of a system shown below. There are 4 lumped masses and 3 linear stiffness of L-l
ID: 2087361 • Letter: O
Question
of a system shown below. There are 4 lumped masses and 3 linear stiffness of L-length each in the model. Obtain free vibration response (frequencies and mode shapes) of a 1-D lumped mass model 2 4 The following [M] ans [K] matrices provided 1 0 0 0 [M]- pAL/2 02 0 0 0 0 2 0 0 0 0 1 an 1-1 0 0 [K]: EA/L-1 2 1 0 0 0 -1 1 (a) Derive the frequency and mode shapes using symbolic computations. (b) Assign the following values to the mass and the stiffness coefficients (pAL/ (EA/L)- 300,000 N/m and ebtain frequency values in Hz and plot mode shapes 2)-400 kg (c) Use 10%, 20%, 30% reduced values of the stiffness coefficients and compare the new frequencies obtained with the ones from part (b). What can you conclude? Explain.Explanation / Answer
% Enter the mass and stiffness matrices
mass=400;
fprintf('Mass matrix is ');
m=mass*[1 0 0 0;0 2 0 0;0 0 2 0;0 0 0 1]
stiff=300000;
fprintf('Stiffness matrix is ');
k=stiff*[1 -1 0 0;-1 2 -1 0;0 -1 2 -1;0 0 -1 1]
% Calculate eigenvectors and eigenvalues %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
[P,lamda]=eig(m^(-1)*k);
[wn,kk]=sort(sqrt(diag(lamda)));
P=P(:,kk);
% Make sure the first entry of every column is positive
for i=1:length(m)
if P(1,i)<0
P(:,i)=-P(:,i);
end
% Normalize so that the top element is 1
phi(:,i)=P(:,i)/P(1,i);
end
disp('%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ')
disp('The natural frequencies are')
disp(' ')
for i=1:length(m)
disp(['f',num2str(i),' = ',num2str(wn(i)/2/pi),' Hz'])
end
disp(' ')
disp('The mode shape matrix of the system is')
phi
fn=wn/2/pi;
plot(1:4,phi)
mass=400;
m=mass*[1 0 0 0;0 2 0 0;0 0 2 0;0 0 0 1];
stiff=270000;
fprintf('Stiffness matrix is ');
k=stiff*[1 -1 0 0;-1 2 -1 0;0 -1 2 -1;0 0 -1 1]
% Calculate eigenvectors and eigenvalues %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
[P,lamda]=eig(m^(-1)*k);
[wn,kk]=sort(sqrt(diag(lamda)));
P=P(:,kk);
% Make sure the first entry of every column is positive
for i=1:length(m)
if P(1,i)<0
P(:,i)=-P(:,i);
end
% Normalize so that the top element is 1
phi(:,i)=P(:,i)/P(1,i);
end
disp('%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ')
disp('The natural frequencies are')
disp(' ')
for i=1:length(m)
disp(['f',num2str(i),' = ',num2str(wn(i)/2/pi),' Hz'])
end
disp(' ')
disp('The mode shape matrix of the system is')
phi
mass=400;
m=mass*[1 0 0 0;0 2 0 0;0 0 2 0;0 0 0 1];
stiff=240000;
fprintf('Stiffness matrix is ');
k=stiff*[1 -1 0 0;-1 2 -1 0;0 -1 2 -1;0 0 -1 1]
% Calculate eigenvectors and eigenvalues %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
[P,lamda]=eig(m^(-1)*k);
[wn,kk]=sort(sqrt(diag(lamda)));
P=P(:,kk);
% Make sure the first entry of every column is positive
for i=1:length(m)
if P(1,i)<0
P(:,i)=-P(:,i);
end
% Normalize so that the top element is 1
phi(:,i)=P(:,i)/P(1,i);
end
disp('%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ')
disp('The natural frequencies are')
disp(' ')
for i=1:length(m)
disp(['f',num2str(i),' = ',num2str(wn(i)/2/pi),' Hz'])
end
disp(' ')
disp('The mode shape matrix of the system is')
phi
results -----------------
As stiffness value reduces natural frequency reduces.Natural frequency is directly proportional to stiffness.
% Enter the mass and stiffness matrices
mass=400;
fprintf('Mass matrix is ');
m=mass*[1 0 0 0;0 2 0 0;0 0 2 0;0 0 0 1]
stiff=300000;
fprintf('Stiffness matrix is ');
k=stiff*[1 -1 0 0;-1 2 -1 0;0 -1 2 -1;0 0 -1 1]
% Calculate eigenvectors and eigenvalues %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
[P,lamda]=eig(m^(-1)*k);
[
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