1、04FTM3A Method to Define Profile Modificationof Spur Gear and Minimize theTransmission Errorby: M. Beghini, F. Presicce, C. Santus, Departimento di IngegneriaMeccanica, Facolt di Ingegneria Universit di Pisa, ItalyTECHNICAL PAPERAmerican Gear ManufacturersAssociationA Method to Define Profile Modifi
2、cation of Spur Gearand Minimize the Transmission ErrorM. Beghini, F. Presicce, C. Santus, Departimento di Ingegneria Meccanica,Facolt di Ingegneria Universit di Pisa, ItalyThe statements and opinions contained herein are those of the author and should not be construed as anofficial action or opinion
3、 of the American Gear Manufacturers Association.AbstractTheobjectofthis articleis toproposeasimplemethod toreduce thetransmission error (T.E.) for a givenspurgear set, at the nominal torque, only by means of the profile modification parameters. Iterative simulationswith advanced software, are needed
4、. A hybrid method has been used, combining the Finite Elementtechnique with semianalytical solutions. A 2D analysis is thought to be adequate for this kind of work, in fact,this software requires both little time for model definition and simulations, with a very high precision of theresults.Copyrigh
5、t 2004American Gear Manufacturers Association500 Montgomery Street, Suite 350Alexandria, Virginia, 22314October, 2004ISBN: 1-55589-826-2A Method to Define Profile Modification of Spur Gear and Minimize the Transmission Error M.Beghini, F.Presicce, C.Santus*: DIMNP - Dipartimento di Ingegneria Meccan
6、ica, Nucleare e della Produzione. Facolt di Ingegneria, Universit di Pisa - via Diotisalvi n.2 Pisa, 56126 Italy Introduction In this paper a simple method to minimize the Peak to Peak Transmission Error (PPTE) for a spur gear set is proposed. A parametric analysis using advanced software was perfor
7、med to obtain a general understanding of the problem. Optimization variables were reduced to the Start Relief Roll Angles. A parabolic topography was assumed for the tip relief and either Low Contact Ratio (LCR) or High Contact Ratio (HCR) spur gears have been considered. Meshing related constraints
8、 on optimization have been taken into account. On the basis of parametric analysis an effective optimization procedure has been suggested for LCR that can be extended even for HCR. Nomenclature P Diametral Pitch b Face width h Tooth height s Coordinate along profile TE Transmission Error PPTE Peak t
9、o Peak TE PPinion start relief roll angle GGear start relief roll angle Pinionev Pinion relief amount Gearev Gear relief amount ev Total relief amount SAP Start Active Profile roll angle EAP End Active Profile roll angle Table 1. Nomenclature. 1 Transmission Error Effect on Gear Noise In a gear set,
10、 Transmission Error (TE) is defined as the difference between the effective and the ideal position of the output shaft with reference to the input shaft. The ideal position represents a condition of perfect gear box, without geometrical errors and deflections. TE can be expressed either by an angula
11、r displacement or, more conveniently, as a linear displacement measured along a line of action at base circle according to 1. TE is typically considered to be the primary cause of whining noise. Indeed, whining noise is produced by: changing tooth load amplitude, changing load position along the too
12、th profile, changing tooth load direction. These changes are consequences of tooth deflection, local contact deformation and body deformation quantities, which are the origin of TE. Several Authors 2-6 studied the correlation between TE and profile modification, in order to reduce the TE. Niemann 4
13、proposed long and short modifications. This different denomination is based on the start point of tip relief modification along the profile. According to experimental results, gears with long modification have the minimum PPTE and the minimum noise level at the design torque. At lower torque this op
14、timum condition was not verified and an intermediate or short modification is suggested. In these works short and long modifications are considered to be technological limits and it is not useful to extend the relief to the pitch radius, as described in 1,3. In the present paper, instead, the start
15、tip relief modifications (P. G) have been considered as design variables ranging from Start Active Profile roll angle (SAP) up to the End Active Profile roll angle (EAP). Total relief ve(at the tooth tip) is imposed equal to the deflection of tooth pairs under the nominal torque. The topography of t
