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h1. What values should I use for stiffnessmass\- and massstiffness-proportional damping coefficients?

*Answer:* Using mass\Mass- and stiffness-proportional [damping|kb:Damping], normally referred to as Rayleigh damping results in a critical damping ratio that varies with frequency according to:

!Figure, is commonly used in [nonlinear-dynamic|kb:Nonlinear] analysis. Suitability for an incremental approach to numerical solution merits its use. During formulation, the damping matrix is assumed to be proportional to the mass and stiffness matrices as follows:

!fig 1.png|align=center,border=0!

where

* *{math}\xi_neta{math}* is the criticalmass-proportional damping coefficient; ratioand

* *{math}\etadelta{math}* is the mass stiffness-proportional damping coefficient.

Relationships between the modal equations and orthogonality conditions allow this equation to be rewritten as:

!Figure 1.png|align=center,border=0!

where
and
* *{math}\deltaxi_n{math}* is the stiffness proportional damping coefficientcritical-damping ratio; and

* *{math}\omega_n{math}* is the circularnatural frequency ({math}\omega_n = 2 \pi f_n{math}).

Here, it can Usually thebe seen that the critical-damping ratio varies with natural frequency. The values of *{math}\eta{math}* and *{math}\delta{math}* are usually selected so, according to engineering judgement, such that the critical -damping ratio is given at two known frequencies. For example, you may specify 5% damping (*{math}\xi{math}* = 0.05{math} ) at the first natural frequency of the structure ( *{math}\omega_i{math}* =  *{math}\omega_1{math} (first natural frequency of the structure* ), and at *{math}\omega_j{math}* = 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.

TheIf twothe Rayleighdamping dampingratios factors( *{math}\etaxi_i{math}\* and *{math}\deltaxi_j{math}\* can) beassociated evaluatedwith bytwo thespecific solutionfrequencies of a pair of simultaneous equations if the damping ratios ( *{math}\xiomega_i{math}* and *{math}\xiomega_j{math} associated with* ), or modes, are known, the two specificRayleigh damping frequenciesfactors (modes) *{math}\omega_ieta{math}* and *{math}\omega_jdelta{math} are known. Mathematically,

* ) can be evaluated by the solution of a pair of simultaneous equations, given mathematically by:

!Figure 2.png|align=center,border=0!

[SAP2000|sap2000:home] allows youusers to either specify coefficients *{math}\eta{math}* and *{math}\delta{math}* directly, or in toterms specifyof the critical -damping ratio at two different frequencies, (*f,* (Hz), or periods, (*T,* (sec).

When damping is set to equal for both frequencies, the conditions associated with the proportionality factors aresimplify as follows:

!Figure 3.png|align=center,border=0!


h1. References

* Wilson, Dr. Edward L. Wilson: _Static & Dynamic analysis of Structures,_. 4th edition ed. Berkeley: Computers and Structures, Inc., 2004;. chapter 19Print.6 (Stiffness and Mass Proportional Damping, p.page 231)
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