Calculation Method for Designing the System Tolerances of a Cylindrical Gear Stage with the Integration of Manufacturing Efforts

Firstly, the calculation process is explained in general terms. This is followed by more detailed explanations of the implementation of the tolerance-cost functions and the optimization criteria. The software MATLAB is used to analyze the production costs and acoustic parameters, while the remaining calculations are carried out in Python. Structure of the Algorithm, Input/Output Variables and Assumptions The aim of the calculation algorithm is to determine a particularly economical tolerance specification for defined correlations between production costs and specifications for the variance of the excitation behavior of the running gearing. The determination of the tolerances is integrated into the development process of an overall gearbox at the point after the micro geometry design. The macrogeometry of the gears, shafts and bearing distances are also taken into account. Furthermore, the displacement behavior of the housing and the bending lines of the shafts are used to consider the load-related displacement in the tooth contact. These parameters are integrated into a model of a finite element-based tooth contact analysis which, together with a software extension, takes system tolerances into account. In detail, the extension reduces important system tolerances such as concentricity deviations of the shafts, the roller bearings or position deviations of the bearing seats in the housing on the tooth contact in terms of axis tilt and skew, wobble, eccentricity and center distance deviation (Refs. 27, 28). It is also possible to consider load-free effective bearing clearances and modified micro-geometries of the tooth flanks that were previously designed. It is assumed that neither the micro geometry of the tooth flanks (change in the resulting force application points) nor the deviations of surrounding components (e.g., change in tilting rigidity) significantly influence the load-related misalignment behavior of the gears. Further input parameters of the method are information on the occurring scatter shape of each geometry deviation, see Figure 3, top left.

The scattering forms are modelled as normally distributed for deviation types that can assume negative and positive values. Concentricity deviations and backlashes, on the other hand, can only take on positive values, so that these deviations are modelled with a first-order normal distribution. For real-world applications, it may be necessary to adapt the distribution shapes, as the dispersion of real processes can deviate from idealized assumptions (Refs. 29–31). In principle, all analytically describable density functions could be implemented here. An analytical description of the probability density functions is required so that the optimization algorithm can generate the variants at a later point in time according to the corresponding distribution and adjust the scattering. In addition, an evaluation of the acoustic behavior and a weighting between excitation behavior and costs are required as input. The reason for this is that there is a design dilemma between low manufacturing deviations or low excitation behavior and low manufacturing costs, which requires a compromise solution. To achieve this design objective, the calculation method iteratively generates a variant plan. Each variant contains a freely definable number n of simulations of virtual overall gearboxes whose geometric scattering corresponds to the previously defined distribution shapes. The excitation behavior is afterwards calculated from the n individual points, see Figure 3 in the center. Each iteration is then checked for fulfilment of the acoustic target criterion. For this purpose, descriptive statistical characteristic values are determined and evaluated for the variant, consisting of n individual points. These are the mean value, standard deviation and various quantiles for selectable order components of the transmission error. This is explained in detail in the following section. In addition, the manufacturing costs are estimated on the basis of the geometric deviations. Depending on the available process types, individual functions are parameterized for each geometric feature, see Figure 3, bottom left. For each iteration cycle, the optimization procedure is used to adapt the variant plan for the geometry data. In addition to the uniform particle-swarm-optimization (PSO), the use of a multi-criteria PSO Pareto optimization is also investigated. The statistical tolerance specifications, which fulfil the excitation requirements with the lowest possible production costs, are derived from the calculation method, see Figure 3, on the right. Depending on the selected weighting and excitation evaluation, different designs result as solutions.