NEWS

Modeling the Cost-Deviation Relationships for Geometric Features

<img src="/ueditor/php/upload/image/20260131/1769818960679428.png" title="1769818960679428.png" alt="2.png"/>The modeling of the cost-tolerance functions is based on the mathematical proposal by Andolfatto, which was presented in section 1 and uses a function comprising four parameters (Ref. 10). To determine a reference cost point, the method by Beckers is applied. It determines the process costs for a single non-purchased component by analyzing their parts. Data from machines from own use as also from machine manufacturers is used to realistically parameterize the equations for example. This applies to acquisition costs of the machines. A lifei of t = 10 years is assumed for all machines. Labor costs are also assumed to be uniform and constant. The costs per component are largely dependent on the individual process time tE,j. These are determined for a reference process on the basis of empirical knowledge or expert surveys.The modeling includes the process combinations shown in Figure 4. For the input and output shafts, the possible variations of soft turning, heat treatment and optional hard turning or external cylindrical grinding treatment are modeled. For the latter, upstream regrinding of the centering bores is simulated as an option, which, based on experience, results in a cost increase of p = +18% in terms of an additional process. The interlinked processes transfer their deviation data to each other. The reason for this is that the effort required to achieve, for example, low concentricity deviations in a subsequent process is significantly greater if the input quality is low due to a poorer turning process. The quality levels are determined from estimates in specialist literature and are an approximation for demonstration (Ref. 34).The manufacture of roller bearings as purchased parts is not simulated in individual processes. The simulated dimensions are divided into quality classes in accordance with ISO 492 (Ref. 35). The relative differentiation of procurement costs from the quality class is based on an expert survey of a bearing manufacturer. Compared to the standard quality class PN, this manufacturer sees a relative cost increase ofp = +10 percent for quality class P5 and p = +15 percent for class P4. It should be emphasized that the internal clearance for a standard deep groove ball bearing is not decisive for costs, as the tolerance zone width remains almost constant in the various internal clearance classes and, in principle, only the tolerance zone position of the ball’s changes. Cost increases of p = 3...5 percent are only to be expected for extreme internal clearance classes (C2, C5). Therefore, the influence of the internal clearance with regard to economic optimization is not considered further in the following. Figure 5, top lists the assumed reference times and tool costs for average qualities for the various processes and geometric features. Their validity assumes that the previous process (hobbing) also fulfils its reference quality (class A 6.5). For turning processes, the values only refer to the finishing of a single feature. The variation for determining the coefficients of the cost-tolerance curves for the features includes the primary process time th, the secondary process time tn and the tool costs KW. Other machine-specific costs such as procurement, depreciation, energy costs, etc. are recognized as non-qualified costs. Energy costs etc. are not considered to be decisive for quality and are kept constant for each process type. The values used correspond to own experience or information from machine manufacturers. The costs are stated in fictious currency (fC).For gear-specific processes, it was taken into account that different machining times and therefore also costs influence geometry parameters such as profile, flank and pitch deviations in various ways. For this reason, different cost-deviation curves (C-T curves) are used for the three deviation groups mentioned, which differ by constant factors for the sake of simplification. In this way, the aim is to map process-specific characteristics. For example, it is assumed for the profile deviations of the honing process that higher tool costs of p = +20 percent are incurred to achieve the same quality class than for flank deviations. The background to this is that the change criterion of the honing ring is the profile deviation. In addition, it is assumed that honing requires p = +40 percent longer idle times due to dressing in order to achieve the same quality level in the pitch deviations as in the profile and flank deviations. The same applies to the primary machining time. The reason for this assumption is that the elimination of pitch deviations on the workpiece or tool is more demanding. The parameterization involves assumptions that should be individually adapted by the user. After setting the reference points of each process concerning its machine times and assumed costs, the C-T curves are parametrized with factors according to the formula of Andolfatto (Ref. 10). For the primary machining time of gear hobbing and finish hobbing, p = +25 percent longer primary machining times are assumed, which lead to better quality classes due to reductions in the feed marks. For generating grinding, in order to achieve better concentricity through an extended infeed process, a idle time that is p = 20 percent longer is assumed to achieve the same quality class level in concentricity/run-out, based on profile and flank deviations. Therefore, time and cost intervals for the tool are used for the reference quality levels, depending on the type of deviation. The characteristic values can be parameterized in the method and can therefore be individually adapted. Figure 5 below shows the C-T function resulting from the assumptions for various machining operations. These also include the five calibration points per curve. It was possible to approximate the Andolfatto description approach with coefficients of determination R2 > 98 percent for all correlations (Ref. 10). For hard finishing of the gears, it can be seen that the C-T curves can also assume values for qualities A < 1. This purely mathematical, although no quality levels are defined in this area in reality. In addition to the aspect that a tolerance design in these particularly precise qualities quickly becomes unprofitable, the final tolerance design must always be checked for plausibility of the required quality class. A relevant aspect for achieving a certain target quality is the pre-machining quality and the clamping situation, particularly for gear machining. For example, the effort required to achieve a low total pitch error after the honing process is only possible in the reference time if the input component has a pitch error that is at maximum two quality levels higher. Otherwise, the pitch error could only be further reduced by extending the primary machining time. Another example is a clamping situation with concentricity errors or wobble due to poor pre-machining quality of the bearing seats or the wheel bore. In this case, there is a helix and profile angle deviation that the hard finishing process cannot compensate for, as this is specific to the component’s clamping. These deviations are therefore subtracted from the process result. The calculation method also takes into account the deviation of the process input quality compared to the reference quality of the previous process (mean values of the intervals specified in Figure 4) in accordance with Equation 4. When calculating the costs for a quality point, the factor fDev, in is multiplied by the hyperbolic and exponential components of Andolfatto’s approach. If the input quality is worse than expected, the factor is fDev, in > 0 and makes the downstream process more expensive, thus reducing its profitability. The sensitivity is assumed to be sWS = 1 for generating grinding and sH = 1.5 for honing, which means that the honing is more dependent on the input quality.As the absolute costs of a single component in a transmission depend on the process and the number of units produced, the costs for purchased parts (roller bearings) must be calibrated according to the number of units (Ref. 36). As there is no data in the state of the art on cost distribution in electric car transmissions, the cost shares determined in a study for a 7-speed dual clutch transmission (DCT) were converted and calculated according to the number of components of an e-drive transmission (Ref. 37). It was assumed that the assembly and testing costs in the EoL test are p = -75 percent lower than for the 7-speed DCT, as fewer components (NE = 16 instead of NDCT = 52 components) are required. Furthermore, the housing can be considered to be about half the size and complexity of the e-drive transmission compared to the 7-speed DCT and the mechatronics are completely eliminated.The cost distribution is subsequently considered to be independent of the number of units. Total fictitious costs of C = 28 fC are assumed for the entire gearbox. This results in a unit price per bearing on the pinion shaft of KBearing,Pinionshaft = 0.40 fC, while that of the wheel shaft is assumed to be p = +15 percent higher due to the dimensions.

27 Jan,2026

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