It has special application for preparing surfaces of two pieces of a composite material (fiber and resin) which have to be glued by means of an adhesive, establishing a perfect mutual contact between them. The preparation of the surface needs to be uniformed and without damaging the piece.It consists of a moving frame (1) which can be displaced along some longitudinal guides (25-26; 27-28) integral with a bench (2) on which the piece (4, 6, 8) to grind is kept immobile. It includes a casing (14) containing a tool-carrier head (15) which rotates inside the casing adopting a floating arrangement for smoothly and uniformly pressing the piece to grind, advancing along the whole of the piece and following the geometry of its surface.The casing (14) incorporates a nozzle (16) for connecting the inside of the casing to a suction hose for the dust and other particles produced. Moreover, said casing (14) can be displaced laterally in order to grind possible lateral surface extensions.