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1、 1 Copyright 2007 by ASME EXPERIMENTAL VALIDATION OF A COMPUTERIZED TOOL FOR FACE HOBBED GEAR CONTACT AND TENSILE STRESS ANALYSIS Andrea Piazza andrea.piazzacrf.it Martino Vimercati Powertrain Research and Technology Centro Ricerche FIAT Strada Torino 50 - 10043 Orbassano (TO), ITALY ABSTRACT While
2、face milled gears have been widely analyzed, about face hobbed ones only very few studies have been developed and presented. Goal of this paper is to propose the validation of an accurate tool, which was presented by the authors in previous works, aimed to the computerized design of face hobbed gear
3、s. Firstly, the mathematical model able to compute detailed gear tooth surface representation on both spiral and hypoid gears will be briefly recalled; then, the so obtained 3D tooth geometry is employed as input for an advanced contact solver that, using a hybrid method combining finite element tec
4、hnique with semianalytical solutions, is able to efficiently carry out both contact analysis under light or heavy loads and stress tensile calculation. The validation analyses will be carried on published aerospace face hobbed spiral bevel gear data comparing measurements of root and fillet stresses
5、. Good agreement with experimental results both in the time scale and in magnitude will be revealed. 1 INTRODUCTION Spiral bevel and hypoid gear drives are widely applied in the transmission of many applications, such as helicopters, cars, trucks, etc. They are manufactured using mainly two cutting
6、processes: face milling or face hobbing method. As well known, face milling process, traditionally adopted by the Gleason Works, utilizes a circular face mill type cutter and employs an intermittent index. On the contrary, during FH process, traditionally adopted by Oerlikon and in the last decades
7、by the Gleason Works as well, the work has continuous rotation and rotates in a timed relationship with the cutter: successive cutter blade groups engages successive tooth slots as the gear is being cut 1. Many studies about tooth surface representation and design of FM spiral bevel and hypoid gears
8、 have been carried out 2-5. On the contrary, about FH process, that is the considerably more complex, only a small number of works are available in the open literature 6-7. The authors of this paper have worked extensively on that topic proposing a mathematical model aimed to the computation of the
9、face hobbed gear tooth surfaces 8; moreover they handled the output of this model in order to carry out a computerized design of these gears 9. Goal of this paper is to provide the validation of that tool. To this end, a comparison with experimental data will be proposed; in particular the results c
10、ollected by Handschuh et al. 10 will be considered. In that reference an experimental evaluation of the performance of an aerospace spiral bevel face-hobbed gear drive, in the following named TEST, is shown. In detail, results in terms of loaded tooth contact analysis, stress calculation and vibrati
11、on/noise measurement are widely discussed. The basic characteristics of the TEST gear drive are summarized in Table 1. Table 1. Basic characteristics of the TEST gear drive. PinionGearModule mm 4.94 Offset mm 0 Shaft Angle 90 Teeth Number 12 36 Mean Spiral Angle 35.000 Hand LH RH Face Width mm 25.4
12、Mean Cone Distance mm 81.05 Nominal Pressure Angle 22.5 The model validation requires the following steps. Starting from the information stored in Table 1, by means of a commercial gear design software, the geometric parameters, the basic machine settings and the cutting blade data will be firstly c
13、omputed; after that, by means of the proposed model, Proceedings of the ASME 2007 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE 2007 September 4-7, 2007, Las Vegas, Nevada, USA DETC2007-35911 2 Copyright 2007 by ASME the geomet
14、ry of the tooth can be calculated and the gear drive performance under load can be evaluated. The main effort is devoted just to validate the model by comparing the stresses experimentally measured in the root and in the fillet area with the one numerically calculated; a qualitative comparison of th
