Is there any analytical study regarding the self-vibration control property of composites?

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A. P. Meilakhs

I know a couple classical papers on phonons in superlattices:

Folded acoustic and quantized optic phonons in (GaAl)As superlattices

PRB, 1985

Optical phonon quantum levels in GaAs/Ga1− xAlxAs supelattices

Superlattices and Microstructures, 1985

Probably it can be of use, or you can try searching in papers that cited those above.

Eric Ancich

Don't quite understand your question. The decay in vibration of the composites (versus non-composites?) sounds like the decay in vibration due to damping. I would expect most composites to be more highly damped than non-composites.