A Comparison of the AGMA Gear Design Stresses, the Lewis Bending Stress, and the Stresses Calculated by the Finite Element Method for Spur Gears by Andrew Wright An Engineering Project Submitted to the Graduate Faculty of Rensselaer Polytechnic Institute in Partial Fulfillment of the Requirements for the degree of Master of Engineering Major Subject: Mechanical Engineering Approved: _________________________________________ Ernesto Gutierrez-Miravete, Thesis Adviser Rensselaer Polytechnic Institute Hartford, CT June, 2013 (For Graduation December 2013) i CONTENTS LIST OF TABLES ............................................................................................................ iv LIST OF FIGURES ........................................................................................................... v ACKNOWLEDGMENT .................................................................................................. vi ABSTRACT .................................................................................................................... vii SYMBOLS AND VARIABLES .................................................................................... viii GLOSSARY ...................................................................................................................... x IMPORTANT KEYWORDS ........................................................................................... xi 1. Introduction and Scope ................................................................................................ 1 2. Theory and Methodology ............................................................................................ 2 3. Analysis ....................................................................................................................... 3 3.1 3.2 Microsoft Excel Analysis ................................................................................... 3 3.1.1 Determination of the Lewis Bending Stress .......................................... 3 3.1.2 Determination of the AGMA Bending Stress ........................................ 3 3.1.3 Determination of the AGMA Contact Stress ......................................... 3 ABAQUS Finite Element Analysis .................................................................... 4 3.2.1 Lewis Bending Stress Model ................................................................. 4 3.2.2 AGMA Design Stresses Model .............................................................. 9 4. Results and Discussion .............................................................................................. 10 4.1 Comparison of Data ......................................................................................... 10 4.2 Sources of Error and Divergence ..................................................................... 12 5. Conclusion ................................................................................................................. 13 6. Appendices ................................................................................................................ 14 6.1 Microsoft Excel Analysis ................................................................................. 14 6.2 Lewis Bending ABAQUS Mesh Convergence Data ....................................... 23 6.3 AGMA ABAQUS Mesh Convergence Data.................................................... 25 ii 7. References.................................................................................................................. 26 iii LIST OF TABLES Table 1 – Symbols and Variables ..................................................................................... ix Table 2 – Lewis Bending Stress by Distance through Tooth as Calculated by Abaqus .. 11 iv LIST OF FIGURES Figure 1 – Cut and Partitioned Pinion Tooth before Meshing ........................................... 4 Figure 2 – Boundary Conditions on the Pinion Gear Tooth .............................................. 5 Figure 3 - Coupling Constraint used to Apply Load ......................................................... 