Enhanced gear efficiency calculation including contact analysis results and drive cycle consideration Dipl.-Ing. J. Langhart, KISSsoft AG, CH-Bubikon; M. Sc. T. Panero, KISSsoft AG, CH-Bubikon Abstract The efficiency calculation and thermal rating for gearboxes is meanwhile a standard analysis which is requested by the customers. The basis for these calculations is the ISO/TR 14179 [1], which includes the power losses for various machine elements as well as the heat dissipation calculation. For the gear meshing losses, the formulas from ISO/TR 14179 are established but the problem remains that no flank modification are considered in these calculation. Also a known issue is the inaccuracy in the losses of oil splashing (churning losses) and other lubrication depending effects. These losses require some correction factors which let the losses adjust based on preceding measurements. These enhanced calculations are applied in KISSsys and make it capable to consider these effects on a system level. Furthermore, the consideration of a drive cycle allows the user to obtain the maximum operating temperature and also the critical load bin for the thermal stress. Additionally when the drive cycle is given the temporal temperature profile is calculated and the critical parts of the drive cycle are determined, together with the temporal losses. Thermal rating in KISSsys Basically the technical report is implemented completely in KISSsoft and KISSsys. However, due to the edition date of 2001, there are some improvements in some power loss calculation due to the current technical state of the calculations, which are considered in the software additionally to the formulas in the ISO. So, the bearing losses are updated according to SKF, methods 1994 and 2013 [2]. For the bevel and hypoid gears, the calculation according to Wech [3] is additionally available. The seal losses are calculated by the ISO/TR 14179. A main task of the calculation in KISSsys is the correct kinematic calculation of the gearbox. This includes an iteration of the torque, as the most of the losses depend on speed and torque, since each shaft needs to achieve a torque equilibrium, which is depending also on the exact input torque from the previous shaft. Having determined the power losses and heat dissipation values, several evaluations are possible as e.g. the cooler dimensioning, thermal rating and maximum permitted input torque
for a given maximum oil temperature. Finally a detailed report is provided which shows many details and the individual losses (Fig. 1). Fig. 1: Thermal rating considering power losses and heat dissipation Correction factors In order to adjust the sometimes simplified calculations in the ISO, in KISSsys several correction factors are applied which allow the user to match the calculation results with the measurements. This provides a very practical approach, which was already introduces in earlier conferences [4]. The correction factors can either be applied for general categories as churning losses, meshing losses, bearing losses and sealing losses (Fig 2). If required, the correction factors can also be applied individually for a specific parameter, so for example the churning losses for a planetary stage can be modified individually. Fig 2: General correction factors for power losses or heat dissipation
Temperature curves based on drive cycles One important add-on in KISSsys is the calculation of temperature curves based on drive cycles. It is a main requirement for a gearbox not to exceed a certain temperature of neither the oil nor the housing surfaces. The reasons typically are that the oil will lose its positive properties or not achieve the required service life time and also the shaft sealing rings are endangered. On the other hand, in some cases the housing surface temperature shouldn t exceed for example 60 C, in order avoid the hazard of burning human skin. During the operation of a gearbox, the temperature typically rises and declines because of the applied drive cycle. In the strength calculation, the drive cycle is represented by a load spectra and the sequence of the bins is neglected. For the operation temperature, the exact sequence has to be considered. Only by using the load spectra, the designer can t estimate the temperature curve and its maximum. In order to determine the temperature curve the temperature gradient is calculated using the heat capacity of the gearbox. The calculation is done in definable temperature step widths, either by defining the maximum allowable temperature change or the maximum number of steps. Application sample with industrial gearbox A typical example for the application of the temperature curve is an industrial gearbox. As reference, a worm gear unit from the company ZAE-AntriebsSysteme GmbH & Co KG [5] will be shown here (Fig. 3, left). The company ZAE is specialized in the development and manufacture of innovative drive systems and components. The product range includes both powerful standard gear units and individual solutions according to customer specifications. In the following, the worm gear transmission for ratio 20:1 and nominal power of about 6 kw on the regarded operating point was closer investigated. Fig. 3: worm gear unit on the test bench (left), KISSsys model (right)
