Method of measuring the load distribution of spur gear stages

Transcription

1 Bulletin of the JSME Journal of Advanced Mechanical Design, Systems, and Manufacturing Vol.11, No.6, 2017 Method of measuring the load distribution of spur gear stages Markus DAFFNER*, Michael OTTO* and Karsten STAHL* *FZG Gear Research Centre, Technische Universität München Boltzmannstraße 15, Garching, Germany Received: 5 April 2017; Revised: 1 June 2017; Accepted: 29 June 2017 Abstract Precise information about the load distribution in the tooth contact can improve the quality of calculation results regarding strength and durability of spur and helical gear stages. A method of measuring the load distribution of spur gear stages in lab conditions has been developed. The basic idea is, that tangential deformations on the backside of a loaded tooth flank are directly linked to the load distribution on the tooth flank. A measuring procedure to detect the aforementioned tangential deformations has been developed. The measurement results can be compared to calculated load distributions with two methods. The first method is to generate tangential deformations out of a calculated load distribution by using the finite element method (FEM). The measurement results can be compared to the so determined tangential deformations. The second method is to directly generate a load distribution out of the measurement results. Calculated load distributions can be directly compared to a so determined load distribution. Therefore, an iterative calculation approach using the FEM has been developed. Using the methods mentioned before, calculated load distributions can be validated. Keywords : Load distribution, Measurement, 3D coordinate measuring machine, Static test rig, RIKOR 1. Introduction In the standardized method ISO 6336, the load distribution of spur gear stages is taken into account by the face load factors KHβ and KFβ. Both face load factors depend on the maximum load per unit face width and the average load per unit face width in the mesh. Regarding more sophisticated local calculation methods, precise information about the load distribution in the tooth contact gets more and more important. The load distribution of spur gear stages depends on a variety of parameters. Main influence factors are gear geometry (such as geometrical parameters, manufacturing accuracy, running-in effects and flank modifications) as well as elastic deformation and deviation (by housing, shafts, bearings, bearing clearances, wheel bodies, teeth, Hertzian contact). The calculation of the load distribution of gear stages requires a detailed analysis of the deformation behavior of the system gear-shaft-bearing-housing. There are several computer programs to calculate load distribution in the tooth contact of cylindrical gear stages. Some of these programs use analytical approaches (Load Distribution Program Ohio State University (Harianto, J., 2006); LVR Technical University of Dresden (Börner, J. et al., 2002); RIKOR Technical University of Munich (Neubauer, B. and Weinberger, U., 2016)), others use FEM approaches (STIRAK RWTH Aachen (Ingeli, J., 2014)). The advantage of these specialized programs in comparison to a full FEM approach is the efficient parameterization of gears and the very short calculation time while delivering high precision results. Measuring the load distribution in tooth contact is difficult. Contact patterns contain no information about the amount of load acting in contact. Using strain gauge sensors to measure the load distribution in tooth contact are state of the art (Arai, N. and Harada, S. Aida, T., 1981; Broßmann, U., 1979; Helmi, I. W. N. et al., 2012; Li, S., 2007; Miyachika, K. et al., 2014). Methods such as pressure measurements in tooth contact or using strain gauge sensors in the tooth root include uncertainties. The here proposed method uses deformation measurements by a coordinate measuring machine. In a research project (Daffner, M., 2017), the here proposed method has been used to validate calculation results of the computer program RIKOR. 2. Measurement technology In order to validate calculated load distributions (e.g. calculated by RIKOR (Neubauer, B. and Weinberger, U., 2016)), deformation measurements of a test gearbox have been taken by using a bridge-type 3D coordinate measuring machine (Fig. 1). Paper No

2 Fig. 1: 3D coordinate measuring machine The accuracy of +-2 μm is specified by the manufacturer and has been checked regularly. Therefore, calibration measurements have been taken and analyzed. Regarding the uncertainty of method proposed in this paper, the accuracy of the coordinate measuring machine is the dominating influencing variable. 3. Test rig Within a previous research project (Fürstenberger, M., 2011) to validate calculation results of the program RIKOR (Neubauer, B. and Weinberger, U., 2016), a static test rig has been built to validate amongst other terms the calculation results of the system shaftbearing. The static test rig is shown in Fig. 2. Fig. 2: Static test rig The static test rig consists of one spur gear and one helical gear stage. The input torque is provided by a lever mechanism. The nominal input torque is 600 Nm. The input torque can be provided continuously by the lever mechanism and is measured by a load cell, which is part of the lever mechanism. The output torque is supported by a bearing block. Elastomer couplings make sure that there is no effect of external forces and external bending torques to the input and output shaft. In order to realize a variety of measuring points, the test rig is not surrounded by a housing. The bearings of the test rig are located in bearing blocks. The main geometry of the spur gear stage is shown in Table 1. 2