6 Figure 4 - Load Position, Magnitude, and Coordinate System .......................................... 6 Figure 5 - Visualization of Completed Pinion Tooth, with Stress Distribution ................ 7 Figure 6 – Example Graph of Stress vs Distance along Path through Gear Tooth ........... 8 Figure 7 - Path Used to Determine Bending Stresses ...................................................... 10 v ACKNOWLEDGMENT Type the text of your acknowledgment here. vi ABSTRACT The Lewis bending stress, the AGMA bending stress, and the AGMA pitting contact stress are calculated for a gear mesh consisting of two spur gears. Gear stresses and factors of safety are calculated both by hand in Microsoft Excel as well as by using the finite element method. The “Rush Gears” website has been used to generate gear CAD files for use in the ABAQUS finite element analysis software package (see References page). With the gear CAD files imported, ABAQUS was used to mesh, constrain, and calculate gear stresses. The stresses calculated by these two methods will be compared in order to determine the effectiveness of the finite element method to design gears. vii SYMBOLS AND VARIABLES Symbol/Variable Cf CH dg dp E F HB I J KB Km Ko KR Ks Kv mG ν φ PdG PdP Qv σcG σcP σG σL σP satG satP ScG ScP SFG SFP SHG SHP SY Description Units Surface condition factor - Hardness ratio factor - Pitch diameter of the gear Pitch diameter of the pinion in in Modulus of elasticity Face width Brinell hardness of the gears psi in HB Pitting resistance geometry factor Bending strength geometry factor Rim thickness factor Load distribution factor Overload factor - Reliability factor Size factor Dynamic factor - Speed ratio Poisson’s ratio of the gears Pressure angle Diametral pitch of the gear rad in-1 Diametral pitch of the pinion in-1 AGMA quality factor AGMA contact stress, gear psi AGMA contact stress, pinion AGMA bending stress on the gear Lewis bending stress psi psi psi AGMA bending stress on the pinion Allowable bending stress number, gear psi psi Allowable bending stress number, pinion psi Contact fatigue strength, gear psi Contact fatigue strength, pinion psi Bending fatigue failure safety factor, gear Bending fatigue failure safety factor, pinion psi psi Wear factor of safety, gear Wear factor of safety, pinion Yield strength of the gears psi viii Equation Used SUT Tg Tp TSG TSP Wt YG YN YP ZNG ZNP Ultimate strength of the gears psi Operational torque transmitted to the gear lbf-in Operational torque transmitted to the pinion Stall torque of the power source, acting on the gear Estimated stall torque of the power source, acting on the pinion Tangential transmitted gear load Lewis form factor of the gear lbf-in lbf-in Stress cycle factor - Lewis form factor of the pinion Pitting resistance stress-cycle factor, gear - Pitting resistance stress-cycle factor, pinion - Table 1 – Symbols and Variables ix lbf-in lbf - GLOSSARY x IMPORTANT KEYWORDS xi 1. Introduction and Scope Use this template to type the text of your thesis, using the various heading styles provided. If your thesis is short and you want it all in one file, you may type all your chapters in one file. If your thesis is long, you will want to put each chapter in a separate file. Start a new file for each chapter, using this template. The chapter numbering and page numbering will start with “1” in each file, but when you assemble the chapters at the end, the numbering will be sequential. When creating your chapters, it is essential to use the heading styles provided by this template! 1 2. Theory and Methodology 2.1 Background of the Lewis Bending Equation and Derivation 2 3. Analysis 3.1 Microsoft Excel Analysis 3.1.1 Determination of the Lewis Bending Stress 3.1.2 Determination of the AGMA Bending Stress 3.1.3 Determination of the AGMA Contact Stress 3 3.2 ABAQUS Finite Element Analysis This section of the report discusses in depth the two Abaqus finite element models used to calculate the different gear stresses used in the report. For both models, the organization of the section will follow the different stages of the model formulation: part(s) creation, material selection, application of boundary conditions, application of the load, and meshing the part(s). Mesh convergence studies were carried out for both models in order to ensure that the final mesh density resulted in sufficiently accurate results. Refer to Appendix 6.2 and 6.3 for the raw data used to carry out the mesh convergence for both models. 