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How can i obtain a damping Matrix C, with an inherent damping matrix of 2% for all the modes ?

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Kimbugwe Daniel
Kimbugwe Daniel el 8 de Jul. de 2023
Comentada: Kimbugwe Daniel el 10 de Jul. de 2023
I have written code to obtain eigen values and vectors using Stiffness K and Mass M and i want to obtain the damping matrix using an inherent damping ratio of 2% of the structure for all the vibration modes. How do i do it? here under is the my code thus far
clc;
h=[5.5]+[0:3.96:20*3.95]; % Total floor height(m)%
%--------------Start of Mass Matrix---------------%
M(1,1)=1.126e+003 ; % unit: ton
M(20,20)=1.170e+003 ; % unit: ton
for i=2:19
M(i,i)=1.100e+003; % unit: ton
end
disp('Mass Matrix(ton):'); disp(M)
%--------------end of Mass Matrix-----------------%
%--------------Start of Stiffness Matrix----------%
for i=1:5
k(1,i)=862.07e+003; % unit: kN/m
end
for i=6:11
k(1,i)=554.17e+003 ; % unit: kN/m
end
for i=12:14
k(1,i)=453.51e+003 ; % unit: kN/m
end
for i=15:17
k(1,i)=291.23e+003 ; % unit: kN/m
end
for i=18:19
k(1,i)=256.46e+003 ; % unit: kN/m
end
k(1,20)=171.70e+003 ; % unit: kN/m
K=zeros(20,20)
for j=1:(20-1)
K(j,j) = k(1,j)+k(1,j+1);
K(j,j+1) = -k(1,j+1);
K(j+1,j) = -k(1,j+1);
end
K(20,20)=171.70e+003 ; % unit: kN/m
%--------------End of Stiffness Matrix--------------%
%----Start of Eigen Values and Eigne Vectors--------%
[eigvec,eigval]=eig(K,M); %Eigen Values and Vectors%
wn = sort(sqrt(diag(eigval)));
f =(wn/(2*pi));
Tn = (2*pi)./wn;
disp('Mode Omega(w) Time_Period ')
for i = 1:length(wn)
fprintf (' %d %.2f %.2f \n',i,wn(i,1),Tn(i,1))
end
%-------End of Eigen values and Eigne Vectors-------%
%-----Start of Normalisation of mode shapes---------%
for j = 1 : length(wn)
eigvec(:,j)=eigvec(:,j)/eigvec(length(wn),j);
end
disp('Mode shapes:'); disp(eigvec)
%-----------End of Normalisation of mode shapes-----------%

