A bearing damping system for damping vibrations and conducting heat from vibration-producing, heat-generating devices, an evacuated energy storage system including such a bearing damping system for damping vibrations produced by the rotating shaft and conducting heat from bearings and/or bearing assemblies; and a method for damping vibrations and conducting heat from vibration-producing, heat-generating devices. The bearing damping system includes one or more flexible bearing dampers in combination with one or more heat transferring rosettes. The heat transferring rosettes comprise a flexible thermally conductive member that transfers heat by conduction from the bearings, bearing assemblies and/or heat generating devices to a remote heat sink.