3.2.1 Lewis Bending Stress Model To begin, the Lewis Bending Stress Abaqus model o Describe the part o imported RUSH gears *step file o Removed bad geometry o Cut into single tooth, partitioned geometry (Fig. 1) Figure 1 – Cut and Partitioned Pinion Tooth before Meshing o Material info o Steel, to match excel analysis o E = 30E6 psi, nu = 0.3 o Isotropic, elastic 4 o BC’s o 3 total o All DOF’s (Fig. 2) o Explanation Figure 2 – Boundary Conditions on the Pinion Gear Tooth o Load o Explain constraint (Fig. 3) o 800 lbf load to match excel (Fig. 4) 5 Figure 3 - Coupling Constraint used to Apply Load Figure 4 - Load Position, Magnitude, and Coordinate System o Mesh o Explain element type (Fig. 5) 6 o No warnings/errors o Convergence study (Fig. 6) o Ref. appendix with data o Results o Explain node path o Screenshot of example graph o Show Figure 5 - Visualization of Completed Pinion Tooth, with Stress Distribution 7 Figure 6 – Example Graph of Stress vs Distance along Path through Gear Tooth 8 3.2.2 AGMA Design Stresses Model 9 4. Results and Discussion 4.1 Comparison of Data Figure 7 - Path Used to Determine Bending Stresses o Excel Lewis bending results o Abaqus Lewis bending results o Explain node path (Fig. 8) o Speak to Table Distance Through Tooth (in) 0.000 35,540 0.025 11,185 0.050 5274 0.075 5811 0.094 8782 0.113 13,046 0.131 20,043 0.150 38,139 Bending Stress (psi) 10 Table 2 – Lewis Bending Stress by Distance through Tooth as Calculated by Abaqus 11 4.2 Sources of Error and Divergence o Mesh elements used? o Lewis bending equation doesn’t take into account some things that Abaqus does (stress concentrations?) o 12 5. Conclusion 13 6. Appendices 6.1 Microsoft Excel Analysis 4.0 CALCULATION / DISCUSSION: The following analysis calculates the AGMA design stresses for the meshing spur gears. 4.1 Input Loads Operational Torques: mG = dg / dp dg and dp are the pitch diameters of the gear and the pinion, respectively. mG [ ] = 3.0 Tg [lbf-in] = 1800 Speed Ratio (Ref. 4, Eq. 14-22) Torque required at the gear to drive the system. This value is the same for the Excel analysis and the ABAQUS analysis. Tp = Tg / mG Tp [lbf-in] = 600 Torque required at the pinion to drive the system. 4.2 Pinion, Idler, and Gear Dimensions Pinion F [in] = d [in] = Pd [in] = dp = PdP = Qv [ ] = φ [rad] = Gear 1.25 1.5 12 1.25 4.5 12 dg = PdG = 5 5 NG = 0.349 54 200 YG = 0.404 N[]= HB [ ] = NP = 0.349 18 200 Y[]= YP = 0.309 Face width Pitch diameter Diametral pitch AGMA quality factor Pressure angle (20 degrees) Number of teeth Brinell hardness of the gears (Ref. 3) Lewis Form Factor (Ref. 4, Table 14-2) The AGMA quality factor is also known as the transmission accuracy grade number, and is a measure of how accurate the gearing is (see Annex A, Ref. 1). Qv ranges from 5 to 11, therefore a quality factor of 5 is a conservative estimate. 14 4.3 Material Properties and Other Input Variables AISI 4140 steel is picked as the gear material. The following material information was retrieved from [INSERT REF]. E [psi] = ν[]= Sy [psi] = SUT [psi] = HB [ ] = 30000000 0.3 61000 95000 197 Modulus of Elasticity of the gears (see Ref. ) Poisson's Ratio of the gears Yield Strength of the gears (Ref. ) Ultimate Strength of the gears (Ref. ) Brinell Hardness of the gears (Ref. ) 4.4 Calculation of Pitch Line Velocity The pitch line velocity is used with the Dynamic factor, Kv, below. Increasing the pitch line velocity of the gear mesh can increase the max stress of the gears. For this analysis, it is assumed that the gears are static to simplify the analysis. The constants affected by the pitch line velocity become unity when the velocity is zero. v = ωr v [ft/min] = 0 Pitch Line Velocity of the geartrain 4.5 AGMA Bending Stress Analysis Section 4.5 calculates