Study On Characteristics Of Curves And Surfaces Of Logarithmic Spiral Bevel Gear

Spiral bevel gears are a sort of space bevel gears what are applied widely to various drive with intersecting shafts and crossed axis under high-speed heavy-load, especially aircraft, ships, precision machine tools, construction machinery, vehicles transmission and so on. Spiral bevel gears have the advantages of smooth transmission and high carrying capacity, but due to very complex and irregular geometry shape of tooth surface, the processing adjustment depends largely on the experience of the operator, and it results in a very difficult precision machining of tooth surface; spiral bevel gears belonged to point contact is very sensitive to the manufacture and installation error. Therefore, the design and manufacturing of spiral bevel gears has occupied a very important position in the machinery industry.At present, ordinary spiral bevel gears'tooth trace are arc, involute and extended epicycloid line, and the helix angle of each point on tooth trace are not equal, well, in theory, it can not guarantee the most rational engagement. Therefore, this project innovatively applies the skill that the helix angles of every point on logarithmic spiral line are equal to the tooth trace of spiral bevel gears with trying to solve the problems that during the drive of the spiral bevel gear unequal helix angle at different meshing point brings shortcomings such as low dynamical transmission efficiency, serious friction and wear, unsmooth transmission and so on. In addition, the fact that the tooth surface of logarithmic spiral bevel gear is a regular space camber built by the conical logarithmic spiral line and the involute provides a practical theoretical basis for the stress analysis on the tooth surface, the strength calculation of the tooth surface, processing and manufacturing and so on.The main research work of this subject can be summarized as follows:1. It analyzes the parameter of conical logarithm spiral equation and the characteristic of conical logarithm spiral, and proves the continuity and smooth of conical logarithm spiral.2. By programming with three dimensional parameterization software of Pro/E and MATLAB, the three dimensional parameterization modeling of conical logarithm spiral is realized, and the characteristic that conical logarithm spiral itself can satisfy uniform distribution on the cone is obtained. Simultaneously using the three dimensional software and the mathematical theory analysis to change partial parameters of conical logarithm spiral equation, then it is obtained that conical logarithm spiral whose beginning position changes and revolving is same or beginning position dose not change and revolving is opposite; Moreover when conical logarithm spiral's beginning radius vector and ending radius vector are determinate, it is obtained that the variation range of unique parameter in the conical logarithm spiral equation, that is establishing the boundary condition of conical logarithm spiral.3. By means the column surface vector equation's construction method, it constructs logarithmic spiral bevel gear's tooth surface vector chart that takes involute as the tooth profile and takes the conical logarithm spiral as the tooth trace, then through the coordinate transformation principle and the characteristic of conical logarithm spiral and the involute it establishes tooth surface equations of the left and right tooth face, and obtains tooth face's three-dimensional model in MATLAB with the M language programming. It proves the tooth face's construction method and the tooth face equation's accuracy, and quantifies tooth face's two parameters, and constructs tooth face's boundary condition completely.4. Based on the differential geometry knowledge, it analyzes the mathematics characteristics such as tooth face's normal curvature, principal curvature, principal direction, geodesic curvature and geodesic torsion and so on, and tooth face's mathematics characteristic is obtained. It lays mathematics foundation for researching new type of bevel gear——logarithmic spiral bevel gear's project significance.