What values should I
...
use for mass- and stiffness
...
-proportional damping
...
?
Using mass Answer: Mass- and stiffness-proportional damping results in a critical damping ratio that varies with frequency according to:
\xi_n = \frac{1}{2 \omega_n} \eta + \frac{\omega_n}{2} \delta
where
- \xi_n is the critical damping ratio
- \eta is the mass proportional damping coefficient, and
- \delta is the stiffness proportional damping coefficient
- \omega_n is circular frequency (\omega_n = 2 \pi f_n)
Usually the values of \eta and \delta are selected so that the critical , normally referred to as Rayleigh damping, is commonly used in nonlinear-dynamic 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:
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:
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, you may specify 5% damping ( \xi ξ = 0.05 ) at \omega_i = \omega_1 (the first natural frequency of the structure ( ω i = ω 1 ), and at \omega_ ω j = 188.5 (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 .
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 \eta and \delta ( η and δ ) can be evaluated by the solution of a pair of simultaneous equations if the damping ratios \xi_i and \xi_j associated with two specific frequencies (modes) \omega_i and \omega_j are known. Mathematically,
\;\;\;\;
\left{
\xi_i
\xi_j
\right}
=
\frac{1}{2}
\;\;
\left[
\begin{array}{cc}
\frac{1}{\omega_i} & \omega_i
\frac{1}{\omega_j} & \omega_j
\end{array}
\right]
\;\;
\left{
\eta
\delta
\right}
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 )
Damping, given mathematically by:
SAP2000 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:
References
- Wilson, E. L. (2004). Static and Dynamic Analysis of Structures (4th ed.). Berkeley, CA: Computers and Structures, Inc. Available for purchase on the CSI Products > Books page