the AGMA gear bending stress, the bending stress number (similar to a material strength), and a factor of safety for bending. The calculated bending stress is based on the assumption that the gear tooth is a cantilevered plate, fixed at the base of the tooth. This bending stress creates fatigue in the gear teeth during operation of the gear mesh. In essence, the AGMA design equations calculate the maximum input load that the gears can withstand over the life of the gears without creating cracking. If the bending stresses do cause cracking in the gears, they usually form at the root fillet because this is where the largest stress is. For gears with small, thin rims, the location of the max stress can change. For this project it is assumed that the rim is of sufficient size to avoid this situation. Overload Factor, Ko Ko [ ] = 1 In most practical purposes the Overlaod Factor is greater than 1 to account for momentary peak torques experienced by most mechanically driven systems. However, in an attempt to get accurate finite element results this value was kept at 1. A constant tangential load will be applied in the model, with no transient peaks. 15 Dynamic Factor, Kv Kv = ((A+V0.5)/A)B A = 50 + 56(1 - B) B = 0.25(12 - Qv)0.66 B[]= 0.90 A[]= 55.4 Kv [ ] = 1.00 See Eq. 14-28 in Ref. 4 See Eq. 14-28 in Ref. 4 See Eq. 14-27 in Ref. 4 Size Factor, Ks The size factor is impacted by many factors, including tooth size, diameter, face width, hardenability, and stress pattern (see Ref. 4, section 14-10). AGMA suggests either using the equation below or simply assuming unity for this factor. I use the equation listed because a size factor greater than 1 is conservative. Ks = 1.192*(F*Y0.5/Pd)0.0535 See section 4.2 above for input variables. Pinion Ks [ ] = KsP = Gear 1.02 KsG = 1.03 Load Distribution Factor, Km The load distribution factor is a ratio of the peak load to the average load applied across the entire face of the gear (Ref. 1, Annex D). When computed analytically, this factor can be very complex. The AGMA gathered empirical data through in service gears and testing to create the equations and variables below that are used to calculate Km. Note: The load-distribution factor is equal to the "face load distribution factor", Cmf, under the conditions listed in section 14-11 of Ref. 4. The gears used herein obey these assumptions, therefore Cmf is used for this factor. Ref. 4, Eq. 14-30 16 Pinion Gear Cmc [ ] = CmcP = 1 CmcG = 1 Cpf [ ] = CpfP = 0.06 CpfG = 0.03 Cpm [ ] = A[]= CpmP = Ap = 1 0.1270 CpmG = AG = 1 0.1270 B[]= C[]= Cma [ ] = Ce [ ] = Cmf [ ] = Km [ ] = Bp = Cp = CmaP = CeP = CmfP = KmP = 0.0158 -0.0001 0.15 1 1.21 1.21 BG = CG = CmaG = CeG = CmfG = KmG = 0.0158 -0.0001 0.15 1 1.17 1.17 Eq. 14-31 from Ref. 4. Equals 1 for uncrowned teeth See Eq. 14-32 in Ref. 4 for 1 < F <= 17 See Eq. 14-33 in Ref. 4 See Table 14-9 from Ref. 4. Commercial, enclosed units See Eq. 14-34 from Ref. 4 See Eq. 14-35 from Ref. 4 Note: Per Ref. 4, for values of F/(10d) < 0.05, F/(10d) = 0.05 is used when computing Cpf above. Bending Strength Geometry Factor, J The bending strength geometry factor, J, is impacted by the shape of the teeth in contact. Figure 14-6 in Ref. 4 is used to estimate this factor based on the number of pinion and gear teeth. This factor assumes spur gears, a 20 degree pressure angle, and full-depth teeth. Pinion J[]= JP = Gear 0.32 JG = 0.40 See Fig. 14-6 in Ref. 4. Note: Assumed that load is applied in highest point of single-tooth contact. Stress Cycle Factor, YN The Stress Cycle Factor alters the design stress based on the number of design stress cycles. The overall "Service Factor" used by AGMA combines the Overload Factor, the Reliability Factor, and the Stress Cycle Factor. AGMA 2001-D04 suggests that if designers are comfortable with the other factors in the Service Factor, unity can be used for the Stress Cycle Factor. Since this is a theoretical problem, and the FEA software will not take into account total stress cycles, the Stress Cycle Factor has been set at 1.00. Pinion YN [ ] = YNP = Gear 1.00 YNG = 1.00 Temperature Factor, KT KT [ ] = 1 This factor is unity unless the working temperature of the gear mesh is higher than 250 degrees Fahrenheit (see section 14-15 of Ref. 4). 