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 1.1 Scope and Background ....................................................................................... 1 2. Theory and Methodology ............................................................................................ 4 2.1 Derivation of the Lewis Bending Equation ........................................................ 4 3. Analysis ....................................................................................................................... 6 3.1 3.2 Microsoft Excel Analysis ................................................................................... 6 3.1.1 Determination of the Lewis Bending Stress .......................................... 6 3.1.2 Determination of the AGMA Bending Stress ........................................ 6 3.1.3 Determination of the AGMA Contact Stress ......................................... 6 ABAQUS Finite Element Analysis .................................................................... 7 3.2.1 Lewis Bending Stress Model ................................................................. 7 3.2.2 AGMA Design Stresses Model ............................................................ 14 4. Results and Discussion .............................................................................................. 15 4.1 Comparison of Data ......................................................................................... 15 4.1.1 4.2 Lewis Bending Model vs Excel Analysis ............................................ 15 Sources of Error and Divergence ..................................................................... 16 5. Conclusion ................................................................................................................. 17 6. Appendices ................................................................................................................ 18 ii 6.1 Microsoft Excel Analysis – Lewis Bending Equation ..................................... 18 6.2 Microsoft Excel Analysis – AGMA Design Equations ................................... 19 6.3 Lewis Bending ABAQUS Mesh Convergence Data ....................................... 28 6.4 AGMA ABAQUS Mesh Convergence Data.................................................... 29 7. References.................................................................................................................. 30 iii LIST OF TABLES Table 1 – Symbols and Variables ..................................................................................... ix Table 2 – Mesh Convergence Study Details.................................................................... 12 Table 3 – Lewis Bending Stress by Distance through Tooth as Calculated by Abaqus .. 13 Table 4 – Comparison of Lewis Bending Stresses .......................................................... 15 iv LIST OF FIGURES Figure 1 – Gear Tooth Material Resisting Bending Load ................................................. 4 Figure 2 - Cut and Partitioned Pinion Tooth before Meshing ........................................... 7 Figure 3 - Boundary Conditions on the Pinion Gear Tooth............................................... 8 Figure 4 – Load Position, Magnitude, and Coordinate System ......................................... 9 Figure 5 – Coupling Constraint used to Apply Wt ............................................................ 9 Figure 6 - Example Graph of Stress vs Distance along Path through Gear Tooth .......... 12 Figure 7 – The Percentage Change in Standard Deviation versus Iteration .................... 12 Figure 8 – Node Path Used to Determine Bending Stresses ........................................... 13 Figure 9 - Visualization of Completed Pinion Tooth, with Stress Distribution .............. 15 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 Pitting resistance geometry factor Bending strength geometry factor Rim thickness factor Load distribution factor Overload factor psi in HB - 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 AGMA quality factor AGMA contact stress, gear in-1 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 Contact fatigue strength, gear psi 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 t W YG YN YP ZNG ZNP Ultimate strength of the gears Operational torque transmitted to the gear psi 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 1.1 Scope and Background There are two primary modes of failure for spur gears in contact with each other: failure by bending and failure by contact stress at the gear tooth surface (Budynas, 2008). The contact stress, or pitting stress, between two contacting gears is a function of the Hertzian contact equation, and is proportional to the square root of the applied tooth load (AGMA 2001-D04). The bending stress is calculated by assuming the gear tooth is a cantilevered beam, with a cross section of face width by tooth thickness. The gear bending stress is directly proportional to the tooth load. In general, bending failure will occur when the stress on the tooth is greater than or equal to the yield strength of the gear tooth material. Pitting failure will occur when the contact stress between the meshing gears is greater than or equal to the surface endurance strength. The objective of this project is to establish the effectiveness of the Abaqus finite element software to calculate bending and contact stresses of spur gears. Gear stresses are calculated in both Microsoft Excel as well as Abaqus software