Understanding the regional rayleigh damping?

[mpnewcombe]

Hello All,

Is there a document/example I can read to understand how regional rayleigh damping works?

I have been modelling a frame with zerolength elements for rotational springs at the beam-column joints with and without rayleigh damping assigned to the rotational springs, which results in 10% difference in displacement (for initial stiffness proportional damping) averaged over 15 records. My understanding is that the damping forces are assigned to the rotational spring elements based on their relative velocity, which explains the 10% reduction in response when I apply damping to the rotational springs. Is this correct?

The rotational springs are MultiLinear elastic and the rest is linear elastic. So, if I want to achieve constant damping for the first mode response (even as the structure goes non-linear) I could apply damping to only the linear elastic elements, but this will under estimate the overall system damping. Is this assertion correct? Is there any way to determine the damping contribution from rotation springs and compensate for this loss of damping by increasing the damping coefficient?

Best regards,
Mike Newcombe

[fmk]

before i begin, rayleigh damping for nonlinear systems is poorly understood. As Ed Wilson puts it in his book "Dynamic Analysis of Structures":

"The use of linear modal damping as a percentage of critical damping has been
used to approximate the nonlinear behavior of structures. The energy dissipation
in real structures is far more complicated and tends to be proportional to
displacements rather than proportional to the velocity. The use of approximate
“equivalent viscous damping” has little theoretical or experimental justification
and produces a mathematical model that violates dynamic equilibrium."


the region command is a way of having different rayleigh factors for different parts of the structures. it is typically used to give the structure and soil part of the model diferent rayleigh factors.

when you use rayleigh damping and uniform excitation, then yes the damping forces are proportional to the relative velocities of the nodes.

not sure i undertsand the question correctly: if you want constant damping use alpha on the initial stiffness as opposed to the current stiffness. if you want it always for he 1'st period and period shifts as goes nonlinear, you could reset the factors at every analysis based on an eigenvalue analysis. it will take longer to run. there is no way to ask the program for the % contributions from the elements.

[mpnewcombe]

Thank you for your thoughts... It is very helpful for me to know that you can do an eigen value analysis at each time step. Have I understood correctly that this will mean that, for example, 5% of critical damping can be maintained for first mode response, even as the 1st mode frequency reduces in the non-linear range?

Also, can you clarify how tangent stiffness rayleigh damping is applied by OpenSees (Kcomm).. are the damping forces at a given timestep a product of the rayleigh damping matrix using the tangent stiffness matrices and the current velocity vector? The reason I ask is that some programs (Ruaumoko) compute the incremental damping forces over a time step by using the tangent stiffness multiplied by the increment in velocity, which can make a big difference.

Regards,
Mike Newcombe

[fmk]

it is the tangent times the current velocity. the tangent can be the initial tangent, tangent at end of last step or the current tangent. the ruaumoko way would be an option that could be added.

most programs use the initial tangent and then it doesn't matter if it is the incremental or the total velocity. damping is rather poorly understood from a modeling pt of view. damping for example should be greater in a damaged structure than an undamaged one. all approaches above would thus be incorrect. as the stiffness of the damaged structure is less than the undamaged one, in general using the current tangent would be even more incorrect than using the initial tangent.

rayleigh damping is a holdover from elastic analysis. even then it was a quick and dirty way to account for damping without having to explicitly model for it.

this is just my 2 cents.

[mpnewcombe]

I appreciate your 2 cents.

But shouldn't the additional damping from a damaged structure be taken into account by the Hysteretic model? Priestley would argue that the elastic damping should reduce because hysteretic energy dissipation begins to account for most of the structural damping...

Fully understand that it is a elastic hangover, but for the structures I am modelling (non-linear elastic) it is particularly problematic.

Again, thanks for your input.

Mike N