Tangent Stiffness Proportional Damping on Updated Frequencies

[suuzie]

Dear Sir/Madam:

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]

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

[suuzie]

Thanks #selimgunay for your response.

In case I want to use tangent stiffness proportional damping with instantaneous frequency, how can I get that instantaneous frequency?
I think usually the modal analysis result gives us the first modal frequency (but not the instantaneous.)

If we take a look at the dynamic analysis example on OpenSees wiki page (https://opensees.berkeley.edu/wiki/inde ... ment_Frame), the modal analysis is conducted using the following sentences:

# Eigenvalue Analysis
set pi [expr 2.0*asin(1.0)]; # Definition of pi
set nEigenI 1; # mode i = 1
set nEigenJ 2; # mode j = 2
set lambdaN [eigen [expr $nEigenJ]]; # eigenvalue analysis for nEigenJ modes
set lambdaI [lindex $lambdaN [expr 0]]; # eigenvalue mode i = 1
set lambdaJ [lindex $lambdaN [expr $nEigenJ-1]]; # eigenvalue mode j = 2
set w1 [expr pow($lambdaI,0.5)]; # w1 (1st mode circular frequency)
set w2 [expr pow($lambdaJ,0.5)]; # w2 (2nd mode circular frequency)
set T1 [expr 2.0*$pi/$w1]; # 1st mode period of the structure
set T2 [expr 2.0*$pi/$w2]; # 2nd mode period of the structure
puts "T1 = $T1 s"; # display the first mode period in the command window
puts "T2 = $T2 s"; # display the second mode period in the command window

Can I still use these sentences to obtain the instantaneous frequency? Or should I place this sentence in other places?

[fmk]

you can issue the eigen command any time you want .. if what you want is to change the rayleigh values after each analysis step you can put the calculations above inside an analysis loop that only does 1 step at a time .. thohj you might leave out the puts if in a loop as this will kill perforrmance .. any you might also check the eigen command works if your model has potential to become unstable.

while {$currentTime < $tFinal} {
# Eigen Analysis
rayleigh ......
analyze 1
}

note: the rayleigh values are for model at end of last converged step and not any trial steps.

[hesamgholamigmail]

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

Hello, can i adopted this value (i.e.1.5 times the first mode period ) considering stiffness-proportional damping for modeling of bridge undergoing earthquakes ?- References mention that only *one* mode of modes must be applied on stiffness-proportional damping, but which mode is best?