17 Reliability Factor, KR The reliability factor takes into account normal statistical material failures that occur in material testing. Table 11 shows some common reliability factors that were calculated from data collected by the US Navy. Unity is picked because this represents a factor in the middle of the range. KR [ ] = 1 See Table 11 of Ref. 1. Rim Thickness Factor, KB The Rim Thickness Factor is an adjustment factor that takes into account gears with smaller "rims", the material in between the bore and the base of the gear teeth. The factor is given in terms of the "backup ratio", the ratio of the rim thickness of the gear to the whole depth (see Fig. X to the right from Ref. 1). For backup ratios of greater than 1.2, the Rim Thickness Factor becomes 1.0. For the sake of making the FEA program simpler and getting more accurate results between the hand analysis and the finite element, I will make the rim thickness large enough for the backup ratio to be greater than 1.2. KB [ ] = Pinion KBP = Gear KBG = 1.00 1.00 See Fig. B.1 from Annex B of Ref. 1. Gear Bending Stress, σ Now that all the factors have been calculated, we can determine the gear bending stress. The stress is calculated below using the equation found in Ref. 4. Wt = 2Tp/d Wt [lbf] = 800 σ = WtKoKvKs(Pd/F)(KmKB/J) Wt is the transmitted tangential load going into the pinion gear. See Ref. 4, Eq. 14-15 Pinion σ [psi] = σP = Gear σG = 29674 18 23251 4.5.1 Calculation of the AGMA Bending Fatigue Failure Safety Factor Allowable Bending Stress Number, sat The allowable bending stress number, sat, is similar to a yield strength except that it goes a step further and takes into account material composition, cleanliness, the presence of residual stresses, heat treatments, and materials processing (see Ref. 1). The AGMA standard contains tables and charts for various common engineering gear materials with their associated bending stress numbers. For AISI 4140, the material of the gears, Fig. 10 is used. Grade 2, the larger stress number, is assumed. This grade is chosen because we will be comparing the gears against gear models that have ideal cleanliness and material properties. sat = 108.6HB + 15890 Pinion sat [psi]4 = satP = Gear 37284 satG = 37284 See Fig. above (from Fig. 10 of Ref. 1) Bending Fatigue Failure Safety Factor, SF In engineering practice, a factor of safety is a design factor that takes into account uncertainty in the calculation of the solution. In general, it is the ratio of the material strength of a component divided by the stress on that component. Depending on the application, the risks involved (whether that be cost, time, or safety), and statistical randomness of the inputs, the engineer may decide to design the component to different factors. For the purposes of the project, I chose to set a factor of safety of 1.5 as a requirement. Not only is this a standard factor of safety for operational loading, but because AGMA has developed so many factors that make the calculated stress more accurate we are getting results with less variance. The AGMA standards use a factor of safety in their gear design process, which can be found in Ref. 4 (see Eq. 14-41). For this project, the equation simplifies to sat divided by the bending stress since YN, KT, and KR are unity. SF = [satYN/(KTKR)]/σ Pinion SF [ ] = SFP = Gear 1.3 SFG = 1.6 19 See Ref. 4, Eq. 1441 4.6 AGMA Pitting Analysis The second important failure mode for gears is pitting, which is a surface fatigue failure that results from progressive contact stress in the meshing gears (Ref. 4, section 14-2). Because pitting is a fatigue phenomenon, it may take many cycles to become serious enough to result in failure of the gear system. AGMA 2001-D04 defines two types of pitting: initial and progressive. In initial pitting, small defects are formed on the surface of the teeth in areas of high stress. These pits will, over time, correct themselves as the surrounding high spots get smoothed out by