Abstract: Apparatus for machining workpieces by abrasion by means of an oscillating ltiple-blade tool, which is moved across the workpiece. The moved masses define an oscillatory system, as a result of which the energy can be stored in the reversal ranges by a buffer system. This apparatus, especially when designed as a lap-cutting machine for cutting semiconductor rods, is preferably driven by the oscillation of the masses to be moved, and is driven in the resonance range. The tool carriages are buffer loaded at each end and are oscillated at or near the natural resonance frequency of the moving masses.