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h1. What values should I use for mass\- and stiffness-proportional damping coefficients?

*Answer:* Mass\- and stiffness-proportional [damping|kb:Damping], normally referred to as Rayleigh damping, 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_}\etaη{math_}* is the mass-proportional damping coefficient; and

* *{math_}\deltaδ{math_}* is the 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

* *{_}ξ{math}\xi_n{math_}* *{_}{~}n{~}{_}* is the critical-damping ratio; and

* *{math}\omega_n{math_}ω{_}* *{_}{~}n{~}{_}* is the natural frequency ({math}\omega_n *{_}ω{_} {_}{~}n{~}{_} = 2 \pi f_n{math}{_}π{_} {_}f{_} {_}{~}n{~}{_}* ).

Here, it can be seen that the critical-damping ratio varies with natural frequency. The values of *{math_}\etaη{math_}* and *{math_}\deltaδ{math_}* are usually selected, according to engineering judgement, such that the critical-damping ratio is given at two known frequencies. For example, 5% damping ( *{math_}\xiξ{math_}* = 0.05 ) at the first natural frequency of the structure ( *{_}ω{math}\omega_i{math_}* *{_}{~}i{~}{_}* = *{_}ω{_}* *{math}\omega_1{math_}{~}1{~}{_}* ), and at *{_}ω{math}\omega_j{math_}* *{_}{~}j{~}{_}* = 188.5 (30 Hz). According to the equation above, the critical-damping ratio will be smaller between these two frequencies, and larger outside.

If the damping ratios ( *{_}ξ{math}\xi_i{math_}* *{_}{~}i{~}{_}* and *{_}ξ{math}\xi_j{math_}* *{_}{~}j{~}{_}* ) associated with two specific frequencies ( *{math}\omega_i{math_}ω{_}* *{_}{~}i{~}{_}* and *{math}\omega_j{math_}ω{_}* *{_}{~}j{~}{_}* ), or modes, are known, the two Rayleigh damping factors ( *{math_}\etaη{math_}* and *{math_}\deltaδ{math_}* ) 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 users to either specify coefficients *{math_}\etaη{math_}* and *{math_}\deltaδ{math_}* directly, or in terms of 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 simplify as follows:

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


h1. References

* Wilson, Dr. Edward L. _Static & Dynamic analysis of Structures_. 4th ed. Berkeley: Computers and Structures, Inc., 2004.
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