contact with the meshing gear. For this reason, the presence of initial pitting is not a failure criteria for gear systems. Progressiv pitting, on the other hand, does not correct itself and can occur when the stresses, lifetime cycles, or other factors are high enough. The AGMA pitting stress equation is designed to calculate the load for which the meshing gears never experience progressive pitting in their usage lifetime (see Ref. 1, section 4.2). This equation is based on the Hertzian contact stress equation, modified to account for the effect of gear teeth sliding. The Elastic Coefficient, Cp, is a term that combines the elastic material constants of the meshing gears. The equation is shown below (see Eq. 14-13, Ref. 4): Elastic Coefficient, Cp Cp = [1/(π((1 - νp2)/EP + (1 - νG2/EG)]1/2 Pinion CPP = Gear 2291 CPG = CP [psi1/2] = 2291 See Ref. 4 Eq. 1413. Surface Condition Factor, Cf The surface condition factor, Cf, is a factor that takes into account surface finish effects such as cutting, lapping, grinding, or work hardening. AGMA suggests that if the meshing gears have detrimental surface finishes caused by one of these processes the surface condition factor should be greater than unity. For our purposes, we will make this factor unity since the finite element model will have idealized gear surface finishes. Cf [ ] = 1 See Ref. 4, Section 14-9 Pitting-Resistance Geometry Factor, I As defined by AGMA, the pitting resistance geometry factor, I, evaluates how the radii of curvature of the contacting tooth profiles of the gear mesh effects the Hertzian contact stress. Shigley's Mechanical Engineering Design, Ref. 4, provides a useful equation for calculating I, shown below. Because the pitting resistance geometry factor only depends on the pressure angle, the load-sharing ratio, and the speed ratio, it is the same for all gears in the mesh. mN [ ] = 1 Load-Sharing Ratio. Equals unity for spur gears. See Ref. 4, Eq. 14-23 I = [(cosφ*sinφ)/2*mN] * (mG/(mG+1)) I[]= 0.121 See Ref. 4 Eq. 14-23 20 Gear Contact Stress, σc The numerical value of the contact stress comes from the equation below, from Ref. 4: σc = Cp(WtKoKvKs(Km / dP*F)(Cf / I))1/2 See Ref. 4, Eq. 14-16. Note: for the gear, dp is actually di in the equation for the contact stress. Pinion σc [psi] = σcP = Gear σcG = 151546 86590 Gear contact stresses 4.6.1 Calculation of the AGMA Wear Safety Factor Similar to the value that was calculated for the bending stress, a factor of safety is calculated for the pitting stress for the pinion and gear. Various factors must be calculated first before solving for the factor of safety. Contact Fatigue Strength, Sc The contact fatigue strength is calculated in a similar manner as the allowable bending stress number (or bending strength), as shown in section 4.5.1. Since the gears are made out of AISI 4140, the contact stress number for nitrided through-hardened steel gears from Table 3 of Ref. 1 can be used directly. Grade 2 is assumed once again in an attempt to get results that match nicely with the results of the finite element analysis. Pinion Sc [psi] = ScP = Gear ScG = 163000 163000 See Table 3 from Ref. 1. Pitting Resistance Stress-Cycle Factor, ZN Similar to how the bending stress cycle factor, YN, was handled, the pitting resistance stress cycle factor ZN will be set at unity for this analysis. This factor alters the pitting strength that AGMA provides based on the number of lifetime stress cycles the gears will encounter. There is no reason to change this factor from unity since there is no way to set the number of cycles in a finite element analysis. ZN [ ] = Pinion ZNP = Gear 1.00 ZNG = 21 1.00 Hardness Ratio Factor, CH The hardness ratio factor, CH takes into account the fact that the smaller meshing gear will see more stress cycles in the lifteime of the gears as a result of it's smaller pitch circle. Because both gears in the mesh have the same hardness (making the ratio 1.0), however, this factor cancels to unity. HBP/HBG [ ] = 1.0 A' = 0 for HBP/HBG < 1.2 Hardness ratio between the pinion and the gear Section 14-12 from Ref. 4. CH = 1.0 + A'*(mG - 