1、01FTM8The Effect of Spacing Errors and Runout onTransverse Load Sharing and the DynamicFactor of Spur and Helical Gearsby: H. Wijaya, D.R. Houser and J. Harianto,The Ohio State UniversityTECHNICAL PAPERAmerican Gear ManufacturersAssociationThe Effect of Spacing Errors and Runout onTransverse Load Sh
2、aring and the Dynamic Factor ofSpur and Helical GearsHusny Wijaya, Donald R. Houser and Jonny Harianto, The Ohio StateUniversityThestatementsandopinionscontainedhereinarethoseoftheauthorandshouldnotbeconstruedasanofficialactionoropinion of the American Gear Manufacturers Association.AbstractThispape
3、raddressestheeffectsoftwocommonmanufacturingerrorsontheperformanceofspurandhelicalgears. Thefirst error to be considered will be tooth spacing error. First the transverse load sharing for two types of worst casescenarios,onewhereonetoothisoutofpositionandthesecondwherethereisastepindexerrorisapplied
4、(allsubsequentteeth areoutofposition).Theresultsoftheseanalysesarethen used asinputto asimulation program that predictsgeardynamic loads, dynamic contact stresses and dynamic root stresses and generalized dynamic factor information isgenerated. Runout, which results in a sinusoidal spacing error is
5、the second type of error that is analyzed. When thedynamic simulation is run with runout, there is much modulation of the meshing forces and again dynamic loads andstresses are presented as a function of rotating speed.Inallcases,resultsarerunforgearsthathaveprofilemodificationsthatinitiallyminimize
6、dynamicload,thusminimizingdynamics due to mesh stiffness variations. This allows the isolation of spacing error effects while at the same timeminimizestoothcornercontact. Sinceitisdifficultinthesesituationstoseparatethetransverseloadsharingeffectsfromthe dynamic effects, they will be simultaneously
7、considered in all of the analyses.CopyrightGe32001American Gear Manufacturers Association1500 King Street, Suite 201Alexandria, Virginia, 22314October, 2001ISBN: 1-55589-787-81The Effect of Spacing Errors and Runout on Transverse Load Sharing and Dynamic Factor of Spur and Helical Gears Dr. D. R. Ho
8、user Professor, Department of Mechanical Engineering The Ohio State University Columbus, OH 43210 and Husni Wijaya Graduate Research Associate, Department of Mechanical Engineering The Ohio State University Columbus, OH 43210 and Jonny Harianto Research Engineer, Department of Mechanical Engineering
9、 The Ohio State University Columbus, OH 43210 INTRODUCTION The dynamic factor used in gear design has been the subject of many studies, most of them being analytical in format. Buckinghams first formulation 1 of a dynamic factor was based on entering contact impacts due to spacing errors and subsequ
10、ent work by Tuplin 2, and Houser and Seireg 3, 4 continued applying this approach. However, in modern gear design, these impacts are eliminated through the use of tip relief and lead crowning so that the later formulations of Harianto and Houser 5, 6 and Lin 7 have been based on errors in the tooth
11、form that result in a transmission error excitation. Three types of dynamic factors defined by Harianto and Houser 5 are the “dynamic load factor” which is the same as the factor used by AGMA 8, the “dynamic root stress factor” and the “dynamic contact stress factor”. In this paper, spacing errors a
12、nd runout will be introduced and further analyzed using these three dynamic factor definitions. However, the static changes in the transverse load sharing (TLS) sometimes are much greater than the dynamic effect, so we shall define two new factors that combine the static and dynamic effects as follo
13、ws: errorwithoutvaluestaticMaximumerrorspacingwithvaluestaticMaximumFactorDynamicFactorSpacing=spacingTLSSpacingTLSrunoutwithoutvaluestaticMaximumrunoutwithvaluestaticMaximumFactorDynamicFactorRunout=runoutTLSrunoutTLSThe “static value” depends on the type of analysis, namely load, pinion root stres
14、s, gear root stress or contact stress, and these two factors give the net static and dynamic effect of the respective spacing errors and runout. The procedure for determining these factors is first to use the Load Distribution Program (LDP) 9 to obtain the static transmission error and mesh stiffnes
15、s as functions of gear rotation. These parameters are then used as excitations to the Dynamic Transmission Error Program (DYTEM) that uses a six degree-of-freedom 2non-linear time domain model to simulate gear dynamics 10. Dynamic loads are predicted from the DYTEM program and then fed back to LDP t
16、o calculate the values of both root and contact stresses at each contact position. The procedures used in this paper for predicting dynamic load, root stress and contact stress are similar to those presented by Harianto and Houser 5, 6. A description of each of these programs is provided in the APPE
17、NDIX. SPACING ERROR AND RUNOUT Spacing error is the circumferential position error of one gear tooth with respect to the previous tooth. Runout is an eccentricity that can be described over one revolution of the gear as a sinusoidal spacing error function. Spacing errors and runout have a significan