15、e loaded tooth contact pattern will be also provided. 2 MODEL DESCRIPTION AND METHOD OF THE ANALYSIS The first step in order to build a reliable numerical model is to get a fine geometrical representation of gear tooth surfaces. This is especially true when one is dealing with complex tooth geometry
16、 such as the face hobbing one. To this aim, a series of algorithms able to compute tooth surfaces of FH gears starting from cutting process has been implemented by the authors 8. The geometry of real FH head cutter (Gleason Tri-Ac) is considered; many kinds of blade configuration (straight and curve
17、 blades, with or without Toprem) are taken into account. Then, according to the theory of gearing 11, FH cutting process (with and without generation motion) is simulated and gear tooth surfaces equations can be computed. The proposed mathematical model is able to provide an accurate description of
18、the whole tooth, including fillet region; it also considers undercutting occurrence, which is very common in FH gears due to uniform depth tooth. The obtained tooth surfaces are used as fundamental input for a powerful contact solver which is based on a semianalytical finite element formulation 12-1
19、3. The gear drive can be study under light load by monitoring, for drive and coast side, the contact pattern and transmission error (i.e. it can be performed the commonly called Tooth Contact Analysis TCA 14). Moreover, with the aim to find out gear drive performance in the real service conditions,
20、a set of torque values can be applied and the influence of the load on contact pattern, on transmission error and on load sharing can be accurately analyzed (Loaded Tooth Contact Analysis LTCA 15). Contact pressure and stress distribution can be also easily evaluated. 2. GEOMETRIC AND MANUFACTORING
21、OF THE TEST GEAR DRIVE Using the data collected in Table 1 as preliminary input for a commercial software for gear design (Gleason T2000), a calculation aimed to reproduce the TEST gear drive has been attempted. Table 2 describes the obtained tooth geometry; Table 3 and 4 show the details regarding
22、the machine setting and the cutting blades: the pinion is generated and the gear is Formate; both the members are cut by means of curved blades using a head cutter with nominal radius equal to 76 mm and 13 blade groups. Table 2. Tooth geometry data of the TEST gear drive. PinionGear Module mm 4.941
23、Offset mm 0 Shaft Angle 90 Teeth Number 12 36 Mean Spiral Angle 35.00035.000Hand LH RH Face Width mm 25.4 25.4 Outer Cone Distance mm 93.743 93.743Pitch Angle 18.43571.565Addendum mm 4.930 2.067 Dedendum mm 2.942 5.805 Table 3. Basic machine settings for the TEST gear drive. Pinion Gear Concave Conv
24、exConcaveConvex Generated Formate Radial Setting mm 91.451 91.451 92.364 92.364 Tilt Angle 20.099 20.099 - - Swivel Angle -25.371 -25.371- - Blank Offset mm 0.000 0.000 - - Machine Root Angle 0.154 0.154 71.565 71.565 Machine Center to Backmm -0.0722 -0.0722-1.509 -1.509 Sliding Base mm 13.865 13.86
25、5 - - Cradle Angle 53.697 49.817 51.405 51.405 Ratio of Roll mm 2.999 2.999 - - Table 4. Cutting blades data for the TEST gear drive. Pinion Gear OB IB OB IB Blade Type Curved Curved Curved Curved Blade Radius mm 75.499 75.758 76.206 75.749 Blade Eccentric 17.832 17.633 17.738 17.846 Blade Height mm
26、 4.363 4.363 4.374 4.374 Blade Angle 25.323 18.122 22.231 21.681 Blade Groups Number 13 13 13 13 Nominal Rake Angle 12.000 12.000 12.000 12.000 Hook Angle 4.420 4.420 4.420 4.420 Cutter Edge Radius mm 0.700 0.700 1.000 1.000 Blade Radius of Curvaturemm 762.000 762.000762.000762.000Toprem Angle - - -
27、 - Toprem Length mm - - - - 3 Copyright 2007 by ASME 3. TOOTH GEOMETRY OF THE TEST GEAR DRIVE Figure 1 illustrates the tooth geometry representation obtained by means of the proposed model for the TEST gear drive. Figure 1. TEST gear tooth geometry representation. Figure 2 describes the fillet area
28、by means of the trend along the face width of the Nominal Root Line NRL, of the Real Root Line RRL and of the UnderCut/Fillet UC/FL line. According to that picture it is possible to note the tooth does not show undercut. Figure 2. Details of the fillet area. Due to the fact that the reference does n
