Rotor support structures including anisotropic foil bearings or anisotropic bearing housings are provided for controlling non-synchronous vibrations of rotating machinery. The rotor support structure comprises a foil bearing adapted to be disposed around a journal of a rotating rotor shaft and a bearing housing disposed around the foil bearing, wherein one of the foil bearing or the bearing housing is configured to be anisotropic. The anisotropic foil bearing and anisotropic bearing housing exhibit a stiffness in a first support direction that varies in magnitude from that in a second support direction that is substantially orthogonal to the first support direction.