3 Table 1 Main geometry of the spur gear stage Spur gear stage Pinion Gear Normal module (mm) 3 Normal pressure angle (deg) 20.0 Centre distance (mm) Helix angle (deg) 0 0 Number of teeth (-) Addendum modification (-) coefficient Pitch diameter (mm) In order to investigate the load distribution of the spur gear stage, there are two different variants. These two variants differ in terms of the flank shape modifications. These modifications have been ground on different angular segments of one single wheel. For this reason, measurements of these two variants can be taken without having to dis- and reassemble the static test rig. The flank shape modifications of the two Variants are shown in Table 2. Both variants are equal in tip relief, root relief and crowning. They differ in terms of the helix angle modifications. Table 2: Flank shape modifications of the spur gear stage Flank shape modification spur gear stage SK0 SK200 Tip relief (μm) Root relief (μm) Crowning over tooth width (μm) Helix angle modification (μm) The main geometry of the helical gear stage is shown in Table 3. Table 3: Main geometry of the helical gear stage Spur gear stage Pinion Gear Normal module (mm) 5 Normal pressure angle (deg) 20.0 Centre distance (mm) Helix angle (deg) Number of teeth (-) Addendum modification (-) coefficient Pitch diameter (mm) The flank shape modifications of the helical gear stage are shown in Table 4. Table 4 Flank shape modifications of the helical gear stage Flank shape modification helical gear stage Tip relief (μm) 40 Root relief (μm) 40 Crowning over tooth width (μm) 15 Helix angle modification (μm) 30 23

4 4. Preliminary investigations Within the scope of a research project, investigations have been made to validate calculated load distributions. The basic idea of measuring the load distribution in the tooth contact in this research project is to detect tangential deformations on the backside of a loaded tooth flank. These tangential deformations are directly linked to the load distribution on the tooth flank (Fig. 3). Fig. 3: Measurement strategy In order to estimate the scale of the tangential deformations, preliminary investigations have been made using the finite elements method. Fig. 4 shows the used FEM-model of the pinion. The side of the lever mechanism (left side, Fig. 4) has been modeled as a fixed support. On the right side, a cylindrical support has been added. Fig. 4: FEM Model of the pinion Regarding the meshing of the FEM-model, it is important to ensure that there are equidistant nodes along the contact line, at the radius of the measured backside of the unloaded tooth flank and at the radius of the measured backside of the loaded tooth flank. The calculation model is loaded by nodal forces. The forces can be determined from calculated load distributions. This proceeding is suitable due to the fact that the deformations are analyzed on the backside of the loaded tooth flank. The load distributions have been calculated with the program RIKOR (Neubauer, B. and Weinberger, U., 2016). Fig. 5 exemplarily shows the tangential deformations on the backside of a loaded and unloaded tooth flank loaded by the nominal input torque of 600 Nm. Fig. 5: Tangential deformation of loaded and unloaded tooth flank The tangential deformation of an unloaded tooth flank can be interpreted as the torsional deformation of the shaft (red line, Fig. 5). By contrast, the tangential deformation of the backside of a loaded tooth flank can be seen as the sum of the torsional deformation 24

5 of the shaft and the bending of the loaded tooth (blue line, Fig. 5). The diameter of the nodes, which were used to analyze the tangential deformations, has been 57.7 mm. Therefore, a tangential deformation of 0.01 deg corresponds to a deformation of 5μm. The results of the preliminary investigations have demonstrated measurable differences between the tangential deformations on the backside of a loaded and the tangential deformations of an unloaded tooth flank. However, the differences are measurable only at nominal input torque (600 Nm) and in singular contact. All calculations have been performed with ANSYS Workbench 16.0 and ANSYS Classic Validation of the calculated load distribution 5.1 Measuring the tangential tooth deformation In order to access the backside of a loaded tooth flank (at single contact), four teeth have been cut out of the wheel. Removing the unloaded teeth from the gear wheel is considered to have no measurable influence on the deformation values that are in focus here. The four teeth are marked in Fig. 6. Fig. 6: Measuring the backside of a loaded tooth flank Fig. 7 shows the coordinate system of the static test rig. The deformation of the backside of a tooth flank includes, apart from the tangential deformations of the tooth, deformations of the system shaft-bearing-housing. For this reason, the tangential deformations have been measured in a cylindrical coordinate system of the pinion shaft. The used cylindrical coordinate system is determined separately for each load. The first step to determine this cylindrical coordinate system is to measure the shaft axis (as shown in Fig. 7). Fig. 7: Measuring the shaft axis Fig. 8 shows the cylindrical coordinate system (rφx). The x-axis corresponds to the measured axis of the pinion shaft. The r-axis is defined by the z-axis of the coordinate system of the test rig. The angle φ is defined positive as shown in Fig

