It's not rigidity that makes CI better at damping, its the boundary layers between dissimilar material within it- i.e. the nodular structure of cast iron. Steel actually has a slightly higher Youngs modulus than CI. There's other factors as per the bridge example, the natural frequency of the structure, but all things being equal, materials with lots of surface area (boundary layers) between dissimilar materials will damp the best (decay the wave, i.e. the amplitude is reduced more quickly)
I don't think anyone here ever said anything about "rigidity" of cast iron making it a better damping material. I certainly didn't - or at least didn't intend to. All of my references to rigidity were in the context of components of a machine and its stand and as such were an outcome or resulting consequence of the overall design.
I agree that it is the grain structure of cast iron that gives it it's high damping. But I'm not sure it's correct to describe the materials as different though. I guess that depends on how strict your definition of different materials is and the individual type of cast iron being considered. Certainly if you want to consider cementite to be a different material than pearlite and that it occurs in grain boundaries then you would be correct. But that's a little deeper into the individual microstructure of cast iron than I thought you were suggesting. Regardless, I don't believe dissimilarity is as important as the fundamental granular nature of cast iron itself.
Looking back at what you said earlier, it's easy to see how we might have needlessly disagreed. Your first comment about the boundary between two materials led me to believe you were talking about the interface between steel and cast iron or some such other dissimilarity on a very large scale when in fact you may have been talking about the molecular structure of the grains that make up the entire mass of cast iron - which I would have agreed with.
Here is a quote from my first response "FWIW, I think there are two kinds of damping being discussed here. One, as you say is at the boundary between two materials, and the other is throughout the grain structure of heavy masses."
Your later comments suggest you were referring to these two as one and the same and I just thought you were talking about the first but not the second.
In any event, this granularity in the second was my whole point. Concrete, epoxy granite, sand, and other materials exhibit a similar energy absorbing behaviour to different degrees for the same reasons. It takes energy to move the particles and grains within a body and that energy is dissipated in the form of heat.
Sometimes I hate the English language.