The kinematic analysis and metrology of cylindrical worm gearing
Worm gearing is very widely used, especially in heavy industry, but due to the complexity of worm gear geometry, worm gear research has lagged behind that for spur and helical gears. In the last decade, however, the potential for significant improvement in worm gearing has dramatically increased: computers have given greater freedom to analyse worm gearing; CNC machines make it possible to aim for optimised worm gear geometries with very high accuracy and the development of synthetic lubricants has substantially improved lubrication conditions. In the UK, over the last few years, research effort in the field of worm gearing has increased considerably. As a part of this recent activity in the UK, the author has been involved mainly in developing the analytical mechanics and metrology of worm gears. A method for the generalised 3D non-elastic worm gear mesh analysis and associated software have been developed and worm wheel metrology software has been implemented on a CNC measuring machine in the UK National Gear Metrology Laboratory, to allow, for the first time, analytical measurement of worm wheel tooth flanks. Combination of the mesh analysis software and CNC measurement of worm wheels has assisted in the design and manufacture of worm gears with modified tooth profiles. Two methods of 3D non-elastic worm gear analysis have been developed for conjugate action and non-conjugate action respectively. The conjugate analysis determines the lines of contact, sliding and rolling velocities, limitations of the working area (the envelope of contact lines on a worm surface and singularities on a wheel surface), principal relative curvatures and the orientations of contact lines. It is based on the B-matrix method [Zhang and Hu, 1989]. The non-conjugate analysis predicts entry and exit gaps, contact ratio, wear marking on the worm flank, instantaneous contact topology on all the engaged tooth flanks, total contact area, contact pattern and transmission error. This is based on numerical simulation of the actual worm gear running process under no-load. Although the non-elastic analysis models have been designed for any type of worm gearing, and have been used to study Cavex (ZC) wormgears and the meshing of a ZA Abstract worm with a helical gear, most of the work has been on involute (ZI) worm gearing, since this is, by far, the most commonly produced type in the UK. This thesis presents the work as follows: 1) The development of the B-Matrix kinematic method for conjugate analysis. The B-Matrix method, presented in chapter 2, elegantly simplifies the derivation and calculation procedures, since the geometric parameters and the motional parameters can be arranged in separate matrices. As a result, the models can be applied to different geometries and coordinate systems with no need for further difficult derivations. The method leads to an easier way of integrating the theory of various types of worm gearing into compact generalised models. It is much more convenient and reliable to let the computer formulate and solve matrix equations numerically, treating each matrix as a simple variable, than to develop analytically the corresponding long tedious non-linear equations. 2) The development of mathematical equations to allow CNC measuring machines to measure cylindrical worm wheels with respect to their mating worms. The measurements are 3-dimensional and absolute, in the sense that the results are the deviations from the theoretical geometries rather than comparative measurements relative to a (necessarily imperfect) master worm wheel. The measurement theory has been implemented on a particular CNC measuring machine. This is presented in chapters 3 and 5. 3) The development of the non-conjugate analysis. The fundamental basis of the non-conjugate analysis presented in this thesis is to rotate the worm wheel to bring its tooth flank into contact with the worm flank at each given angle of worm rotation, so that the no-load transmission error and gap contours can be determined. This method is suitable for both cylindrical and globoidal worm gears, since the rotation angles of worm and wheel are used to simulate the running process directly. Abstract The method also allows the wheel tooth flank to be obtained either by conjugate analysis of the hobbing process, or by analytical measurements or other methods (for example, when a theoretically-generated involute helical gear is used to mesh with a worm). This work is presented in chapter 4. 4) implementation of the non-elastic analysis theory. The non-elastic analysis software has been written for personal computers. In addition, dimensional calculations specified by BS 721 and commonly used hob design methods have been added to the non-elastic analysis software to increase user-friendliness. The software has been used to investigate the effects on the worm gear contact and performance of machining errors and profile deviations or modifications. The structure of this analysis software allows for the inclusion of new modules for other types of worm gearing without in any way disturbing the integrity of the program's existing abilities. The non-elastic analysis software is user-friendly with a "Windows" graphical user interface. Software reliability and error tolerance have been of particular concern during program development. This work is presented in chapter 5. 5) The software has been thoroughly validated against other published results and/or actual production. The software has been used extensively for both research and commercial purposes, and the user interface developed further in response to user feedback. Examples of these applications are given in chapter 7.