6 Fig. 8: Cylindrical coordinate system of the pinion shaft The mesh position can be measured by using the centering function of the 3D coordinate measuring machine. The mesh position is uniquely defined by the angular position of the pinion, the working pressure angle and the direction of loading. In order to eliminate the share of the tangential deformation, which depends on the torsional deformation of the shaft, from the measured tangential deformation on the backside of a tooth flank, two teeth have been measured. The tangential deformation has been measured on the backside of a loaded and on the backside of an unloaded tooth flank. So, the angular distance of a loaded and an unloaded tooth flank (Δφl,u) as a function of the position on the face width can be determined. Fig. 9 shows the determination of the angular distance of a loaded and a unloaded tooth flank. Fig. 9: determination of the angular distance of a loaded and an unloaded tooth flank Main influence parameter on the determined angular distance is the load distribution on the tooth flank. Deviations due to the manufacturing process can be eliminated by an additional analysis of the angular distance at 0% of the nominal input torque. Therefore, all measurements have been taken at 0% and 100% of the nominal input torque. Fig. 10 exemplarily shows the measurement on the backside of an unloaded tooth flank. Fig. 10: Measuring the backside of a unloaded tooth flank The here described investigations have been taken by measuring 20 equidistant measuring points on the backside of the loaded and unloaded tooth flanks. In the following, the tangential deformation of a loaded tooth without the share of the torsional deformation of the shaft is called tangential tooth deformation uz (equation (1)). 26

7 u Z = φ l,u,100 φ l,u,0 (1) u Z (deg) Tangential tooth deformation φ l,u,100 (deg) Angular distance at 100% of the nominal torque φ l,u,0 (deg) Angular distance at 0% of the nominal torque The tangential tooth deformation allows an actual determination of the load distribution in the tooth contact. 5.2 Calculation the tangential tooth deformation The first method to compare measured load distributions to calculated load distributions is to generate tangential tooth deformations out of the calculated load distribution by using the FEM. Therefore, the tangential tooth deformation due to a calculated load distribution has to be determined. Knowing the actual mesh position and the load distribution, the tangential deformations on the backside of a loaded and an unloaded tooth flank can be calculated by using the above mentioned FEM model. Therefore, the radius on which the contact line is located has to be determined. This radius can be determined by the measured angular position of the pinion (mesh position). Regarding the meshing of the FEM-model, it is important to ensure, that there are equidistant nodes on the tooth flank along the contact line at the radius of the measured backside of the unloaded tooth flank and at the radius of the measured backside of the loaded tooth flank. The calculation model is loaded by nodal forces. The forces can be determined from calculated load distributions. The tangential tooth deformation can be calculated as shown in equation (2). u Z = u t,l u t,u (2) u Z (deg) Tangential tooth deformation u t,l (deg) Tangential deformation on the backside of a loaded tooth flank u t,u (deg) Tangential deformation on the backside of a unloaded tooth flank 5.3 Iterative approach to determine the load distribution The second method to compare measured load distributions to calculated load distributions is to directly generate a load distribution out of the measurement results. Therefore, an iterative calculation approach using the FEM has been developed. The so determined load distributions can be compared to calculated load distributions. This method also requires the determination of the mesh position. The mesh position is uniquely defined by the angular position of the pinion, the working pressure angle and the direction of loading. For the iterative calculations, the above mentioned FEM-model has been used. For each step of the iteration, the nodal forces on the contact line are adjusted. Step 1: In the first iterative step, the FEM-model has to be loaded by constant nodal forces (Fig. 11 left chart). The sum of the nodal forces has to generate the input torque, which has been adjusted before measuring. The calculation results for the tangential tooth deformation for this load state can be compared to the measured tangential tooth deformation (Fig. 11 right chart). Fig. 11: Iterative approach (1 st Step) Nodal forces F of the 1 st step of the iteration (left chart). Comparison of the measured tangential tooth deformation along face width with the calculated tangential tooth deformation (right chart). 27

