Superconducting Radio Frequency (SRF) structures used for particle accelerators typically operate between 4 2 K, i-e- cryogenic temperatures, and are sensitive to microphonics that create length oscillations in the accelerating structures- To keep the SRF cavities on resonance under the influence of distortions in the SRF cavities caused by microphonics, RF power above what is required for accelerating the particles is supplied to the cavities- This additional RF power is both a significant capital expense and operational expense- Boron nitride nanotube (BNNT) mats, the size of small postage stamps, have been observed to have exceptional viscoelastic behavior- Preliminary results on larger samples have demonstrated that this viscoelastic behavior is also present in the larger mats, and very importantly, this behavior is observed at both liquid nitrogen and elevated temperatures- Consequently, these novel BNNT mats are anticipated to be able to provide vibration damping for both high-temperature and low/cryogenic-temperature structures- This includes fully assembled cryo-modules with SRF cavities, their superfluid liquid helium vessels, tuners, and cryostats thereby reducing capital cost and operating expenses for particle accelerators-