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Using mass and stiffness proportional damping results in a critical damping ratio that varies with frequency according to:

{math:display=block}\xi_n = \frac\{1\}\{2 \omega_n\} \eta + \frac\{\omega_n\}\{2\} \delta

{math}

where

* {math}\xi_n{math} is the critical damping

 ratio
* {math}\eta{math} is the mass proportional damping coefficient,

 and
* {math}\delta{math} is the stiffness proportional damping

 coefficient
* {math}\omega_n{math} is circular frequency ({math}\omega_n = 2 \pi f_n{math})

Usually the values of {math}\eta{math} and {math}\delta{math} are selected so that the critical damping ratio is given at two known frequencies. For example, you may specify 5% damping ({math}\xi = 0.05{math}) at {math}\omega_i = \omega_1{math} (first natural frequency of the structure), and at {math}\omega_j = 188.5{math} (30 Hz). It is up to you, the engineer, to make this choice. According to the equation above, the critical damping ratio will be smaller between these two frequencies, and larger outside of them.

The two Rayleigh damping factors *{math}\eta{math}* and *{math}\delta{math}* can be evaluated by the solution of a pair of simultaneous equations if the damping ratios {math}\xi_i{math} and {math}\xi_j{math} associated with two specific frequencies (modes) {math}\omega_i{math} and {math}\omega_j{math} are known. Mathematically,

{math:display=block}
((\xi_i

),(\xi_j

)) = 
1/2 [[1/\omega_i

, \omega_i

], [1/\omega_j

, \omega_j

]]
((\eta), (\delta))
{math}

SAP2000 allows you to specify \eta and \delta directly, or to specify the critical damping ratio at two different frequencies (f, Hz) or periods (T, sec).

*References:*
* Edward L. Wilson: Static & Dynamic analysis of Structures, 4th edition, 2004; chapter 19.6 Stiffness and Mass Proportional Damping, p.231 (the book is available from CSI at http://orders.csiberkeley.com/ProductDetails.asp?ProductCode=ELWDOC-3D )

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