Research On Transmission Theory Of Spherical Gear With Ring-Involute Tooth

With the development of modern gear transmission technology, in some rising fields which need multi-degree-of-freedom transmission, such as the fields of robotics, bionics, and vector propulsion, and so on, the traditional single-degree-of-freedom gear mechanisms increase the difficulty about the research work. So, Trallfa spherical gear, the first 2-DOF gear mechanism, is focused on in engineering immediately. However, since Trallfa spherical gear with discrete cone-shaped concave teeth has two disadvantages: difficulty to manufacture and low load capacity, it hasn't been used widly. Afterword, some researchers redesigned the tooth of Trallfa spherical gear, and some 2-DOF gear mechanisms appeared, such as spherical gear with discrete circular arc teeth, spherical gear with discrete ring-involute teeth, spherical gear with discrete cone teeth and so on. These teeth designs haven't broken through the limitation of discrete teeth; therefore the difficulties in manufacture and low load capacity haven't been improved evidently. A spherical gear with continuous ring-involute tooth studied in this paper overcomes the defects of discrete teeth and has the advantages of easy manufacture and good load capacity. Some basic researches about its transmission theory and kinematic analysis have been made. However, as a 2-DOF gear driven mechanism, the transmission characteristics of the spherical gear with continuous ring-involute tooth are more complex than the traditional gears, and therefore it is very necessary to make in-depth research in theory for the progress of its transmission technology. This thesis is dedicated to solving some problems in theory about the spherical gear transmission, in order to set up a relatively whole system of transmission theory of the spherical gear, and promote the development of 2-DOF gear transmission technology. The major research efforts include the following aspects.1. The 2-DOF meshing theory of conjugate surfaces of the spherical gear pair is studied. The two-parameter kinematic model of the spherical gear pair is established, and the kinematic analysis of two basic mechanisms, the ideal mechanism and the gimbal mechanism, is performed. And then, the relative movement between two teeth surfaces in mesh is analyzed and based on the two-parameter meshing model, the equations of the meshing cone surface and conjugate surface are established. The simulation of the conjugate meshing principle of the ideal mechanism and the gimbal mechanism indicates that the teeth surfaces in mesh are the conjugate surfaces.2. Sliding ratio is an important index which can evaluate the wear of spherical gear tooth surface. According to the 2-DOF meshing theory of the spherical gear and the basic definition of sliding ratio, the calculating method of the sliding ratio of the spherical gear is studied, and the specific equation is received, which is a dual function with the swaying angleφ_y1 and the azimuth angleφ_z1 and relates to the angle velocity ratioεalong with two respective motion directions. Then, the sliding ratios of the teeth surfaces in mesh in the ideal mechanism and the gimbal mechanism are computed and analyzed and according to the distribution of sliding ratio, the following rules can be gotten: wear on tooth middle part is slight; wear on tooth tip is relatively severer; wear on tooth base is the severest. In addition, in the gimbal mechanism, the installed axes have a good effect on sliding wear of tooth surface of the spherical gear.3. The tooth contact characteristics analysis and stress analysis for the spherical gear are investigated. Theoretical derivation validates that tooth contact form of the meshing spherical gears is a point contact between a convexity and a saddle surface except the position in which the polar axes of two spherical gears are collinear to each other. Then, the meshing model of the spherical gear pair for computer simulation is set up, and the position of instantaneous contact point of two spherical gears can be gotten using the nonlinear solver composed of five equations. Furthermore, the contact point paths of the spherical gear in mesh are studied. Combined the curvature analysis method with the tooth surface separation topology method, the contact ellipse of the spherical gear pair is calculated and simulated, and the distributing rule is gained. Finite element analysis is applied to perform contact stress analysis and bending stress analysis of the convex tooth, the weakest tooth. The result indicates that: the distributing form of contact stress is an ellipse; regardless of contact ratio, contact stress is large on tooth base and tooth tip, and contact stress is little on tooth middle part; considering contact ratio, contact stress is evidently decreased on tooth base and tooth tip; bending stress is little, and the maximum of bending stress occurs on tooth middle part.4. The effect of assemble errors on the transmission error of the spherical gear mechanism is analyzed and simulated. The transmission error model and pointing error models of two mechanisms of the spherical gear sets in common use, the gimbal mechanism and the pointing mechanism, are set up, respectively. After that, the kinematic models of two mechanisms with assemble errors are built, and the assemble errors comprise the vertical and horizontial axial misalignment of the fixed axis and rotating axis, and the center distance error of two spherical gears, respectively. With these models above, the transmission errors of the gimbal mechanism are simulated and analyzed, and using two computation methods, the pointing errors of the pointing mechanism are simulated and analyzed. The results are useful for the installation and measurement of the spherical gear pairs in practice.5. Grinding method is the key for solving the finishing machining problem of the spherical gear. According to tooth surface characteristics of the spherical gear, with forming method and generating method, respectively, the grinding manufacture of the spherical gear are researched. To grinding with the forming method, according to the required degrees of freedom, a grinding machine and a fingerlike grinding wheel are designed, then the relationship of motions of the machine is analyzed and the grinding principle of the fingerlike grinding wheel is studied. To grinding with the generating method, similarly, a grinding mchine and plate grinding wheels are designed, and then the conjugate meshing kinematic equations are set up. The generating process of the involute tooth surface and the tooth base surface with the plate grinding wheels are introduced in detail. In the end, based on the mathematical models of the plate grinding wheels, the mathematical models of the spherical gear pair are received and the machining parameters can be set and adjusted.