The worm gear drive unit is designed as double side input shaft and single side output shaft. In this test bench arrangement, on one side of the input shaft the fan was mounted for ventilation. The power losses in this gear box are caused by 4 roller bearings, the sealing on the shafts and the worm meshing with the worm wheel. The housing was defined in KISSsys according to the dimensions of length, width and height, and also with the measured surface area from the CAD model. As usual, the values of the calculated area from the (simplified) dimensions didn t match with the measured surface data, so the priority was to use the exact surface area according to CAD model (Fig. 4). Fig. 4: Surface definition according to ISO/TR 14179 (left), surface determination in CAD (right) According to the ISO/TR 14179-2 the heat dissipation calculation considers the influences of finnings, foundation and outcoming parts. The data for the finnings are very detailed and consist of the total surface, the projected surface as well as the height and length of the finnings. All these data are derived from the CAD model. The diameter and length for the outcoming parts (shafts and couplings) can be defined comfortably in KISSsys (Fig. 5). Fig. 5: Automatic definition of shaft and coupling diameter and length in KISSsys
The foundation is defined according to its real dimensions and as heat transfer up- and downwards. The ventilation speed is 1.4 m/s, as the speed of the input shaft is 1000 rpm and the diameter of the fan is 163 mm. Two important parameters for the heat dissipation are the heat transfer coefficient k* and the emmision ratio ε. The heat transfer coefficient k* is either calculated by the ISO or defined by own input. The emmision ratio ε is the ratio between oil and housing temperature. As a first approach, the heat transfer coefficient k* was used as calculated value, and the emmision ratio ε was defined as 1. Adaption of heat dissipation by using correction factors The worm gear drive was applied in a test bench with an input speed of n 1=1000 rpm and an output torque of T 2=1020 Nm. The temperature measurement (Fig. 6) was recorded for the oil temperature (red) and the ambient temperature (blue). Fig. 6: Temperature curves for measurement and calculation results The green line represents the calculated oil temperature which is based on mean thermal coefficients, which matches very well in the main part of the measurement. However, in the beginning of the warming-up phase the difference is slightly bigger, but still acceptable. The figure 7 below shows the results of the calculation and the measurements. The first temperature calculations using the calculated k* = 36 W/m 2 K and the emmision ratio ε = 1 were deviating by max. 4K. Using optimized values for k* = 40.9 W/m 2 K and emmision ratio ε = 0.925, the correlation between measurement and calculation matches within 1K, which is a very good basis for further calculations. The power losses were not modified in this case.
Fig. 7: Comparison of KISSsys calculation to the measurement of the test bench Drive cycle and temperature curve The designer has the task to proof the drive cycle and to make sure that the temperature of the worm gearbox doesn t exceed the limit of the oil temperature of about 90 C at an ambient temperature of maximum 40 C. The customer wants an oil changing interval of at least 10 000h and doesn t want to endanger the shaft sealing s life time. The drive cycle is a power-on/power-off cycle, having a shorter power-off period. The calculated temperature profile, using the gearbox data from test bench, shows that the oil temperature will not exceed the limit of 90 C (Fig. 8). This result was convincing the customer to bring this gearbox in use. Fig. 8: Initial (orange) and improved (grey) drive cycle for a maximium permitted oil temperature of 90 C
Meshing losses from gear contact analysis The gear meshing losses are calculated in ISO/TR 14179 according to Niemann. This is a reliable, but nowadays simplified method, as the calculation doesn t considering the micro geometry of the gears. In noise-optimized transmissions typically the gears are calculated with the gear contact analysis, considering the gear misalignments, based on the deformations and resiliencies from the shafts, bearings and housings. The target is to find the optimal modifications such as lengthwise crowning, helix angle modification and tip relief in order to get the optimal noise behaviour, but also the minimum power losses of the gear pair (Fig. 9). Fig. 9: Gear contact analysis with stress distribution represented on gear (left) and power loss along the path of contact (right) Application sample with an automotive transmission The automotive transmissions focus on power losses, as the CO 2 reduction is a declared target and oblige the design to improve in that field. On the other hand, the transmissions are rather noise sensitive and hence have quite high demands in NVH. The xdct transmission from FEV GmbH [6] is shown here as an example. The xdct family is a series of dual clutch transmission concepts, setting a new benchmark regarding number of gears per mechanical complexity, means, the 7-xDCT features 7 forward speeds by using only three shift sleeves and 14 gear wheels in total (Fig. 10). Herewith the weight, cost and installation space is significantly decreased. Because of its favorable arrangement and short length, the gearset arrangement is extremely robust with small values for shaft bending [7].