1.0) CH [ ] = 1.0 See Ref. 4, Eq. 14-36. This factor is only used for the gear. Wear Factor of Safety, SH SH = [ScZNCH / (KTKR)] / σc Pinion SH [ ] = SHP = Gear 1.1 SHG = 22 1.9 See Ref. 5.2, Eq. 1442 6.2 Lewis Bending ABAQUS Mesh Convergence Data *First column is the thru-thickness distance, and the second column is the bending stress. Iteration 1 Iteration 2 Iteration 3 0 34858.4 0 35311.9 0 35782.1 0.007505 25571.7 0.007504 27392.3 0.007505 26555.9 0.015009 18201 0.015009 20739.2 0.015009 19130.2 0.022514 12746.3 0.022513 15352.4 0.022514 13504.4 0.030019 9207.72 0.030018 11232 0.030018 9680.22 0.037523 7585.35 0.037522 8379.7 0.037523 7656.52 0.045028 5895.34 0.045027 6354.17 0.045028 6014.95 0.052533 4877.98 0.052531 5065.21 0.052532 4996.13 0.060037 4532.59 0.060036 4510.88 0.060037 4599.05 0.067542 4859.15 0.06754 4691.16 0.067541 4823.79 0.075047 5857.29 0.075044 5605.31 0.075046 5669.93 0.082552 6827.63 0.082549 6593.34 0.082551 6714.4 0.11257 12446.5 0.090053 7840.42 0.090055 7902.6 0.120075 14327.2 0.112567 13137.9 0.112569 12330.3 0.142589 28336.8 0.142584 28786.5 0.142587 28333.3 0.150094 35796.1 0.150089 35972.1 0.150092 33884.1 Iteration 4 0 0.007505 0.015009 0.022514 0.030019 0.037523 0.045028 0.052533 0.060037 0.067542 0.075047 0.082551 0.090056 0.097561 0.11257 0.150093 Iteration 5 36895.9 26400.2 18312.4 12632.4 9585.79 7512.37 5989 5040.63 4705.5 4945.32 5759.78 6871.66 8149.43 9593.35 12899.1 37862.5 Iteration 6 0 0.007505 0.015009 0.022514 0.030019 0.037523 0.045028 0.052533 0.060037 0.075047 0.090056 0.11257 0.120075 0.127579 0.142589 0.150093 23 35548 25615.6 17968.4 12603.2 9677.79 7608.42 6014.46 5003.01 4714.17 5728.19 8163.64 12826.8 14958.9 17372.4 28288.4 37805.4 0 0.007505 0.015009 0.022514 0.030019 0.037523 0.045028 0.052533 0.060037 0.067542 0.075047 0.082552 0.11257 0.120075 0.135084 0.150094 34858.4 25571.7 18201 12746.3 9207.72 7585.35 5895.34 4877.98 4532.59 4859.15 5857.29 6827.63 12446.5 14327.2 22272.2 35796.1 Iteration 7 0 0.007505 0.015009 0.030019 0.037523 0.045028 0.052533 0.060037 0.067542 0.075047 0.090056 0.097561 0.105066 0.11257 0.120075 0.150094 Iteration 8 35446.6 25233.7 17537.5 9586.04 7576.18 6041.08 5061.26 4763.46 5014.41 5813.85 8117.91 9531.8 11177.3 13053.7 15203.6 38000.7 0 0.007505 0.015009 0.022514 0.030019 0.037523 0.045028 0.052533 0.060038 0.067542 0.075047 0.090056 0.097561 0.11257 0.120075 0.150094 24 35540.1 25166.9 17422.6 12305.7 9580.99 7571.49 6034.73 5052.7 4755.72 5008.73 5811.47 8106.95 9517.99 13046.1 15172.1 38139 6.3 AGMA ABAQUS Mesh Convergence Data 25 7. References 1. AGMA 2001-D04, “Fundamental Rating Factors and Calculation Methods for Involute Spur and Helical Gear Teeth.” 2. AGMA 908-B89, “Geometry Factors for Determining the Pitting Resistance and Bending Strength of Spur, Helical, and Herringbone Gear Teeth.” 3. Budynas, R. G., & Nisbett, J. K. (2008). Shigley's Mechanical Engineering Design. New York: McGraw-Hill. 4. Cavdar, K., Karpat, F., & Babalik, F. C. (2005). Computer Aided Analysis of Bending Strength of Involute Spur Gears with Asymmetric Profile. Journal of Mechanical Design, 127(3), 477. 5. Kawalec, A., Wiktor, J., & Ceglarek, D. (2006). Comparative Analysis of ToothRoot Strength Using ISO and AGMA Standards in Spur and Helical Gears With FEM-based Verification. Journal of Mechanical Design, 128(5), 1141. 6. Li C.-H., Chiou H.-S., Hung C., Chang Y.-Y., & Yen C.-C. (2002). Integration of finite element analysis and optimum design on gear systems. Finite Elements in Analysis and Design, 38(3), 179–192. 7. Li, S. (2002). Gear Contact Model and Loaded Tooth Contact Analysis of a Three-Dimensional, Thin-Rimmed Gear. Journal of Mechanical Design, 124(3), 511. 8. Sfakiotakis V.G., Vaitsis J.P., & Anifantis N.K. Numerical simulation of conjugate spur gear action. Computers and Structures, 79(12), 1153–1160. doi:10.1016/S0045-7949(01)00014-1 9. Zhang-Hua F., Ta-Wei C., & Chieh-Wen T. (2002). Mathematical model for parametric tooth profile of spur gear using line of action. Mathematical and Computer Modelling, 36(4), 603–614. RUSH GEARS WEBSITE USED TO GENERATE CAD FILES? 26