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Anuj
Anuj el 10 de Jul. de 2023
Ran in:
Hi Kimbugwe,
To obtain the damping matrix using an inherent damping ratio of 2% for all vibration modes, you can modify your existing code by following these steps:
  1. Calculate the damping coefficient (c) using the formula: c = 2 * damping_ratio * sqrt(mass * stiffness), where damping_ratio is the desired inherent damping ratio (2% in this case), mass is the mass matrix, and stiffness is the stiffness matrix.
  2. Multiply the damping coefficient by the mass matrix to obtain the damping matrix (C).
Here's the modified code to incorporate these changes:
clc;
h = [5.5] + [0:3.96:20*3.95]; % Total floor height(m)
% Start of Mass Matrix
M(1,1) = 1.126e+003; % unit: ton
M(20,20) = 1.170e+003; % unit: ton
for i = 2:19
M(i,i) = 1.100e+003; % unit: ton
end
disp('Mass Matrix(ton):');
Mass Matrix(ton):
disp(M);
Columns 1 through 16 1126 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1100 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1100 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1100 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1100 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1100 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1100 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1100 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1100 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1100 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1100 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1100 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1100 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1100 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1100 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1100 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Columns 17 through 20 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1100 0 0 0 0 1100 0 0 0 0 1100 0 0 0 0 1170
%--------------Start of Stiffness Matrix----------%
for i=1:5
k(1,i)=862.07e+003; % unit: kN/m
end
for i=6:11
k(1,i)=554.17e+003 ; % unit: kN/m
end
for i=12:14
k(1,i)=453.51e+003 ; % unit: kN/m
end
for i=15:17
k(1,i)=291.23e+003 ; % unit: kN/m
end
for i=18:19
k(1,i)=256.46e+003 ; % unit: kN/m
end
k(1,20)=171.70e+003 ; % unit: kN/m
K=zeros(20,20)
K = 20×20
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
for j=1:(20-1)
K(j,j) = k(1,j)+k(1,j+1);
K(j,j+1) = -k(1,j+1);
K(j+1,j) = -k(1,j+1);
end
K(20,20)=171.70e+003 ; % unit: kN/m
% Calculate the damping coefficient (c)
damping_ratio = 0.02; % 2% inherent damping ratio
mass = M;
stiffness = K;
c = 2 * damping_ratio * sqrt(mass * stiffness);
% Multiply the damping coefficient by the mass matrix to obtain the damping matrix (C)
C = c .* M;
% Start of Eigen Values and Eigen Vectors
[eigvec, eigval] = eig(K, M); % Eigen Values and Vectors
wn = sort(sqrt(diag(eigval)));
f = wn / (2*pi);
Tn = (2*pi) ./ wn;
disp('Mode Omega(w) Time_Period ')
Mode Omega(w) Time_Period
for i = 1:length(wn)
fprintf(' %d %.2f %.2f \n', i, wn(i,1), Tn(i,1))
end
1 1.79 3.51 2 4.65 1.35 3 7.73 0.81 4 10.45 0.60 5 13.16 0.48 6 16.07 0.39 7 18.51 0.34 8 21.48 0.29 9 23.70 0.27 10 26.15 0.24 11 27.84 0.23 12 30.07 0.21 13 31.42 0.20 14 34.49 0.18 15 37.31 0.17 16 39.31 0.16 17 41.60 0.15 18 43.90 0.14 19 48.23 0.13 20 53.93 0.12
% Start of Normalisation of mode shapes
for j = 1:length(wn)
eigvec(:,j) = eigvec(:,j) / eigvec(length(wn),j);
end
disp('Mode shapes:');
Mode shapes:
disp(eigvec);
1.0e+11 * Columns 1 through 19 0.0000 -0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0157 0.0000 -0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0000 0.0000 -0.0000 0.0000 -0.0163 0.0000 -0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0000 -0.0000 -0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0000 0.0001 0.0000 -0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0000 0.0000 -0.0000 0.0162 0.0000 -0.0000 0.0000 -0.0000 0.0000 -0.0000 -0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0000 -0.0000 -0.0000 0.0000 -0.0000 0.0000 -0.0000 -0.0157 0.0000 -0.0000 0.0000 -0.0000 0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0000 0.0000 -0.0000 0.0000 0.0073 0.0000 -0.0000 0.0000 -0.0000 -0.0000 0.0000 -0.0000 0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0000 0.0000 -0.0000 0.0000 0.0000 -0.0000 -0.0034 0.0000 -0.0000 0.0000 0.0000 -0.0000 0.0000 -0.0000 -0.0000 0.0000 -0.0000 0.0000 0.0000 -0.0000 0.0000 0.0000 -0.0000 -0.0000 0.0000 0.0016 0.0000 -0.0000 0.0000 0.0000 -0.0000 0.0000 0.0000 -0.0000 0.0000 0.0000 -0.0000 0.0000 -0.0000 -0.0000 0.0000 0.0000 -0.0000 -0.0000 -0.0007 0.0000 -0.0000 0.0000 0.0000 -0.0000 -0.0000 0.0000 -0.0000 -0.0000 0.0000 -0.0000 -0.0000 0.0000 -0.0000 -0.0000 0.0000 0.0000 0.0000 0.0003 0.0000 -0.0000 -0.0000 0.0000 -0.0000 -0.0000 0.0000 0.0000 -0.0000 0.0000 0.0000 -0.0000 0.0000 0.0000 0.0000 -0.0000 -0.0000 -0.0000 -0.0001 0.0000 -0.0000 -0.0000 0.0000 0.0000 -0.0000 -0.0000 0.0000 -0.0000 -0.0000 0.0000 0.0000 -0.0000 -0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0000 -0.0000 0.0000 0.0000 -0.0000 -0.0000 0.0000 0.0000 -0.0000 -0.0000 0.0000 0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 0.0000 -0.0000 -0.0000 -0.0000 0.0000 0.0000 -0.0000 -0.0000 0.0000 0.0000 -0.0000 -0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0000 -0.0000 -0.0000 0.0000 0.0000 0.0000 -0.0000 -0.0000 0.0000 0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 0.0000 0.0000 -0.0000 -0.0000 -0.0000 0.0000 0.0000 0.0000 -0.0000 -0.0000 -0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0000 -0.0000 -0.0000 -0.0000 0.0000 0.0000 0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 0.0000 0.0000 0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 Column 20 -3.9705 7.1448 -8.2594 6.9930 -3.7107 1.0641 -0.3051 0.0875 -0.0251 0.0071 -0.0019 0.0004 -0.0001 0.0000 -0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0000
By adding the calculation of the damping coefficient and multiplying it with the mass matrix, you will obtain the desired damping matrix (C) using the inherent damping ratio of 2% for all vibration modes.
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Piotr
Piotr el 10 de Jul. de 2023
Are sure about this ? My question is why c = 2 * damping_ratio * sqrt(mass * stiffness) is additionally multiplied by M. I am not sure but maybe I will look for inverse eigenproblem when we have eigenvectors and eigenvalues available and we want to get the origin matrices including damping matrix ? Have you checked if the above-mentioned solution gave you the proper damping ratios of desired modes ?
Kimbugwe Daniel
Kimbugwe Daniel el 10 de Jul. de 2023
Well , his solution is right and part of the solution I think . It’s called a mass damping matrix . There is also a stiffness damping Matrix . The sum of mass damping matrix and stiffness damping matrix gives us Rayleigh damping which appears to be the full solution . So mass damping matrix is 2*damping ratio * wn * phi’*M*phi whereas stiffness damping matrix is 2* damping ratio /wn *phi’*K*phi
Note ‘ means transpose , phi are eigen values, wn denotes circular natural frequency

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