If you know the material properties, I think it may possible to calculate theoretically. Since, the critical damping coefficient and its variation with frequency is a property of the material (like Young’s Modulus).

You may use this formula:

critical damping coefficient Cc=2*sqrt(km).

For calculating actual damping coefficient 'c', you must have to perform either simulation with proper material properties or experiments. From there you can find the natural frequency and damping ratio. Finally, you can calculate damping coefficient.

The damping coefficient depends on the material, on the shape of the structure and the boundary conditions so it is something that cannot be calculated with accuracy with the present knowledge. Also, damping varies (increase) with the vibration amplitude, and quite significantly, so it is also significantly nonlinear. In conclusion, experimental data are necessary for that component or something similar installed in similar conditions. Otherwise numbers are really fantasy.

It is easy to model (calculate) the stiffness (restoring or conservative force ), if geometry and elastic constants (E, G, mu) of the material are known. E.g. in axial direction, stiffness = A E / L.

But it is very difficult to model the damping (non-conservative force) from physical parameters of the system.

It is possible to predict the damping approximately, In those cases only in which we intentionally design and attach the dampers externally e.g. hydraulic dampers in vehicle suspension. In case of structural analysis its impossible to predict damping from physical parameters or becomes highly complicated as we have to go at micro or molecular level.

The only way, is experimental identification. Two approaches can be used- i) test the material with harmonic excitation at different frequencies and amplitudes of practical range and obtain its stiffness and damping properties. ii) Obtain FRF of the material with some mass as a vibration system.

Dear Sina,

I think it depends on the vibration mode. (Distortional, flexural, longitudinal, etc.)The vibration damping of the system depends on many boundary conditions. I am not sure, but SMA actuators may also have elasto-plastic behavior. But if you are in the first phase of material design, I think the attenuation constant (factor) of an infinite beam (or infinite rod) can be a vibration damping index. The attenuation constant is related to the loss factor. The loss factor is related to the damping ratio if the damping ratio of the mode is sufficiently small.

I attached some notes for my own learning. There may be some misunderstandings and wrong description. Dear readers, please send me your correction.

I am looking forward to hearing from you.

Sincerely,

Ryuzo Horiguchi (Kao Corporation)

Dear readers,

some spoke of simulation to find the damping coefficient. I have all other information about the system. I just wanted to ask if there are any vibration/dynamic analysis softwares which could give me the damping coefficient.

I look forward to hearing from you.

If you have all the information about your system, you can simulate and obtain the damping behavior of the system, i mean the effects of damping variation, but about the damping value it is copmlex because the damping is nonlinear process of energy dissipation.

Regards

Dear Gasagara

Thank you for the reply.

Which Software is suitable for obtaining damping behavior of the system using simulation?

Sincerely

@Sina Keshavarz, First You design your system (SMA actuator) in any modelling software like Solidworks, Catia, CAD etc. And simulate the designed model in either ANSYS or Abaqus for better results with all known properties.

By MATLAB/SIMULINK also you can simulate it

It is interesting to note the points mentioned by different people. However, modeling of SMA using ANSYS not straight forward. It will be better to adopt MATLAB/SIMULINK as mentioned by @ Gasagara . You will not find any details regarding SMA modeling using any of the software instead you can find some own code based analysis. You may find the modeling steps for other advanced materials except SMA.

Dear Javad,

the damping ratio for your beam will be different for each vibration mode. So the correct answer is: you need experimental modal analysis data in order to determine those parameters. If you have in formation on the loss tangent of the plate material, you can have a rough idea of the damping ratios to insert in your model. However, the damping ratio is different for different vibration mode shapes and takes into account also boundary conditions. So it cannot be exactly calculated from the loss tangent of the material.

I take this opportunity to say that damping is also a function of the vibration amplitude. In case of large-amplitude vibrations (e.g. for a plate, when the vibration amplitude is comparable to the plate thickness) the damping largely increases. This is shown by experiments. I have done recently quite a bit of work on the subject. For those interested:

M. Amabili, 2018, Journal of the Mechanics and Physics of Solids, vol. 118, pp. 275-292. Nonlinear damping in nonlinear vibrations of rectangular plates: derivation from viscoelasticity and experimental validation.

M. Amabili; 2018, Nonlinear Dynamics. Derivation of nonlinear damping from viscoelasticity in case of nonlinear vibrations. doi:10.1007/s11071-018-4312-0

M. Amabili; 2018, Nonlinear Dynamics, vol. 93, pp. 5-18. Nonlinear damping in large-amplitude vibrations: modelling and experiments. doi:10.1007/s11071-017-3889-z

Dear Javad,

I am very interested in your answer and question. So I wrote some notes for my own learning. (“Thin viscoelastic cantilever beam_wk1a5.pdf”) The contents are mainly restricted to linear flexural vibration of a thin viscoelastic beam. While writing the note, I felt the following things.

(1) If we deal with viscoelasticity, it is difficult to express the exact free edge conditions. (Especially for the case of larger loss factor)

(2) When the width becomes wider, many flexural modes appear. (Torsional motion will also appear.)

(3) It is difficult to express the free edge conditions analytically. I think that damping coefficient of a cantilever beam is a vibration damping index. I prefer to use frequency response functions. The analytical expression can be derived by using the attenuation constant (factor). There might be some mistakes in my notes. So please send me your correction. I am looking forward to hearing from you.

Sincerely,

Ryuzo Horiguchi (Kao Corporation)