16、he tip relief is parabolic. The conditions for obtaining the minimum PPTE configuration are found and discussed. 2 Methodology The quantities Pinionev ( ) and GearevP(G) are defined in figure 1. The ranges for the two Roll Angle variables are the Start Active Profile roll angle (SAP) and the End Act
17、ive Profile roll angle (EAP) for each gear (figure 2). Meshing gears simulations have been carried out by means of a hybrid method, combining the Finite Element technique with a semi-analytical solution 7,8. The main assumptions for the analysis are the following: 1Figure 1. Definition of ( ) and Pi
18、nionevGearevP(G). Figure 2. Simbols SAP and EAP. Plain strain conditions: suggested by the spur gear geometry (high ratio b/h and so not affecting bending deformation behavior). 2D plane strain analysis is adequate for this kind of tooth. Moreover the bi-dimensional version of the software requires
19、little time both for model definition and simulations, with a very high precision of the results. Static analysis: static TE was determined neglecting rotational speed and inertia forces. Friction neglected: in order to get rid of the uncertainness about Coulomb frictional coefficient. The friction
20、force is assumed not affecting the TE shape function substantially. Spacing error and pitch error not considered: the nominal TE was calculated, and no statistical consideration have been performed in the analysis. Otherwise required if geometrical errors considered. Two kinds of spur gear set have
21、been analyzed, in which the TE function is rather different: Low Contact Ratio gears (LCR), High Contact Ratio gears (HCR). For the LCR gear contact ratio is between 1 and 2 and when the contact is at the pitch position, only one tooth pair is in contact, while for HCR, 3 tooth pairs are in contact
22、at the pitch position. At the beginning each spur gear set was analyzed by assuming no modification. The maximum deflection of the gear pair tooth, TEmax(evaluated along the pinion base circle) was then calculated and imposed equal to the total relief amount. ve= TEmaxThe amount of the tip relief wa
23、s equally shared between pinion and gear assuming ong modification suggested in 4,5. l2evGearePinionevv = Successively, contact pressure was calculated along the meshing zone in order to detect possible corner contacts. As shown in figure 3 the corner contact can be observed at both sides. has to be
24、 increased in order to avoid corner contact at the start of meshing, likewise has to be increased to avoid the corner contact at the end of meshing. GearevPinionev Figure 3. Adapting veto avoid Corner Contacts: (a) Corner Contact both at Start and End meshing, (b) Corner Contact only at Start meshin
25、g, (c) No Corner Contact detected. When corner contact was eliminated, pinion relief ( ) and gear relief ( ) were held fixed through the subsequent calculations. PinionevGearevTo perform the minimization, an object function has to be defined along the variables and the ranges in which the optimum is
26、 searched. In a similar analysis 9 the object function was related to the Fourier expansion of the TE and the first three harmonics considered. However as shown in figures 4 and 5, in this analysis the first three harmonics could not be the main part of the signal reproducing the TE function. 2Figur
27、e 4. Decomposition of the PPTE according to FFT and signal reproduction. Figure 5. FFT harmonics components of total TE (t) signal The reason is that at the instances when the contact passes from different tooth pairs the TE is not regular (C0condition satisfied only). Thus according to these consid
28、erations the PPTE was considered as the object function, and the independent variables were reduced to the start Roll Angles for gear and pinion P , G. The input torque applied to the pinion is the nominal value of the mission profile. For every configuration of (P,tG) the following outputs were con
29、sidered: Transmission Error, tooth load, contact pressure, bending principal stress at tooth root. Each of them were calculated as a function of the Contact Length along the meshing zone. The following limits were then imposed as boundaries of the optimization domain: Corner Contact. For particular
30、starting relief Roll Angle combinations (PG) corner contact can reappear even if total relief amount is set as discussed above. Contact Pressure. Due to profile modification, relative curvatures are modified and contact pressure can be locally increased, exceeding the Pitting limit, Bending Stress.
