Thermodynamic uncertainty relation for underdamped dynamics driven by time-dependent protocols

The thermodynamic uncertainty relation (TUR) for underdamped dynamics has intriguing problems while its counterpart overdamped recently been derived. Even the case of steady states, a proper way to match and TURs not found. We derive TUR systems subject general time-dependent protocols, that covers by using Cram\'{e}r-Rao inequality. show resultant give rise inequality product variance entropy production. prove it approach known result large viscosity limit. present three examples confirm our rigorous result.