18、t effect on gear dynamics, and many studies have been conducted in this area. Houser and Seireg 3, 4 have done experimental and analytical studies on the influence of different pitch errors and profile modifications on the dynamic factors of spur and helical gears. The studies were limited to gear s
19、ystems that operate at low speed or non-resonant speeds and concentrated on the entering tooth impact that occurs due to corner contact. Today, this corner contact effect can be avoided or minimized by providing adequate tip relief and lead crowning. Umezawa, Sato and Kohno 11 studied the rotational
20、 vibration of gears with pitch errors on every other tooth and on every third tooth, and their results show that low contact ratio gears are more sensitive to pitch error than are high contact ratio gears. Kohler 12 examined the effect of spacing error on the gear transmission error. Padmasolala and
21、 Lin 13 investigated the effectiveness of profile modification for reducing dynamic loads on the gears using long span and short span spacing error over several teeth, but the magnitudes of spacing errors used are very low and are close to the highest quality (quality 15) rated gear tolerances presc
22、ribed by AGMA. Briere and Sabot 14 investigated the eccentricity faults in spur gears and their results show amplitude modulation and a resulting periodicity of kinematic transmission errors. The proper means of simulating spacing errors would be to apply the errors over the total rotation of both p
23、inion and gear. This would require running the gears through their total hunting period. Instead, we have found that the worst-case spacing errors occur when the respective worst-case errors of the pinion and gear match up. Therefore, we are only going to study the worst-case combinations of spacing
24、 errors, first with errors in the positive direction and second with errors in the negative direction. Even so, there are two ways errors can occur that give high dynamic responses. The first, called case A, is when one tooth is out of position and all other teeth are in their original locations. Th
25、is means that if one tooth is given a spacing error, then the opposite magnitude spacing error is applied to the next tooth to relocate the next tooth back to its original position. The second, called case B, is a step index error. This means that if a tooth is out of position because of a spacing e
26、rror, then all subsequent teeth are also out of position by the same amount. Positive spacing error means the tooth with error is closer from the previous tooth (added material), and negative means the tooth is farther from the previous tooth. Since only the worst-case is considered, the magnitudes
27、of spacing error and runout used for all simulations will be taken as double the tolerances prescribed by AGMA quality standards. In this study, gears of AGMA quality 10, 12, and 14 are used in the simulations. Schematics of negative spacing errors modeled as case A and case B are shown respectively
28、 in Fig. 1. Fig. 1 Schematic of teeth with negative spacing error OPTIMIZED PROFILES AND LEADS In order to isolate the effects of spacing errors and runout from other factors such as profile errors, it is important to start with gears that have minimum dynamic effects due to profile errors. It is as
29、sumed that the effect of each type of error must be superimposed in order to obtain a complete dynamic factor. In Fig. 2, the dynamic load factors are shown for a perfect involute spur gear and a spur gear whose profile has been optimally modified. The geometry of the spur gear pair is given in Tabl
30、e 1. The dynamic factor of the perfect involute spur gear pair exceeds 2.0 for several operating speeds and the normal tooth spacing error tooth3dynamic factor for the optimally modified spur gear pair is virtually 1.0 at all operating speeds. Fig. 2 Dynamic load factors comparison of the perfect in
31、volute spur gear pair and the optimum spur gear pair SPUR GEAR RESULTS A spur gear pair whose geometry data is given in Table 1 is used for this analysis. The gear pair has the same 1:1 ratio geometry as was studied by Harianto and Houser 5 and by Rebbechi 15. The tolerances of spacing errors and ru
32、nout for quality 10, 12 and 14 gears are also given, respectively, in Table 1. Table 1 Spur Gear Data Number of teeth pinion/gear 28 Normal diametral Pitch 8 (1/in) Center distance 3.5 in Face width 0.25 in Helix angle 0 Pressure angle 20 Theoretical contact ratio 1.64 in Outside diameter pinion/gea
33、r 3.75 in Root diameter pinion/gear 3.1875 in Torque 1128 lb-inMesh frequency 0.467*RPM Quality 14: spacing error / radial runout 100 in / 379 in Quality 12: spacing error / radial runout 202 in / 744 in Quality 10: spacing error / radial runout 408 in / 1457 in Fig. 3 Light torque (1 lb-in) and rat
