Dear Reseem

You mentioned you have a damping ratio and wish to form the damping matrix. The nature of damping in structural systems is very complex and not properly understood. Some experts suggest that damping is hysteristic in nature and frequency independent and propose the use of a complex stiffness defined by K* = K(1 + 2βi) where i = √(-1) and β = the loss factor. Others suggest that damping is viscous in nature and that the resistance due to damping is given by c(du/dt) where c is the damping coefficient. The damping ratio is then given by r = c/(2Mω). The idea of a single damping ratio is strictly correct for a SDOF system for it appears that it depends on mass alone. In order to simplify the modal analysis of damped MDOF systems a so called proportional or Rayleigh damping is assumed whereby c = αM + βK, that is damping is dependent on both mass M and stiffness K. It can then be shown that the damping ratio varies with frequency and is given by ri = (α + βω2)/(2ω), where in modal analysis ω = ωi = the ith natural frequency. It is obvious that the basic assumption of proportional damping is that c are constants such that the damping ratio varies. I believe that you adopt this damping model with the remaining task being how to estimate α and β. This can be done experimentally by determining the damping ratios of a structure at two separate frequencies and solving for α and β. However, for structures made up of more than a single type of material, where the different materials provide drastically differing energy ­loss mechanisms in various parts of the structure, the distribution of damping forces will not be similar to the distribution of the inertial and elastic forces; in other words, the resulting damping will be nonproportional. A nonproportional damping matrix that will represent this situation may be constructed by applying procedures similar to those discussed above in developing proportional damping matrices, with a proportional matrix being developed for each distinct part of the structure and then the combined system matrix being formed by direct assembly (Clough and Penzien 1995, page 242). I advise Reseem to consult this textbook by Clough and Penzien for more information on synthesis of damping matrices.

Clough, R. W. and Penzien, J. (1995) Dynamics of structures,Computers and Structures, Berkeley, CA

From the Equation of motion Damping coefficients are grouped in a matrix this will give the damped matrix . Again this matrix is used for estimating the eigen values and eigen vectors

In case of modal damping you might use

D = \sum\limits_{j=1}^n \left ( M x_j \right) \frac{d_j}{m_j^2} \left(M x_j \right)^T

d_j = 2 m_j \omega_j \xi_j

m_j : j-th modal mass

x_j : j-th eigenvector

\omega_j : j-th eigenfrequency [rad/s]

\xi_j : j-th damping ratio

If you have an incomplete set of eigenvectors you can use r

Dear Reseem

You mentioned you have a damping ratio and wish to form the damping matrix. The nature of damping in structural systems is very complex and not properly understood. Some experts suggest that damping is hysteristic in nature and frequency independent and propose the use of a complex stiffness defined by K* = K(1 + 2βi) where i = √(-1) and β = the loss factor. Others suggest that damping is viscous in nature and that the resistance due to damping is given by c(du/dt) where c is the damping coefficient. The damping ratio is then given by r = c/(2Mω). The idea of a single damping ratio is strictly correct for a SDOF system for it appears that it depends on mass alone. In order to simplify the modal analysis of damped MDOF systems a so called proportional or Rayleigh damping is assumed whereby c = αM + βK, that is damping is dependent on both mass M and stiffness K. It can then be shown that the damping ratio varies with frequency and is given by ri = (α + βω2)/(2ω), where in modal analysis ω = ωi = the ith natural frequency. It is obvious that the basic assumption of proportional damping is that c are constants such that the damping ratio varies. I believe that you adopt this damping model with the remaining task being how to estimate α and β. This can be done experimentally by determining the damping ratios of a structure at two separate frequencies and solving for α and β. However, for structures made up of more than a single type of material, where the different materials provide drastically differing energy ­loss mechanisms in various parts of the structure, the distribution of damping forces will not be similar to the distribution of the inertial and elastic forces; in other words, the resulting damping will be nonproportional. A nonproportional damping matrix that will represent this situation may be constructed by applying procedures similar to those discussed above in developing proportional damping matrices, with a proportional matrix being developed for each distinct part of the structure and then the combined system matrix being formed by direct assembly (Clough and Penzien 1995, page 242). I advise Reseem to consult this textbook by Clough and Penzien for more information on synthesis of damping matrices.

Clough, R. W. and Penzien, J. (1995) Dynamics of structures,Computers and Structures, Berkeley, CA

Hi Hameed,

If you have mass and stiffness matrices, you can find damping matrix easily using Rayleigh or Caughey damping. If you use Rayleigh damping, damping coefficients for two modes is sufficient regardless of mode numbers. Belows are calculation steps;

C=X*m+Y*k; you should find X and Y, using two damping coefficients.

Where X=2*((si)i*Wi*Wj)/(Wi+Wj) and Y=2*(si)j/(Wi+Wj), i and j = different mode numbers

C=X*m+Y*k, we got it

If you use Caughey damping you can use more than two modes to obtain damping matrix. If you are interested I can explain it.

Sincerely yours,

thanks buddies..its working...and am getting clasical damping :)..tanku every1

How to get damping element wise instead of alpha* [Global M]+ Beta* [Global K]?