ORIGINAL ARTICLE Localised tooth contact analysis of single envelope worm gears with assembly errors A. H. Falah & M. A. Alfares & A. H. Elkholy Received: 19 December 2011 / Accepted: 28 January 2013 / Published online: 14 February 2013 # Springer-Verlag London 2013 Abstract This paper presents the basis of loaded tooth contact analysis and predicts the influence of assembly errors on localised contact stress distribution in single envel- oping (cylindrical) worm gearing during a meshing cycle. A method for loaded tooth contact analysis, geometry and kinematics of such gear pairs is developed. The method accounts for the effects of tooth composite deflection caused by bending, shearing, foundation, tooth contact deformation and initial profile separation due to assembly errors. The method includes the determination of contact lines, load and stress distribution due to assembly errors. Because of the complex geometry of worm gear teeth, the tooth bending stiffness is calculated using the slicing technique developed earlier by the authors. Classical Hertz theory is used for calculating contact stress and deformation. A computer pro- gram based on the presented method has been developed and used to study the influence of errors on mating teeth contact. It is shown that the governing factors in loaded gears with assembly error are the mesh stiffness and the amount of error which is linked to load sharing between adjacent tooth pairs. A numerical example is presented to further clarify the outlined method. Keywords Worm gearing . Assembly error . Contact stress . Tooth stiffness Nomenclature c Half-length of the contact area between two cylindrical elements sliding against each other E Modulus of elasticity F Total load normal to profile for all contact lines f i Applied load per slice in case of error h Subscript refers to the wheel gear k Slice stiffness in case of no errors k eq Equivalent instantaneous meshing stiffness at contact k i Instantaneous meshing stiffness of contact point in case of error n i Unit normal load in case of no error p o Maximum Hertz contact pressure (stress) R Radius of curvature at the contact point on the slice R * Equivalent radius of curvature R 1 , R 2 Contact radii of worm r Subscript refers to the worm gear δ The total slice deflection δ b Bending deflection δ g Foundation deflection δ p Contact deflection between the teeth δ s Shear deflection δ t Torsional deflection ε Transmission error due to both assembly and profile errors θ Roll angle of each spur gear slice 1 Worm gear lead angle ν Poisson’ s ratio Δ Total slice deformation Δα Angular misalignment Δx Wheel axial offset due to assembly errors ΔC Center distance error 1 Introduction Surface fatigue failures due to rolling and sliding contact are the most important problems in gear industry. Under rolling A. H. Falah (*) : M. A. Alfares : A. H. Elkholy Department of Mechanical Engineering, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait e-mail: abdulazim.falah@gmail.com Int J Adv Manuf Technol (2013) 68:2057–2070 DOI 10.1007/s00170-013-4821-4