34、ed torque (1128 lb-in) comparison for the perfect involute spur gear pair with case A, negative spacing error 0 1 2 3 4-5000500Light TorqueTransmission error(in)0 1 2 3 4-2000-1500-1000-5000Rated TorqueTransmission error(in)0 1 2 3 400.20.40.60.8Mesh cycleToothforce (lbf)0 1 2 3 40200400600800Mesh c
35、ycleToothforce (lbf)0.511.522.530 5000 10000 15000Pinion Speed (rpm)DynamicLoad FactorPerfect involuteOptimum 4Fig. 3 shows a comparison of static conditions for a case A, negative spacing error under both light torque (1 lb-in) and the rated torque (1128 lb-in). Both cases were analyzed with the pe
36、rfect involute spur gear pair and 500 in spacing error. The top part shows the transmission errors, and the bottom part shows the tooth forces. For the light torque case, it is interesting to note that the transmission error is equal to the spacing error, and the duration of each tooth pair in conta
37、ct changes significantly. Under the rated load, the contact durations do not change, but the load sharing in the two teeth pair contact region does shift a bit. Fig. 4 shows the transmission error, mesh force, tooth force and pinion root stress results for the perfect involute spur gear pair with no
38、 spacing error. The thinner line is the static condition and the thicker line is the dynamic response when operated at 4100 rpm. Throughout the analysis presented here, the gear root stress results are very similar to those for the pinion so will not be shown. Fig. 5 shows similar results for the op
39、timally modified spur gear pair with no spacing error. Note that the tooth force and pinion root stress show that the transition from the one tooth pair contact region to the two tooth pair contact region is very gradual, thus resulting in an extremely small dynamic effect so that the static and dyn
40、amic curves virtually coincide. As mentioned earlier, all of the spacing errors and runout simulations are made after the gear teeth have been optimally modified. Fig. 6 shows results for the optimally modified spur gear pair with quality 10, case A, negative spacing error. The maximum dynamic mesh
41、force occurs at the mesh cycle about 1.36 where the third tooth pair comes into contact. However, the maximum dynamic tooth force occurs at the mesh cycle 2 location where the second tooth pair leaves contact. Fig. 7 shows transmission error and torque results for the optimally modified spur gear pa
42、ir with quality 10 runout. The simulation is run for the whole rotation of pinion gear. Modulation of the dynamic mesh force is observed. Combining the two types and magnitudes of spacing errors, we have four possible worst-case combinations as shown below. The notations that are shown will be used
43、in all subsequent figures. AP: case A, positive spacing error AN: case A, negative spacing error BP: case B, positive spacing error BN: case B, negative spacing error Fig. 4 Static and dynamic (4100 rpm) results for the perfect involute spur gear pair with no spacing error 0 1 2 3 4-2000-1500-1000-5
44、000Transmission error(in)static dynamic0 1 2 3 440060080010001200Meshforce(lbf)0 1 2 3 402004006008001000Mesh cycleTooth force (lbf)0 1 2 3 40204060Pinionrootstress (ksi)Mesh cycle5Fig. 5 Static and dynamic (4100 rpm) results for the optimum spur gear pair with no spacing error Fig. 6 Static and dyn
45、amic (4100 rpm) results for the optimum spur gear pair with quality 10, case A, negative spacing error (AN) 0 1 2 3 4-1250-1240-1230-1220-1210-1200Transmission error(in)static dynamic0 1 2 3 4660680700720Meshforce (lbf)0 1 2 3 402004006008001000Mesh cycleTooth force(lbf)0 1 2 3 40204060Pinionrootstr
46、ess (ksi)Mesh cycle0 1 2 3 4-2500-2000-1500-1000-500Transmission error(in)static dynamic0 1 2 3 44006008001000Meshforce (lbf)0 1 2 3 402004006008001000Mesh cycleTooth force(lbf)0 1 2 3 40204060Pinionrootstress (ksi)Mesh cycle6Fig. 7 Static and dynamic (4100 rpm) results for the optimum spur gear pai
47、r with quality 10 runout Fig. 8 Dynamic factor comparison of the optimum spur gear pair with the four worst-case combinations of quality 10 spacing errors 0 5 10 15 20 25-2000-1500-1000-5000Transmission error(in)static dynamic0 5 10 15 20 25660670680690700710Meshforce (lbf)Mesh cycle0.801.001.201.40
48、1.601.800 5 10 15Pinion speed (krpm)DynamicLoad FactorAN AP BNBP Opt4000 rpm9000 rpm0.901.001.101.201.300 5 10 15Pinion speed (krpm)DynamicPinionRoot StressFactor0.901.001.101.201.30051015Pinion speed (krpm)DynamicGearRootStress Factor0.951.001.051.101.151.20051015Pinion speed (krpm)DynamicContact S
49、tressFactor7Fig. 8 shows a comparison of the four worst-case combinations of quality 10 spur gear pairs along with the same results for the optimally modified spur gear without error. Case A shows higher dynamic factors than case B at most operating speeds, especially around the “resonant-like” speeds. An exception to this is the low speed values of gear root stress where the BN values of 1.1 are quite a bit higher than the other values. Each of the dynamic factors shown in Fig. 8 has significant variability with changes in oper
