  Precise control of the optical path differences (OPD) in the Very Large
Telescope Interferometer (VLTI) was critical for the characterization of the
black hole at the center of our Galaxy - leading to the 2020 Nobel prize in
physics. There is now significant effort to push these OPD limits even further,
in-particular achieving 100nm OPD RMS on the 8m unit telescopes (UT's) to allow
higher contrast and sensitivity at the VLTI. This work calculated the
theoretical atmospheric OPD limit of the VLTI as 5nm and 15nm RMS, with current
levels around 200nm and 100nm RMS for the UT and 1.8m auxillary telescopes
(AT's) respectively, when using bright targets in good atmospheric conditions.
We find experimental evidence for the $f^{-17/3}$ power law theoretically
predicted from the effect of telescope filtering in the case of the ATs which
is not currently observed for the UT's. Fitting a series of vibrating mirrors
modelled as dampened harmonic oscillators, we were able to model the UT OPD PSD
of the gravity fringe tracker to $<1nm/\sqrt{Hz}$ RMSE up to 100Hz, which could
adequately explain a hidden $f^{-17/3}$ power law on the UTs. Vibration
frequencies in the range of 60-90Hz and also 40-50Hz were found to generally
dominate the closed loop OPD residuals of Gravity. Cross correlating
accelerometer with Gravity data, it was found that strong contributions in the
40-50Hz range are coming from the M1-M3 mirrors, while a significant portion of
power from the 60-100Hz contributions are likely coming from between the
M4-M10. From the vibrating mirror model it was shown that achieving sub 100nm
OPD RMS for particular baselines (that have OPD$\sim$200nm RMS) required
removing nearly all vibration sources below 100Hz.