8 Step 2: The constant nodal forces of the first step have to be modified. For this purpose, the percentage deviation of the tangential deformations of the first step is calculated. The nodal forces are getting modified according to the percentage deviation of the tangential deformations. Then, tangential tooth deformations for the new loads are calculated and compared to the measurement results again. Fig. 12: Iterative approach (2 nd Step) Nodal forces F of the 2 nd step of the iteration (left chart). Comparison of the measured tangential tooth deformation along face width with the calculated tangential tooth deformation (right chart). Step 2 is repeated until a convergence criterion is fulfilled. The result of the iterative approach is a load distribution. 6. Results The aforementioned two variants of flank shape modifications have been measured. Both variants contain tip relief, root relief and crowning. They differ in terms of the helix angle modification (SK0 and SK200). In order to validate the calculation results of the computer program RIKOR (Neubauer, B. and Weinberger, U., 2016), measurement results and calculation results for these two variants are compared to each other in the chapter below. Therefore, the tangential tooth deformation and the load distributions have been determined. 6.1 Flank shape modification SK0 The tangential deformations for the variant SK0 have been measured at a diameter of mm. Due to the actual mesh position, the contact line has been located at a diameter of mm. The FEM model has included 294 equidistant nodes along the contact line. Fig. 13: Comparison of the measured and calculated tangential tooth deformation along face width for Variant SK0 Fig. 14: Comparison of the nodal forces of the calculated load distribution by RIKOR and the measured load distribution (iterative approach) along face width for Variant SK0 6.2 Flank shape modification SK200 The tangential deformations for the variant SK200 have been measured at a diameter of mm. Due to the actual mesh position, the contact line has been located at a diameter of mm. The FEM model has included 300 equidistant nodes along the contact line. 28

9 Fig. 15: Comparison of the measured and calculated tangential tooth deformation along face width for Variant SK200 Fig. 16: Comparison of the nodal forces of the calculated load distribution by RIKOR and the measured load distribution (iterative approach) along face width for Variant SK Discussion of the results 7.1 Chart tangential tooth deformation The calculation results of the tangential tooth deformation are determined by using the FEM. The FEM-model has been loaded by a calculated load distribution. The load distribution has been calculated by the computer program RIKOR (Neubauer, B. and Weinberger, U., 2016). The calculation results of the tangential tooth deformation are compared to the measured tangential tooth deformation. The here used unit is (deg). All measuring points on the backside of a tooth flank have been taken at a diameter of about 57.7 mm. Therefore, a tangential deformation of 0.01 deg corresponds to a deformation of 5μm. 7.2 Chart load distribution The load distribution can be directly generated by using the aforementioned iterative approach. The load distribution charts show the nodal forces of the FEM-model as a function of the position on the face width. Depending on the meshing, there are about 300 equidistant nodes along the contact line. In order to compare calculated load distributions to measured load distributions, the calculated load distributions have been converted to nodal forces. The here used calculation results have been determined by the computer program RIKOR (Neubauer, B. and Weinberger, U., 2016). 7.3 Comparison of the results Regarding the variant SK0, the measured and calculated tangential tooth deformations show a good correlation (Fig. 13). The differences between the calculated and the measured tangential tooth deformations are comparable to the measuring accuracy. Comparing the nodal forces, the calculated and the measured load distribution show a good correlation (Fig. 14). Regarding the variant SK200, the load distribution is shifted to the right side. This effect is caused by the helix angle modification of 200 µm. The measured and calculated tangential tooth deformations also show a good correlation (Fig. 15). The differences are also comparable to the measuring accuracy. Comparing the nodal forces, the calculated and the measured load distribution show a good correlation (Fig. 16). The test results show, that the influence of flank shape modifications on the load distribution is modelled correctly in the computer program RIKOR (Neubauer, B. and Weinberger, U., 2016). 8. Summary and outlook Precise information about the load distribution in the tooth contact can improve the quality of calculation results regarding strength and durability of spur gear stages. Within the scope of a research project, investigations have been made to validate calculated load distributions. Measuring the load distribution in tooth contact is difficult. Contact patterns contain no information about the load distribution. Methods such as pressure measurements in tooth contact or using strain gauge sensors in the tooth root include 29

