The workpiece clamps of at least one finishing station are mounted above the workpieces so that the workpieces hang down during the finishing process. A collecting chamber (12) for the chippings is mounted underneath the workpieces free of clamping devices and finishing heads so that all the chippings can be collected. The chippings are removed through transport discharge devices. The workpieces can be mounted on pallets held by the clamps. Transfer members for example rhythmic rods (5) act from above on the workpieces or pallets.