29、ot provide any topological data, just a qualitative comparison between the real tooth geometry and the one calculated by means of the numerical model is feasible (Figure 3). Figure 3. Qualitative comparison between the real pinion tooth geometry and the calculated one. 3.1 Evaluation of actual TEST
30、gear fillet radius Starting from the picture of the real pinion tooth (Figure 3 above), a rough measurement of the radius of the fillet has been also attempted. Doing this way, referring to the toe of the concave side, a value about equal to 0.94 mm is obtained. When the same zone of the numerically
31、 computed tooth is considered, a value equal to 1.26 mm in correspondence of the maximum curvature point between the middle of inner surface and the contact surface is evaluated. The difference may be quite large (+34%) and, as it will be shown later, this evidence will have a significant influence
32、on the fillet state of stress. As known the fillet radius is strictly related to the edge radius of the cutting blade. The value used to cut the real tooth is unknown while in the numerical model it is assumed to be equal to 0.7 mm. In order to achieve a finer correspondence, models considering othe
33、r edge radius values have been built. Namely, 0.5 mm and 0.3 mm have been tried obtaining the results summarized in Table 5 and Figure 4 (the points used for the radius calculation are highlighted). It can be noted that using an edge radius equal to 0.3 mm the best correspondence can be achieved. 4
34、Copyright 2007 by ASME Table 5. Comparison between the photo measured and the numerical fillet radius by varying edge radius. Cutter Edge Radius mm Pinion Fillet Radius mm Photo-measured Pinion Fillet Radius mm Difference % 0.70 1.26 0.94 34.04 0.50 1.10 0.94 17.02 0.30 0.98 0.94 4.26 Figure 4. Comp
35、arison between the numerical pinion concave side profile and the photo-measured one (note that the reference systems are different). 4. STRESS CALCULATION Referring to the experimental investigation, the stresses are evaluated by means of strain gages in the fillet area. In detail, referring to the
36、sketch depicted in Figure 5, one strain gage at the heel position in the fillet and three strain gages (at heel, mid and toe positions) in the root (i.e. on the root cone). On the other hand, with the aim to numerically compute the stresses, it is necessary to define a set of coordinates which are a
37、ble to straightforwardly provide the stress measuring point on the tooth. Here, the curvilinear coordinate t which runs along the face width (-1 t +1 in Figure 6.a) and the curvilinear coordinate s which runs along the tooth profile (0 s 48 in Figure 6.b) have been defined. According to this schemat
38、ization it is possible to affirm that the heel position corresponds to t = +0.5, the mid one to t = 0 and the toe one to t = -0.5; the root area is located in the range 0 s 2 while the fillet one in the range 5 s 7. Figure 5. Sketch used in the TEST reference for location of the strain gages. Figure
39、 6.a. Schematization for defining the stress measuring section along the face width of the model. Figure 6.b. Schematization for defining the stress measuring point on a generic section of the model. 5 Copyright 2007 by ASME In Figure 7 the results of the TEST reference at a level of torque equal to
40、 269 Nm are shown. In detail, the trend of bending stress vs time (during a whole meshing cycle) in the fillet and in the root region of the real pinion tooth is drawn. Referring to the same tooth position, Figure 8 reports the numerical results. According to the coordinates previously defined in Fi
41、gure 6, the trend of the bending stress vs time in the fillet (s = 6) at the heel section (t = +0.5) and in the root (s = 1 o 2) in the mid, toe and heel section (t = 0, t = -0.5, t = +0.5) is shown. Figure 7. Pinion bending stress vs time as reported in the reference 10. Figure 8. Pinion bending st
42、ress vs time as computed by the numerical model at an edge radius = 0.7 mm. By analyzing these graphs and by considering Table 6 which summarizes the maximum/minimum stresses for each tooth position, it is possible to affirm that the differences between numerical results and the experimental one are