Fig. 10: 7-xDCT prototype and main technical data Efficiency calculation using correction factors For the kinematic layout the transmission was modelled in KISSsys (Fig. 11). As a basis for the efficiency calculation, the ISO/TR 14179-2 was applied. Assuming that the technical report may be too simplified, the target was to find suitable correction factors for the power losses, so that the results finally will deliver useful results and can be applied for future designs as well. In this case, the 7 th speed was calculated with the combination of 5 torques (50, 75, 100, 150, 240 Nm) and 5 speeds (1000, 1500, 2000, 3000, 4000 rpm). Note, that the 7 th speed is a generic speed, means, 4 of the total 8 gear pairs are in action. Fig. 11: KISSsys model with power flow of 7 th speed (red gears are in the power flow, grey gears not in power flow)
The calculated power loss values are compared to the measurements and the results are shown in the table below. In order to achieve the match of the results, the calculated results are modified by individual correction factors per speed, for gear meshing losses and churning losses. Finally, the calculation match astonishingly well with the measured data, and is capable to predict the power losses for any other speeds and similar transmission designs (Fig. 12). Fig. 12: Comparison of KISSsys calculation to the measurement of the test bench Looking at the correction factors, it can be found that the optimal correction factors for churning losses, they are nearly constant and obviously independent of the applied speed and torque. In contrary the optimal correction factors for the meshing losses are decreasing by approximately 20%, between speed n 1=1000 rpm and n 1=4000 rpm (Fig. 13).
Fig. 13: Correction factors varying depending on speed Contact analysis and gear modification sizing The final drive of the xdct transmission was analyzed in order to optimize the gear meshing losses. Therefore, the KISSsoft modification sizing was used, where the profile modification was varyied in value and length (begin of modification). The length (coefficient) of the tip relief was varied between long, short, or no tip relief. The value was varied between 10 and 20 μm. As the value and coefficient were cross-varied within the same gear, there were 81 variants calculated. The results are shown in the radar charts for efficiency and peak-to-peak transmission error (Fig. 14). Fig. 14: Evaluation of efficiency (left) and peak-to-peak transmission error (right)
The evaluation of the best solution depends on the priorities of the designer. Considering the efficiency as the highest priority, the variants 72, 78 and 81 are preferred. If the criterias efficiency and PPTE are to be balanced, the designer may prefer the solutions 65, 66, 67. Conclusion It is shown that the ISO/TR 14179-2 can be applied for industrial as well as for automotive transmissions. With the help of correction factors, both the efficiency and the thermal rating calculation results are precisely matching with measurement data, which allows a fast and precise prediction for similar transmissions. As the industry transmissions require more and more the application of load spectra and drive cycles, a reliable method is presented how the temperature curve and hence the maximum temperature can be determined. Further on, the gear meshing losses are to be considered using the gear contact analysis, in order to have an exact evaluation of gear flank modifications. With the KISSsoft contact analysis the modification sizing can be performed and the results easily evaluated for the best suitable solution. [1] ISO/TR 14179:2001-07(E), Gears Thermal capacity, Berlin [2] SKF, Main Catalog, 1994 and 2013 [3] Wech, L.: Untersuchungen zum Wirkungsgrad von Kegel- und Hypoidgetrieben, Diss. TU München, 1987 [4] Langhart, J.: How to get most realistic efficiency calculation for gearboxes, International Gear conference Lyon, France, 2014 [5] Website ZAE-AntriebsSysteme GmbH & Co KG, www.zae.de, 2015 [6] Website FEV GmbH, www.fev.com, 2015 [7] Hellenbroich, G.: FEV's Extremely Compact 7-xDCT - First Test Results, 22. Aachener Kolloquium Fahrzeug- und Motorentechnik, 2013