Steady State speed of rotor in following doubly excited system
Assuming that this is a synchronous motor, when AC stator power and DC rotor power is applied with no torque, the rotor will accelerate or decelerate very briefly to bring the rotor into alignment with the stator field.
When load is applied, the rotor will decelerate very briefly to achieve a torque angle required for the motor driving torque to match the retarding torque.
Re comment:
The motor torque is proportional to the product: the air gap flux X field winding magnetomotive force X sin torque angle. The torque angle is the phase angle between the rotor and stator fluxes. With the motor rotating without power applied, the initial torque would be determined by the the random angle of the rotor and would accelerate or decelerate the rotor to achieve a stable torque angle. That motion would be determined by the total load torque and inertia. It would tend to be a harmonic motion with a decreasing amplitude.
A lightly loaded motor would oscillate longer, but there would always be some  friction load from motor bearing friction and aerodynamic drag even if it is not coupled to a load.