A resistive torsional mode damping system for a shaft of a machine includes: a sensor configured for sensing a signal representative of torque on the shaft; a controller configured for using the sensed signal for detecting a presence of a torsional vibration on the shaft corresponding to a natural frequency of the shaft and for generating control signals for damping the torsional vibration; and a damper including a damping converter and resistor coupled to a DC output of the damping converter, the damping converter being coupled to the machine through a power bus and having a power rating on the order of less than or equal to about five percent of a nominal power of the machine.