43、 quite small. Table 6. Comparison between experimental and numerical analysis at 269 Nm. Fillet - Heel Max Min TEST MPa 440.57 -39.30 Model MPa 296.35 -45.37 Difference % 32.73 -15.45 Root - Heel Root - Mid Root - Toe Max Min Max Min Max Min 222.70 -384.73 258.55 -284.06 248.90 -221.32 206.87 -256.3
44、6 247.25 -227.79 240.11 -141.01 7.11 33.36 4.37 19.81 3.53 36.29 As stated in Table 7, similar evidences are collected at a lower level of torque (166 Nm). Table 7. Experimental vs numerical stress results. Edge radius = 0.7 mm 166 Nm Fillet - Heel Max Min TEST MPa 278.55 -27.58 Model MPa 190.00 -35
45、.00 Difference % 31.79 -26.91 Root - Heel Root - Mid Root - Toe Max Min Max Min Max Min 139.96 -236.49 164.09 -239.25 166.85 -167.54 138.00 -287.00 176.37 -176.51 157.98 -104.71 1.40 -21.36 -7.48 26.22 5.32 37.50 It is reasonable to believe that the largest error value, which happens in the fillet-h
46、eel, is mainly due to two reasons. Firstly, it is quite difficult to find the exact correspondence between the experimental and the numerical stress measuring point; in fact, by studying Figure 9 which reports the numerical computed trend of the maximum principal stress in the fillet (s = 6) vs the
47、position along the face width at 269 Nm, it is possible to note the value of the stress is significantly affected by the position along the face width. 050100150200250300350400450-1-0.75-0.5-0.2500.250.50.751Position along face width tStress MPa Figure 9. Numerically computed maximum principal stres
48、s vs position along face width in the fillet position. 6 Copyright 2007 by ASME Another issue to investigate is the influence of the value of the fillet radius on the bending stress. In fact, as previously detected, when a cutting blade edge radius value equal to 0.7 mm is used, the numerical pinion
49、 fillet radius (1.26 mm) is quite larger (+34%) than the real one (0.94 mm). In order to achieve a finer stress correspondence, models considering other edge radius values have been built. Namely, 0.5 mm and 0.3 mm have been tried obtaining the results summarized respectively in Table 8 and 9. Figur
50、e 10 summarizes the trend of the error between the maximum fillet-heel stress superimposed to the error between the fillet radius vs the cutting blade edge radius. According to these results, it seems that for a value of cutting blade edge radius equal to 0.3 mm good agreement between numerical and
51、experimental data is achieved. Table 8. Experimental vs numerical stress analysis. Edge radius = 0.5 mm 269 Nm Fillet - Heel Max Min TEST MPa 440.57 -39.30 Model MPa 322.92 -58.95 Difference % 26.70 -49.99 Root - Heel Root - Mid Root - Toe Max Min Max Min Max Min 222.70 -384.73 258.55 -284.06 248.90
52、 -221.32176.49 -227.92 204.18 -221.57 213.33 -155.2920.75 40.76 21.03 22.00 14.29 29.83 Table 9. Experimental vs numerical stress analysis. Edge radius = 0.3 mm 269 Nm Fillet - Heel Max Min NASA MPa 440.57 -39.30 Model MPa 351.98 -74.66 Difference % 20.11 -89.98 Root - Heel Root - Mid Root - Toe Max
53、 Min Max Min Max Min 222.70 -384.73 258.55 -284.06 248.90 -221.32 141.76 -196.90 159.53 -187.75 172.01 -134.15 36.34 48.82 38.30 33.91 30.89 39.39 4.2617.0234.0420.1126.7032.730.005.0010.0015.0020.0025.0030.0035.0040.0000.20.40.60.8Edge Radius mmDifference %Fillet RadiusMaximum Fillet Stress Figure
54、10. Differences between numerical and experimental results vs cutting edge radius at 269 Nm. Similar behaviour is obtained at a level of torque equal to 166 Nm (Table 10 and 11 and Figure 11). Table 10. Experimental vs numerical stress results. Edge radius = 0.5 mm 166 Nm Fillet - Heel Max Min NASA
55、MPa 278.55 -27.58Model MPa 201.27 -41.00Error % 27.74 -48.66 Root - Heel Root - Mid Root - Toe Max Min Max Min Max Min 139.96-236.49164.09 -239.25 166.85-167.54111.31-159.91142.98 -149.18 138.56-96.0020.4732.3812.87 37.64 16.9642.70 Table 11. Experimental vs numerical stress results. Edge radius = 0
56、.3 mm 166 Nm Fillet - Heel Max Min NASA MPa 278.55 -27.58Model MPa 219.51 -52.80Error % 21.19 -91.45 Root - Heel Root - Mid Root - Toe Max Min Max Min Max Min 139.96-236.49164.09 -239.25 166.85-167.5489.66-138.22111.69 -128.12 111.65-83.0135.9441.5531.94 46.45 33.0850.45 7 Copyright 2007 by ASME 4.2
