement column vector: ');
xd0=input('Enter the initial velocity column vector: ');
tf=input('Enter the final time: ');
t=0:0.1:tf;
q=tf/0.1;
x=zeros(size(n,q));
% Applying Equation 7.183.
%
for j=1:n
w(j)=sqrt(l(j,j))
xt=u(:,j)*(u(:,j)'*M*x0*cos(w(j).*t)+u(:,j)'*M*xd0/...
w(j)*sin(w(j).*t));
x=x+xt;
end
% Plotting the modes in a subplot format.
% Note that, for more than 3 or 4 degrees
% of freedom, the plots will become nearly
%
for r=1:n
subplot(n,1,r)
plot(t,x(r,:))
xlabel('Time, seconds');
ylabel(['Response x',num2str(r)]);
end
for gh=1:n
ans=x(gh,:);
end
ans3=ans(21)