This paper aims to study the influence of tooth errors and shaft misalignments on the tooth contact in hourglass worm gearing with spherical meshing elements. The geometries of the worm and worm-gear tooth surfaces are described, and the distribution characteristic of contact lines is explored. Based on the common moving frame of conjugate tooth surfaces, the errors and the variations are quantified, and the error-variation equation is developed; in order to evaluate the influence levels of different errors, the error-sensitivity formulas are deduced and illustrated by a numerical example. The results provide a theoretical basis for the manufacture and the tooth contact analysis of hourglass worm pair with spherical meshing elements.
Worm gearing is one of most important mechanisms for transmitting the rotation and the torque between spatial crossed axes, and it is widely used in industrial gear systems that demand high transmission ratio, steady working situation and compact structure. There are three different types of worm gearing: (i) non-throated cylindrical worm drive, (ii) single-throated conical worm drive, and (iii) double-throated hourglass worm drive, and their theoretical and technical problems in design, manufacture and analysis have been the subjects of intensive researches of many scholars. In recent years, Fang et al. proposed some tooth profile modification methods for ZE, ZN and ZK-type worm gearings, and studied the effects on the meshing performance of these worm gearings [1], [2], [3] and [4]. In order to lessen the transmission error in the manufacture and assembly processes of conical worm drive, Litvin and Donno presented a method to modify the tooth surfaces with localized bearing contact [5]. By means of the meshing simulation, Zhang and Xu obtained accurate conditions for the formation of contact envelope and tooth undercutting in conical worm gear drive, which presents a basis for the proper design of worm gear blank and tooth geometry [6]. Due to high load-carrying capacity, hourglass worm gearing draws increasing attention from the researchers. Shi et al. applied the finite element method (FEM) to study the localization of the contact zone in the planar double-enveloping hourglass worm gearing [7]. Wang et al. presented a parameter optimization approach for the non-backlash double-roller enveloping hourglass worm gearing, in which the contact and lubrication performances were taken into consideration [8]. Chen et al. explored the real tooth surface of toroidal worm gearing with spherical meshing elements machined by means of the forming method [9].
Due to the impacts of elastic deformation, manufacture and assembly errors, the transmission error cannot be completely avoided, and then the meshing performance of gear pair will turn faulty. If the transmission error is a continuous linear function, a high level of gear vibration and noise will be caused. In order to reduce the negative influence of errors, a high-order discontinuous function of transmission error is usually chosen to absorb the linear error. Livtin et al. applied a parabolic transmission error function as the basis of tooth profile modification, and related findings for traditional gears have been presented [10], [11], [12], [13] and [14]. Stadtfeld and Gaiser applied a fourth-order function of transmission error to reduce the gear noise and to increase the gear strength of bevel and hypoid gear sets [15]. Wang and Fong presented a synthesis modification methodology for the tooth surfaces of a face-milling spiral bevel gear set by means of a predetermined fourth-order polynomial function of transmission error [16]. Xu et al. discussed the contact problem of conjugate surfaces with the effect of assembly errors, and verified the error compensation property of mismatched teeth [17]. Wu et al. studied the error-sensitivity of the mounting error in the point-contact spiral bevel gear pair [18]. Aiming to different types of cylindrical worm gears, Simon conducted an intensive investigation about the influence of tooth errors and shaft misalignments on loaded tooth contact [19]. All these efforts have contributed significantly to the progress of gear design, manufacture and analysis technologies.
The methods to study the transmission error can be roughly classified into two groups: (1) methods in which the effect of tooth deflection under load is taken into account, and (2) methods which are conducted from the kinematics point of view. In this thesis, hourglass worm gearing with spherical meshing elements is considered under the rigidity condition without taking the load-dependent deformation into account. The geometries and the contact characteristics of conjugate tooth surfaces are described, and the error-variation equation, reflecting the inherent relationships between the errors and the variations, is developed. In order to evaluate the influence levels of different errors, the error-sensitivity formulas are deduced and illustrated by a design example of mismatched hourglass worm pair with spherical meshing elements.
On the basis of obtained results, the following conclusions can be drawn:
(1)
In hourglass worm gearing with spherical meshing elements, the rotation angle variation is only involved in the position difference between the directrix of the worm surface and the center of the steel ball, and is irrelevant to actual shapes of the worm surface and the steel ball; the position variation is a synthetic result of multi-factor co-action, but at the tooth depth direction it is irrelevant to the angular error.
(2)
The error-sensitivities related to the angular error are far less than the error-sensitivities related to the linear error. The tooth contact is severely influenced by the radial and tangential errors of the worm gear, however it is insensitive to the axial error of the worm gear and the perpendicularity error of axle holes.
(3)
The induced tooth pitch cumulative error of the worm gear is generated by the rotation angle variation and it influences the number of teeth being in mesh at every instant; the contact area is influenced by the position variation along the tooth depth direction.