57、617.0234.0421.1927.7431.790.005.0010.0015.0020.0025.0030.0035.0040.0000.20.40.60.8Edge Radius mmDifference %Fillet RadiusMaximum Fillet Stress Figure 11. Difference between numerical and experimental results vs cutting edge radius at 166 Nm. 5. LOADED TOOTH CONTACT ANALYSIS Figure 12 shows the tooth
58、 contact pattern which was experimentally measured at a level of torque equal to 269 Nm compared with the one numerically computed by means of the model. In both of the cases the pattern enlarges nearly on the whole face width assuming a similar shape. Figure 12. Comparison of the experimental loade
59、d tooth contact pattern (above) with the numerical one (below). 6 CONCLUSIONS In this paper the validation of a tool previously developed by the author for computerized design of face hobbed hypoid gears has been proposed. A reference case has been firstly chosen: experimental data collected on an a
60、erospace spiral bevel face-hobbed gear drive by Handschuh et al. has been considered. Then, by means of the proposed model, the geometry of the tooth has been calculated. Focusing the attention on the fillet radius, a comparison between the real tooth geometry and the numerical one has been attempte
61、d finding an acceptable correspondence, even if the result are strictly related to the value of the edge radius of the cutting blade (in the best case differences are lower than 5 %). Next, the stresses experimentally measured in the root and in the fillet area with the one numerically calculated ha
62、ve been compared. While a satisfactory agreement has been achieved in the root area (both in the time scale and in magnitude), in the tooth fillet some discrepancies has been revealed and some additional consideration has to be done. Firstly, it is not so straightforwardly to catch the exact corresp
63、ondence between the experimental and the numerical stress measuring point; this is a significant consideration being the value of the stress notably affected by the position along the face width. Another issue to point out is the influence of the value of the fillet radius on the bending stress. In
64、fact, as previously mentioned, differences in the numerical fillet radius vs the real have been detected. Finally, loaded tooth contact pattern has been also compared finding a reasonable agreement. All these considerations allow to conclude that the model can be considered a reliable numerical tool
65、 for studying face hobbed hypoid gear drive. ACKNOWLEDGMENTS The authors would like to sincerely thank Robert F. Handschuh of the U.S. Army Research Laboratory, Glenn Research Center, Cleveland, Ohio for his kind support. REFERENCES 1 Stadtfeld, H.J., 2000, Advanced Bevel Gear Technology, The Gleaso
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69、Dafoe, R. S. and Swanger Jr., J, 2005, “Development of bevel gear face hobbing simulation and software”, International Conference on Gears, Munich, Germany, (September 2005). 7 Yi-Pei Shih, Zhang-Hua Fong, and Grandle C. Y. Lin, 2007, “Mathematical Model for a Universal Face Hobbing Hypoid Gear Gene
70、rator”, Journal of Mechanical Design, Vol. 129, pp.38-47. 8 Vimercati, M., 2006, “Mathematical Model for Tooth Surfaces Representation of Face-Hobbed Hypoid Gears and its Application to Contact Analysis and Stress Calculation”, in press on Mechanism and Machine Theory, available on line August 2006.
71、 9 Vimercati, M. and Piazza, A., 2005, “Computerized Design of Face Hobbed Hypoid Gears: Tooth Surfaces Generation, Contact Analysis and Stress Calculation”, AGMA Fall Technical Meeting 2005, Detroit, Michigan, (October 2005). 10 Handschuh, R. F., Nanlawala, M., Hawkins, J. M. and Mahan D., 2001, “E
72、xperimental Comparison of Face-Milled and Face-Hobbed Spiral Bevel Gears”, NASA/TM-2001-210940, ARL-TR-1104. 11 Litvin, F.L., 1994, Gear Geometry and Applied Theory, Prentice Hall, Englewood Cliffs, New Jersey. 12 Vijayakar, S.M., 2003, Calyx Users Manual, Advanced Numerical Solution, Hilliard, Ohio
73、. 13 Vijayakar, S.M., 1991, “A Combined Surface Integral and Finite Element Solution for a Three-Dimensional Contact Problem”, International Journal of Numerical Methods Engineering, Vol. 31, pp. 525545. 14 Understanding Tooth Contact Analysis, Gleason Works Publication, SD3139, 1978 15 Krenzer, T.J., 1981, Tooth Contact Analysis of Spiral Bevel and Hypoid Gears Under Load, The Gleason Works, Rochester, New York.