31、Principal stress at the tooth root can exceed the Fatigue strength limit. Each of the boundary are here better discussed. Corner Contact. Corner contact is produced when the contact region includes zones of the fillet of the tooth tip and the contact pressure rises locally at the tip fillet, as in f
32、igure 6. Furthermore as a consequence of the teeth deflection the effective contact ratio is greater than that found according to rigid geometry hypothesis and so corner contact can be detected in instances when the contact should not appear (figure 7). This definition of Corner Con act can be explo
33、ited only if a Loaded Tooth Contact Analysis (LTCA) is performed. When the corner contact is detected, the calculated pressure peak was not considered reliable, as in this situation the maximum is strongly affected by the radius of the fillet 10, which is a very unpredictable quantity for its techno
34、logical generation. (a) (b) Figure 6. (a) No Corner Contact detected. (b) Corner Contact detected It is worth noting that starting from long profile modification by lowering Pinion and Gear relief Roll Angles (P,G) even if amounts ( , ) PinionevGearev3are kept fixed, corner contact can reappear. Thi
35、s is due to the fact that if pinion and gear tooth flanks in contact are both modified, the effective total relief is lessened. To understand this an example is given by the green curve of the third graph of figure 8. In that situation the nominal relationship between Pinion and Gear Roll Angles has
36、 been claimed, but according to the aforementioned figure 7 the teeth deflection generates angular shift which is related to the torque applied, and then earlier tip relief overlap has to be expected. This is the cause of Corner Contact reappearing. Figure 7. Comparison between effective Contact Pre
37、ssure and rigid geometry Contact Pressure estimation. The strong contact length enlargement is here depicted. Configurations producing Corner Contact were obviously considered outside the boundary of the minimization domain. Figure 8. Examples of three different Tip Relief profiles combinations. Con
38、tact Pressure. Local contact pressure along the profile, can change considerably due to profile modifications. Actually, by changing the Start Relief point along the profile, relative curvature considerably changes as well. n figure 9 it is shown a contact pressure rise with the curvature discontinu
39、ity inside the contact region. This numerical result has been confirmed theoretically in 10. A worse condition (continuity C0condition satisfied only) is found when the topography is linear 11 but it is not treated in this paper. Figure 9. (a) Contact pressure with rise of curvature, on the right. (
40、b) Contact pressure in the subsequent tooth pair, with no curvature discontinuity inside the contact region. Bending Stress. Bending stress at tooth root is another important issue in high performance gear design. By applying profile modifications, load transfer between teeth pairs, contact points a
41、nd load directions can change. As a consequence, different bending stress 1at tooth base can be produced. If a LCR gear set were considered, bending stress variations would not be greater than 5%, as shown in figure 10, however for HCR gear set variation is greater because with three tooth pairs in
42、contact the sharing factor is more sensitive to profile modification. Anyway it can be convenient to consider bending stress as a possible penalty for the object function instead of a boundary. Figure 10. Effect of different profile modification on bending stress on a LCR gear set. 2.1 Computational
43、 performances To perform the minimization analysis, following the steps presented in the previous section, a hardware platform PC was used with the following characteristics 4 CPU 2.6 GHz RAM 1 GB Plane strain analysis was performed by an advanced hybrid FEM analysis software, whose references are 1
44、2,13. Analysis were automatically performed in about 12 CPU hours, simulating 50 time steps for each meshing, for 600 different relief (P,G) configurations 3 Results 3.1 LCR spur gear set The analyzed LCR gear set design parameters are summarized in table 2. Pinion N. of teeth 80 Gear N. of teeth 80
45、 P 0.571 mm-1 Pressure angle 22.5 deg Table 2. LCR gear set parameters. The object function PPTE for the LCR gear set is shown in figure 11. No boundaries are presented yet. It is worth noting that: The PPTE minimum is unique inside the ranges for the two variables. Near the minimum the Hessian matr
46、ix is positive defined. According to these conditions the minimum could be found by adopting a classical gradient derived method. Figure 11. Three-dimensional plot of the PPTE, LCR gear set. As pinion and gear have the same number of teeth, meshing properties are symmetric about the domain diagonal
47、defined by the equation P=Gand the absolute minimum is on this diagonal (figure 12). The numerical values are the following: Pmin= 23.035 deg Gmin= 23.035 deg PPTEmin=1.8 mThe TE with minimum PPTE is shown in figure 14. In order to analyze this minimum, TE functions for P= at different minP Gare plo
48、tted. It is remarkable that the shape of the TE functions is different between configurations with . This is due to the fact that for , profile configurations cause a vGminG GminGGminGedrop. In fact in the analyzed configuration, start relief roll angles couple (P,G) for the minimum PPTEproduces ove
49、rlapping modified profiles and a reduced effective total amount relief as already discussed and shown in figure 13. Figure12. Symmetry locus of PPTE(P,G) function for LCR gear set. This is due to an overestimation of the total relief amount calculated at the beginning. For example in this configuration the total amount is reduced from 25 m to 23.3 m. And this result can be an insight for better evaluating ve. Figure13. Effective relief diagram, with evidence to the relief overlap and it
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