What values should I enter for stiffness and mass proportional damping coefficients?
Using 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
- Unknown macro: {math}is the critical damping ratio
\xi_n
- Unknown macro: {math}is the mass proportional damping coefficient, and
\eta
- Unknown macro: {math}is the stiffness proportional damping coefficient
\delta
- Unknown macro: {math}is circular frequency (
\omega_n
Unknown macro: {math})\omega_n = 2 \pi f_n
Usually the values of
\eta
and
\delta
are selected so that the critical damping ratio is given at two known frequencies. For example, you may specify 5% damping (
\xi = 0.05
) at
\omega_i = \omega_1
(first natural frequency of the structure), and at
\omega_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 two Rayleigh damping factors *
\eta
* and *
\delta
* 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,
((\xi_i),(\xi_j)) =
1/2 [[1/\omega_i, \omega_i], [1/\omega_j, \omega_j]]
((\eta), (\delta))
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 )