How to get damping matrix for structural model in FE analysis

I need to implement in C a method of obtaining transient solution Rdm of damped FE models based on modal results R for a structural model model defined with hysteretic (structural) damping, available in Matlab by using

Rdm = solve(model,tlist,"ModalResults",R);

However, the main problem is how to get damping matrix for this model. Every route I took from the below listed in solving this problem, finally appers to be problematic:

1. I cannot find any information how the above mentioned application of solve function was implemented in Matlab,

2. I tried to get this damping matrix using assembleFEMatrices function but it does not give damping matrix despite its usage in structural model as hysteretic damping,

3. I tested reduce function based on Craig-Bampton reduction method, but it similarly does not give damping matrix, only stiffness and mass matrices,

4. I was checking if modal analysis performed for a damped model (hysteretic damping) can return any information about damping, but I suppose it does not.

Do you have any clue how to solve this problem and get the full data about the model or the algorithm for C implementation ?

Thanks, Piotr

Edit: The most promising idea so far it seems to me to get stiffness, mass and damping matrices from geometry mesh in the same way as matlab function assemblefematrices assuming hysterestic damping for the model. I believe it was already solved and used in many tools but I am not able to reach the proper document.

Answer 1

[Sahaj]

Hi Piotr.

To generate the damping matrix of Finite Element (FE) models, you can adopt the following methods:

1. Rayleigh Damping: It assumes that the damping matrix is a linear combination of the mass and stiffness matrices. The Rayleigh damping matrix is given by C = α*M + β*K, where C is the damping matrix, M is the mass matrix, K is the stiffness matrix, and α and β are coefficients that can be determined based on the desired damping characteristics.
2. Modal Damping: In this method, the damping matrix is expressed in terms of the modal coordinates of the system. The modal damping matrix is diagonal and each element corresponds to the damping ratio associated with a specific mode of vibration. The damping ratios can be estimated based on experimental data or analytical models.

Note that you can use the damp() function in MATLAB is used to extract the natural frequencies and damping ratios of a linear dynamic system.

Hope this helps.

[Piotr]

Thank you for your answer. That's right that in the case of proportional damping assumed I can create the damping matrix based on stiffness and mass matrices. The damping ratios can be also further used for simulation.

However, as I understand correctly, since version 2023a Matlab allows to define hysteretic damping. Further, these parameters accompanied with Poisson, Yongs and density including geometry can be used to solve modal analysis. And further transient simulation can be performed based on such modal results. I such a procedure no additinal and other damping is used as I understand correctly.

Thus, my question can be precised as how the hysteretic damping is included and taken into account in such a simulation.

Thanks