Design method of end worm gear drive, including tooth curve geometry and improved lubrication angle formula

End worm gear drives are characterized by their multi-tooth contact, compact contour, and theoretical potential to overcome some inherent flaws of cylindrical worm drives. However, quantitative basic research on end worm gear drives is currently inadequate, which hinders the development of this transmission. This work focuses on the computational design of end worm gear drives and proposes a new Niemann-type design. Meshing models of the proposed drive are established, and its engagement theory is deduced systematically. Based on the derived tooth surface equations, an innovative research methodology for the tooth curve geometry of the end worm gear is created, and the tooth curve in the worm gear reference plane is proved to be a spiral. An improved formula for the lubrication angle is developed, which can provide more rational numerical results for the angle. Theoretically, the modified formula is universally applicable for line contact drives and can be used to quantitatively investigate the lubrication level between the teeth for the proposed drive. Simulation outcomes demonstrate the favorable characteristics of the transmission, including broad conjugate areas, even contact lines, and fine global lubrication state.