Hi Bashar Owaid. To begin with, I am not an expert in damping but for an unknown system, I try to get the damping ratio by finding the logarithmic decrement. I know this is not a perfect method for a multi DOF system but I get reasonable values for the initial modes.
As described above, the logarithmic decrement is one way. As with most things, the way you measure damping very much depends on what is its intended use, i.e. is it for the material itself or perhaps, for a built up construction.
Here are some other ways
Reverberation time - https://en.wikipedia.org/wiki/Reverberation
Some of the standards that refer to experimental damping measurement are the following:
ASTM E756, ASTM E1876, ISO 16940, ISO 6721, J671, J1637.
The one I have used is ISO 16940. It is quite simple, but you will need the relevant equipment (electrodynamic shaker, signal processor). Additionally, impact hammer is also a straight forward way to measure damping, examining the specimen's response and using 3db bandwidth method (check ISO 16940 for this). In this case you have to take into consideration the effect of the suspending method of the specimen on the damping behaviour of it.
Also, keep in mind that damping measuring is most of the times statistically uncertain, especially for materials that have weak damping performance.
Hope I helped somehow,if you have any queries concerning damping measurement feel free to add them here.