packages. Two Abaqus models will be created: one to model the Lewis Bending equation, and another to model the AGMA gear design equations. The Abaqus analyses will be static (as opposed to dynamic) in order to simplify the analysis. For the AGMA Abaqus model, two gears will be modeled: a pinion (smaller) gear and a driven gear. These gears are cut from solid gears to single teeth in order to simplify the analysis as well as greatly reduce computation time. In order to match the equations detailed in Ref. 1, various other assumptions must be made. These assumptions include full-depth teeth, spur involute gears operating on parallel axes, undamaged gear teeth, elastic isotropic materials, and gear contact ratios between 1.0 and 2.0. For a list of all assumptions built into the AGMA equations, see section 1.2 of Ref. 1. The Lewis Bending equation is one of the oldest and yet most important design equations to consider when sizing gears (especially spur gears). The equation was formulated by Wilfred Lewis in 1892, and was the first of its kind to take into account specific geometric aspects of the tooth profile to determine tooth stresses (Ref. 3). It remains one of the primary ways to size gears for bending loads, and is by far the easiest 1 way to get reasonable results. Lewis derived his equation by making a few assumptions. Firstly, he assumed that each gear tooth could be treated separately from the gear mesh. t Next, he applied the transmitted load (W in the table of variables) to the tip of the tooth. This is ideally the most conservative place to apply the load, however it doesn’t quite match reality. In the instant that a pair of gear teeth comes into contact in a gear mesh, an adjacent tooth pair is still in contact. Therefore, when contact is created at the tip of a pair of teeth the load is shared by multiple contact points. It is therefore conservative to apply the full transmitted load to the tip of the gear tooth. In reality, the full load should be applied somewhere in the middle of the tooth (say, at the pitch circle). This is the point of contact on the gear teeth when only one pair of teeth is contacting (Ref. 3). Lewis assumed that the largest stresses in the gear tooth would be bending, and therefore modeled the tooth as a cantilevered beam (see Fig. 1 below). Based on this assumption, the largest stress is located in the root of the tooth at the base, since this location is furthest away from the neutral axis of bending. Section 2.1 contains a derivation of the Lewis Bending equation, and appendix 6.1 contains the Lewis Bending Excel analysis. • What do the AGMA equations take into account that the Lewis Bending equation does not? (which equation is more conservative, and where is the predicted failure point?) • Lewis bending equation is theoretical, AGMA is empirical – experimentation + trial / error • Using Rush Gear CAD files to import geometry into Abaqus 2 3 2. Theory and Methodology 2.1 Derivation of the Lewis Bending Equation (retrieved from http://en.wikipedia.org/wiki/File:To http://en.wikipedia.org/wiki/File:Tooth_as_beam.png) Figure 1 – Gear Tooth Material Resisting Bending Load 4 5 3. Analysis 3.1 Microsoft Excel Analysis 3.1.1 Determination of the Lewis Bending Stress See section 1.1 for a brief history of the Lewis Bending equation, section 2.1 for a derivation, and appendix 6.1 for the Lewis Bending analysis in Excel. The values of the face width, “F”, diametral pitch, “P”, and Lewis Form factor, “Y”, can be found in appendix 6.2. The transmitted load and yield strength used are consistent between the Excel and Abaqus analyses. 3.1.2 Determination of the AGMA Bending Stress 3.1.3 Determination of the AGMA Contact Stress 6 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 applica 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 ou the mesh convergence for both models. 3.2.1 Lewis Bending Stress Model The first Abaqus model described in this report was created in order to simulate bending of a single gear tooth, and to compare results with the Lewis Bending equation as calculated in Excell (see Appendix 6.2). This was a static analysis, with simplified geometry. The development of this model is described below, with content organized by the different “windows” of the Abaqus software (Part, Material, Load, etc.). etc The pinion (smaller) gear was chosen (instead of the larger gear) for this analysis. The CAD file was imported as a *step file, and acquired from the Rush Gears website (Ref. 10). The geometry required some fixing in order to result in a meshable part. The “geometry edit” tools were used to remove features like redundant edges, small faces, and invalid features. Once the geometry was cleaned up, all but one tooth were removed using datum planes and extruding commands. Using only one tooth for this is analysis allows for more accurate results since a higher density of mesh becomes available for the same memory requirement. The last step taken in the Parts w window was to create a few partitions in the gear tooth to make Figure 2 - Cut and Partitioned Pinion Tooth the mesh elements more symmetri symmetric. Figure 2 7 before Meshing shows the final geometry of the pinion gear tooth, before meshing. The material chosen for this gear is AISI 4140 steel, to match the corresponding analysis performed in Excel ((INSERT REF FOR MATERIAL). ). The material is assumed to be isotropic and elastic. The modulus of elasticity, E, for the material is 30E6 psi, and nd the poisson’s ratio is 0.3. The