Spindle for carrying out machining assisted by non-ultrasonic axial oscillations, including a tool-bearing shaft, and an exciting portion, for subjecting the shaft to non-ultrasonic axial oscillations, especially during its rotation. The exciting portion including a first exciting stage, having at least one piezoelectric actuator, and a second exciting stage, having at least one piezoelectric actuator, having a non-zero axial overlap with the first exciting stage, the actuators of the two stages being arranged so that their effects add.