A container bodymaker gearbox (20, 130) drives rams (60, 66) linearly by use of a hypocycloid drive that controls operation of a rotating and orbiting output shaft (40) by synchronizing the rotating and orbiting motion such that a tracking point at a predetermined radius from the shaft 40 tracks a straight line. A coaxial sleeve (124) is spaced around shaft (40) and carries an annular crank hub (140) on bearings (142). The crank hub (140) is eccentrically mounted on the sleeve (124) with the hub centerpoint positioned to track the straight line tracking point. A rotary ring (166) mounted around the periphery of the crank hub (140) supports diametrically opposed connections (176) to rams (60, 66). The ring (166) and rams (60, 66) translate on the straight-line path determined by the centerpoint, with the hub (140) rotating as required within the ring (166) to cancel rotary motion of hub (14) from the translational motion of the ring (166).