10 uncertainties. The here developed method of measuring the load distribution of spur gear stages allows a verified statement about the load distribution of spur gear stages in lab conditions. For this purpose, deformation measurements of a test gearbox have been taken by using a bridge-type 3D coordinate measuring machine. The test gearbox is static and includes two spur gear stages. The input torque is provided by a lever mechanism. Regarding the mentioned method to measure the load distribution of spur gear stages, the basic idea is, that tangential deformations on the backside of a loaded tooth flank are directly linked to the load distribution on the tooth flank. Deviations due to the manufacturing process can be eliminated by an additional analysis of an unloaded tooth flank. A measuring procedure to detect the aforementioned tangential deformations has been developed. The measurement results can be compared to calculated load distributions with two methods. The first method is to generate tangential deformations out of the calculated load distribution by using the Finite Element Method (FEM). The second method is to directly generate a load distribution out of the measurement results. Therefore, an iterative calculation approach using the FEM has been developed. Both methods require a determination of the contact point. This paper shows measurements of two different variants. The two variants differ in terms of flank shape modifications. The next step was to compare the measured load distributions to results of the computer program RIKOR (Neubauer, B. and Weinberger, U., 2016). The test results showed, that the influence of flank shape modifications on the load distribution is modelled correctly in the computer program RIKOR (Neubauer, B. and Weinberger, U., 2016). Using the methods mentioned before, calculated load distributions can be validated. Influence of factors, such as housing stiffness and wheel body geometry can be analyzed and compared to calculation results. Furthermore, a test set up for helical gear stages with small helix angles is conceivable. Acknowledgement The contents of this publication are based on a research project, which is supported by the Forschungsvereinigung Antriebstechnik e.v. (FVA-Nr. 592 II "Weiterführende Validierung der Verformungsrechnung in RIKOR - Detaillierte Betrachtung einzelner Getriebeelemente ). The authors would like to thank the Forschungsvereinigung Antriebstechnik e.v. and the working groups AK Berechnung und Simulation and AG RIKOR for the financing and the support of this research project. This work was supported by the German Research Foundation (DFG) and the Technical University of Munich (TUM) in the framework of the Open Access Publishing Program. References Arai, N. and Harada, S. Aida, T., Research on Bending Strength Properties of Spur Gears with a Thin Rim. Bulletin of the JSME. Vol: 24 (1981), p Börner, J. et al., Effective Analysis of Gears with the Program LVR (Stiffness Method). VDI-Berichte. Vol: 1665 (2002), p Broßmann, U., Über den Einfluss der Zahnfußausrundung und des Schrägungswinkels auf Beanspruchung und Festigkeit schrägverzahnter Stirnräder. Dissertation, Technische Universität München (1979). Daffner, M., FVA-Nr. 592 II- Validierung RIKOR II - Weiterführende Validierung der Verformungsrechnung in RIKOR - Detaillierte Betrachtung einzelner Getriebeelemente (2017). Fürstenberger, M., FVA-Nr Heft 987- Validierung RIKOR - Validierung und Untersuchung von Anwendungsgrenzen des FVA Getriebeprogramms RIKOR anhand von Verformungsmessungen. Forschungsvereinigung Antriebstechnik e.v., Frankfurt/Main (2011). Harianto, J., Information about Load Distribution Program (WindowsLDP). Ohio State University: Department of Mechanical Engineering (2006). Helmi, I. W. N. et al., Effects of Rim and Web Thicknesses on Root Stresses of Thin-rimmed Helical Gear. Applied Mechanics and Materials. Vol: 165 (2012), p Ingeli, J., FVA-Nr. 127/VIII- Heft STIRAK/Innenverzahnung - Erweiterung der FE-basierten Zahnkontaktanalyse zur Berechnung von Innenverzahnungen. Forschungsvereinigung Antriebstechnik e.v., Frankfurt/Main (2014). Li, S., Finite element analyses for contact strength and bending strength of a pair of spur gears with machining errors, assembly errors and tooth modifications. Mechanism and Machine Theory. Vol: 42 (2007), p Miyachika, K. et al., Root stresses and bending fatigue strength of thin-rimmed helical gears. In International Gear Conference Lyon (2014), p Neubauer, B. and Weinberger, U., FVA-Nr. 571/II- Heft Berechnung der Lastverteilung in Getriebesystemen mit beliebig angeordneten Planetenradstufen. Forschungsvereinigung Antriebstechnik e.v., Frankfurt/Main (2016). 10 2