Abstract: Machine for machining an envelope surface of a cylindrical work piece includes a tubular cylindrical housing with a longitudinal axis and having an inner space extending from a first end of the housing, and with an axis offset from the housing axis, a shaft angularly displaceable in the inner space and provided with a recess, and a motor in the recess. A spindle is coupled to the motor and carries a first rotatable tool, a rotatable outer casing encloses the housing, and a lid is arranged for co-rotation with the casing and extends radially over the first end of housing with the lid having a central through opening with the peripheral surface. A second rotated tool is disposed on the peripheral surface, and a support device is adapted to hold the work piece between the tools. The shaft is arranged to cause the first tool to effect radial displacement relative to the second rotated tool.