Rotor systems having lead-lag damper cooling

A rotorcraft comprising:

• a fuselage;
• a powertrain disposed at least partially within the fuselage and including a mast;
• and
• a rotor system coupled to the powertrain, the rotor system including:
• a rotor hub coupled to the mast and rotatable therewith;
• a plurality of blade grips coupled to the rotor hub;
• a plurality of rotor blades each coupled to a respective one of the blade grips;
• a fairing disposed at least partially around the rotor hub;
• a plurality of lead-lag dampers each coupled to at least a respective one of the blade grips, each lead-lag damper having a damper heat exchanger and a fluid pump operably associated therewith;
• and
• a fairing heat exchanger in fluid communication with the damper heat exchangers and the fluid pumps;
• wherein, each lead-lag damper is configured to drive the respective fluid pump responsive to damping operations to pump a cooling fluid from the respective damper heat exchanger to the fairing heat exchanger.