## Tangent Stiffness Proportional Damping on Updated Frequencies

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suuzie
Posts: 5
Joined: Tue Jun 13, 2017 3:51 pm
Location: SUNY Buffalo

### Tangent Stiffness Proportional Damping on Updated Frequencies

I would like to know how to write in .tcl file tangent stiffness proportional damping on updated frequencies?
This tangent stiffness proportional damping on updated frequencies is discussed in references such as:
• Finley A. Charney, "Unintended Consequences of Modeling Damping in Structures", J. Struct. Engrg. Volume 134, Issue 4, pp. 581-592 (April 2008).
• Anil K. Chopra and Frank McKenna, "Modeling viscous damping in nonlinear response history analysis of buildings for earthquake excitation", J. Earth Eng & Struct. Dyn,. Volume 45, Issue 2, pp. 193-211 (February 2016).

I know how to write initial stiffness proportional damping, that is (based on the opensees wiki page "https://opensees.berkeley.edu/wiki/inde ... ng_Command"):
rayleigh \$alphaM \$betaK \$betaKinit \$betaKcomm

where (f1, f2: frequencies; zeta1,zeta2:damping ratios):
• \$alphaM = 4pi*f1*f2*(zeta1*f2-zeta2*f1)/(f2^2-f1^2)
• \$betaK = 0
• \$betaKinit = (zeta2*f2-zeta1*f1)/(pi*(f2^2-f1^2))
• \$betaKcomm = 0

Can you please tell me how to specify \$alphaM \$betaK \$betaKinit \$betaKcomm for the tangent stiffness proportional damping on updated frequencies in OpenSees?

selimgunay
Posts: 879
Joined: Mon Sep 09, 2013 8:50 pm
Location: University of California, Berkeley

### Re: Tangent Stiffness Proportional Damping on Updated Frequencies

To be able to use tangent stiffness proportional damping, you need to use \$betaK or \$betaKcomm, where \$betaKcomm is preferred for numerical purposes and better convergence. Instead of computing \$betaKcomm for each time step and instantaneous frequency, you can approximately compute it for an elongated period, for example 1.5 times the first mode period. This is the recommended method in some practical nonlinear modeling guidelines. If you want to update \$betaKcomm every time step, you can conduct an eigenvalue analysis each time step to compute f1 and f2, from which you can compute \$alphaM and \$betaKcomm