yield strength of the gear, SY, is 61,000 psi, and the ultimate strength, SUT, is 95,000 psi. The Brinell hardness of this steel is 197. See the SYMBOLS AND VARIABLES section for the explanation of all variables used in this report. Fixed boundary conditions (BC), with all degrees of freedom (DOF) restricted, were imposed on three faces of the he pinion tooth: on the bottom face and on each side (see Figure 3). ). There are three important assumptions of the Figure 3 - Boundary Conditions on the Pinion Gear Tooth Lewis Bending equation, as stated in Ref. 3: the gear tooth is treated as a cantilevered bbeam, eam, it is assumed that other teeth in the mesh do not share the load, and the max stress will be bending and occur in the root of the tooth. The fixed BC on the bottom face emulates a cantilevered condition. The BC on each side face of the tooth restric restrictt lateral material movement, forcing the tooth to be rigid and thus resulting in the max stress forming in the root of the tooth. 8 The next step was to apply t the 800 lbf transmitted load (W ), to match the applied load in the Excel analysis. In order to apply the load, a continuum distributing coupling with a reference point and custom datum coordinate system are used (see Figure 4). This coupling applies the load to the reference point then redistributes it to the entire surface surfa (shown in pink in Figure 6). Figure 4 – Load Position, Magnitude, and Coordinate Shigley states that the Lewis Bending equation assumes that the System load is applied completely at the top of the tooth, evenly distributed across the face width, F (Ref. 3). While the use of the coupling ling doesn’t exactly match the intent of the equation, I found that the stress distribution that results from it more closely matches the actual state of stress in the gear tooth. The stress distribution created using this method is symmetric, with the largest l values in the root on either side of the tooth. Figure 5 – Coupling Constraint used to Apply W t 9 Once the gear tooth was simplified, partitioned, bounded, and had an applied load, the next step was to mesh the geometry. Originally I used Tet elements for the mesh, however the partitions I created after I cut the tooth enabled the use of Hex elements. This elements tend to results in more accurate stresses (INSERT REF.?). The exact element type is a standard, quad, non-reduced integration element (C3D20). I chose quadratic elements vice linear elements because of the increased number of nodes and therefore higher accuracy. Figure 10 shows the resulting gear tooth mesh. It was very symmetric, with zero percent mesh warnings and errors. A mesh convergence study is an important tool that should be used in any finite element model to determine when the mesh density is sufficient enough to provide accurate results. Starting with a course mesh, the model is compiled and submitted and a result is found. The mesh is then made denser, the model resubmitted, and a new result documented. This process is iteratively repeated until the result (or some statistical variable that makes use of the resulting data) shows a minimal percentage change between iterations. I performed a mesh convergence study with eight iterations and finished with a seed size (mesh density) in Abaqus of 0.0175. According to Kawalec and Wiktor (Ref. 5), a successful mesh convergence study will result with stresses changing by less than 0.4% at the last iteration. Even though the error percentage calculated in my study was slightly larger (0.57%), I decided to keep the 0.0175 mesh density. The computer that ran the analysis did not have enough memory to run the model at a higher number of elements, and a decrease in error of 0.1% will not largely impact my results. For each iteration I created a node path through the thickness of the tooth, and then created a data set of the stress along this path (see Table 3). Appendix 6.3 contains this stress versus distance data for all eight iterations, as calculated by Abaqus. Table 2 shows information for each iteration of the mesh convergence study: the number of elements, mesh size, max stress, standard deviation, and percentage change in standard deviation of the data relative to the last iteration. The percentage change in standard deviation versus iteration from this Table is shown visually in Figure 7. As you can see, with each successive iteration the distribution of stress through the tooth face reaches a steady state. Table 3 shows the stress versus distance data for iteration 8, the 10 final mesh density. The max stress in the root on one side of the tooth was 35,540 psi, and on the other side it was 38,139 psi. In section 4.1, Comparison of Data, this result will be compared to the results calculated in Excel. 