Homogeneous geometry calculation of arbitrary tooth shapes - Mathematical approach and practical applications

As an extensive machine element to transfer and convert rotational movement, gears meet high requirements for construction and assembly. Due to existing modern production techniques, more sophisticated gear types can be produced with high precision and maintainable financial effort. The benefits of traditional gear profiles, such as an involute, are thus no longer of major importance. In particular, for gear types such as bevel, worm, and hypoid gears, but also for non-standard gear types (e.g., beveloid gears, crown gears, or spiroid gearings), modern gear production systems ensure high quality and reliability to the operator. Depending on the context of application, different gear types have advantages and disadvantages concerning load carrying capacity, effectiveness, or noise excitation. Supported by various calculation software tools for the particular gear type, it is possible to create the optimal gear design, depending on the respective application. A homogeneous calculation software for ubiquitous gear geometries-irrespective of the gear type, and especially for analyzing non-standard gears-would be preferable. This paper provides a mathematical framework and its implementation for calculating the tooth geometry of arbitrary gear types, based on the basic law of gear kinematics. The rack or gear geometry can be generated in two different ways: by calculating the conjugate geometry and the line of contact of a gear to the given geometric shape of a known geometry (e.g., a cutting hob), or by prescribing the surface of action of two gears in contact and calculating the correspondent flank shapes. Besides so-called standard gears like involute spur and helical gears, bevel or worm gears, it is possible to analyze the tooth geometry of non-standard gears (e.g., non-involute spur, conical, or spiroid gears). Depending on the type of gear, a distinction is made between tool-dependent and tool-independent geometry calculation. The described mathematical algorithms are summarized in implemented software modules for the particular gear types. Two practice-oriented examples are presented to illustrate the calculation model: beveloid gears for use in vehicle or marine gear boxes as well as rack-and-pinion meshing with variable ratio, as it is used for steering systems on automobiles. Since the geometry is exported as a point cloud, a further analysis of the generated gear types is possible, e.g., by computer-aided design or finiteelement software tools as well as manufacturing on 5-axis CNC or forging machines. Thus, a detailed analysis-especially of non-standard gears-is feasible that currently cannot be calculated and evaluated with common industrial gear calculating software. The project is funded by the Forschungsvereinigung Antriebstechnik e.V. (FVA).