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

*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:

* *{_}η{_}* is the mass-proportional damping coefficient; and

* *{_}δ{_}* 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{_}* *{_}{~}n{~}{_}* is the critical-damping ratio; and

* *{_}ω{_}* *{_}{~}n{~}{_}* is the natural frequency ( *{_}ω{_}* *{_}{~}n{~}{_}* *= 2* *{_}π{_}* *{_}f{_}* *{_}{~}n{~}{_}* ).

Here, it can be seen that the critical-damping ratio varies with natural frequency. The values of *{_}η{_}* and *{_}δ{_}* are usually selected, according to engineering judgement, such that the critical-damping ratio is given at two known frequencies. For example, 5% damping ( *{_}ξ{_}* = 0.05 ) at the first natural frequency of the structure ( *{_}ω{_}* *{_}{~}i{~}{_}* = *{_}ω{_}* *{_}{~}1{~}{_}* ), and at *{_}ω{_}* *{_}{~}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{_}* *{_}{~}i{~}{_}* and *{_}ξ{_}* *{_}{~}j{~}{_}* ) associated with two specific frequencies ( *{_}ω{_}* *{_}{~}i{~}{_}* and *{_}ω{_}* *{_}{~}j{~}{_}* ), or modes, are known, the two Rayleigh damping factors ( *{_}η{_}* and *{_}δ{_}* ) 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 *{_}η{_}* and *{_}δ{_}* directly, or in terms of the critical-damping ratio either at two different frequencies, *{_}f{_}* (Hz), or at two different periods, *{_}T{_}* (sec).

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

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


h1. References

* Wilson, DrE. Edward LL. (2004). _Static &and Dynamic Analysis of Structures_. (4th ed.). Berkeley, CA: Computers and Structures, Inc., 2004.
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