11 # of Mesh Max Stress % Iteration Elements Size [psi] STDEV change 1 240 0.065 35796 10389 N/A 2 460 0.05 35972 10117 -2.694 3 1334 0.035 35782 10947 7.582 4 3120 0.025 37863 10338 -5.889 5 3872 0.0225 37805 10562 2.119 6 3900 0.02 35796 10389 -1.661 7 6642 0.0185 38001 10563 1.647 8 7410 0.0175 38139 10503 -0.570 Table 2 – Mesh Convergence Study Details Figure 6 - Example Graph of Stress vs Distance along Path through Gear Tooth FEA Lewis Bending Analysis Mesh Convergence Study % Difference 10.000 5.000 0.000 1 2 3 4 5 6 -5.000 -10.000 Iteration No. Figure 7 – The Percentage Change in Standard Deviation versus Iteration 12 7 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) Table 3 – Lewis Bending Stress by Distance through Tooth as Calculated by Abaqus Figure 8 – Node Path Used to Determine Bending Stresses 13 3.2.2 AGMA Design Stresses Model 14 4. Results and Discussion 4.1 Comparison of Data 4.1.1 Lewis Bending Model vs Excel Analysis Pinion σLP = 24854 Excel analysis σLP2 = 35540 Abaqus analysis Table 4 – Comparison of Lewis Bending Stresses Table 4 shows the final results for the Lewis Bending stress as calculated by both Microsoft Excel and Abaqus. As stated in section 3.2.1, the bending stress in Abaqus was determined by finding the stresses along the node path shown in Fig. 7. The location of the node path was chosen based on proximity to the tooth root and the strength of the mesh in the area. It was important to compare the Excel bending stress with the Abaqus stress in the root of the tooth, since that is where the Lewis Bending equation assumes the maximum stress will occur. The bending stress as calculated by Abaqus is 30.7% higher than the stress calculated in Excel. There are various reasons why this error exists between the two solutions. A few sources of error are discussed below in section 4.2. Figure 9 - Visualization of Completed Pinion Tooth, with Stress Distribution 15 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 Abaqus load distributed to entire face of tooth 16 5. Conclusion 17 6. Appendices 6.1 Microsoft Excel Analysis – Lewis Bending Equation 4.7 Lewis Bending Analysis The gears are analyzed for stall torque using the Lewis Bending equation. The transmitted load for section 4.7 is assumed to act at the top of the tooth, evenly distributed along the face width. t W [lbf] = 800 Tangential transmitted gear load. See section 4.1 and 4.2 for inputs. Pinion σL [psi] = σLP = Gear 24854 σLG = Factors of safety are calculated on material yield strength. FSL = Sy / σL Pinion FSL [ ] = 19010 FSLP = Gear 2.5 FSLG = 18 3.2 Lewis Bending stress. 6.2 Microsoft Excel Analysis – AGMA Design Equations 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. 19 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. 20 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 21 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). 22 Reliability Factor, KR The reliability factor takes into account normal statistical material failures that occur iin n 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 facto in the middle of the range. KR [ ] = 1 See Table 11 of Ref. 11. 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 progra 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 t σ = W KoKvKs(Pd/F)(KmKB/J) Wt is the transmitted tangential load going into the pinion gear. See Ref. 4, Eq. 14 14-15 Gear Pinion σ [psi] = σP = 29674 σG = 23 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 erial 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 ifferent 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 t 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). 14 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 24 See Ref. 4, Eq. 1414 41 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 CP [psi1/2] = Pinion CPP = Gear 2291 CPG = 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 25 Gear Contact Stress, σc The numerical value of the contact stress comes from the equation below, from Ref. 4: t σc = Cp(W KoKvKs(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 151546 86590 σcG = 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 = 26 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 = 27 1.9 See Ref. 5.2, Eq. 1442 6.3 Lewis Bending ABAQUS Mesh Convergence Data Iteration 1 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 0.12758 0.135084 0.142589 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 17602.3 22272.2 28336.8 35796.1 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 0.105065 0.11257 0.120075 0.150093 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 11082.9 12899.1 15148 37862.5 Iteration 2 0 0.007504 0.015009 0.022513 0.030018 0.037522 0.045027 0.052531 0.060036 0.06754 0.075044 0.082549 0.090053 0.097558 0.105062 0.112567 0.120071 0.127575 0.142584 0.150089 35311.9 27392.3 20739.2 15352.4 11232 8379.7 6354.17 5065.21 4510.88 4691.16 5605.31 6593.34 7840.42 9347.05 11113.2 13137.9 15086.4 18343.7 28786.5 35972.1 Iteration 5 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 0.135084 0.142589 0.150093 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 21477.4 28288.4 37805.4 28 Iteration 3 0 0.007505 0.015009 0.022514 0.030018 0.037523 0.045028 0.052532 0.060037 0.067541 0.075046 0.082551 0.090055 0.112569 0.142587 0.150092 35782.1 26555.9 19130.2 13504.4 9680.22 7656.52 6014.95 4996.13 4599.05 4823.79 5669.93 6714.4 7902.6 12330.3 28333.3 33884.1 Iteration 6 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 0.12758 0.135084 0.142589 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 17602.3 22272.2 28336.8 35796.1 6.4 AGMA ABAQUS Mesh Convergence Data 29 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. (2001). Analysis of strength of tooth root with notch after finishing of involute gears. Archive of Mechanical Engineering, 48(3), 217248. 6. 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. 7. 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. 8. 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. 9. 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 10. 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. 11. Rush Gears Part Search and CAD. Retrieved July 9, 2013, http://www.rushgears.com/Tech_Tools/PartSearch8/partSearch.php. 30 from