An ultrasonically vibrated die comprises a generally cylindrical die ring supported by a coaxial resonant mounting tube. The die ring is vibrated in a radial bending mode of vibration, in which an end surface of the die ring oscillates between a concave and a convex state. The mounting tube joins the end surface of the die ring at a radius R where the amplitude of the oscillation of the end surface is at a minimum, in order to reduce transmission of the vibration into the mounting tube.