Program Gear Load, Stress and Life Analysis

Program 60-540 Gear Load, Stress and Life Analysis Introduction This TK Model calculates load ratings for gearsets according to American Gear Manufacturers Association Standard AGMA 2001-B88, Fundamental Rating Methods for Involute Spur and Helical Gear Teeth. This standard provides a method by which different gear designs can be compared. It is not intended to assure the performance of assembled gear drive systems. NOTE: The symbols and definitions used in this Standard may differ from other AGMA Standards. The user should not assume that familiar symbols can be used without a careful study of these definitions. Where applicable AGMA Standards exist, they should be used in preference to this Standard. Miner's Rule for duty cycle analysis with steel gears is available in the model for all AGMA steel materials except steels that have been nitrided. (The life factor/cycle curves in the standard do not apply to nitrided gears.) Duty cycle analysis may be used for non-agma materials added to the material table by the user if a stress/cycle curve is entered. Help information for most of the variables in the model is available in the Help file. This model does not address scoring or scuffing problems. Appendix A of the standard may be of help in assessing choice of lubrication. UTS Models 60-560 and 60-5408 may also be of help. You may wish to use UTS Program 500, Models 60-102 and 60-5406 to obtain input data for this program. Caution THIS MODEL IS MEANT FOR USE BY EXPERIENCED GEAR DESIGNERS--NOT BY THE GENERAL ENGINEERING PUBLIC. Use of data from this model requires experience with the manufacture and application of parallel axis spur and helical involute gears of all types. SUGGESTED AND DEFAULTED VALUES PROVIDED BY THE MODEL ARE FROM AGMA 2001-B88, BUT THE STANDARD SHOULD BE CONSULTED BEFORE USING THESE VALUES. 1

UTS Integrated Gear Software IMPORTANT: IF YOU CHANGE ANY OF THE DEFAULT VALUES YOU MAY NOT BE IN ACCORDANCE WITH AGMA 2001-B88. CAUTION IS ADVISED IF YOU MAKE A CHANGE. Reference: Data has been extracted from AGMA Standard 2001-B88, Fundamental Rating Methods for Involute Spur and Helical Gear Teeth with the permission of the publisher, American Gear Manufacturers Association, 1500 King Street, Alexandria, Virginia 22314. 2

60-540 Gear Load, Stress and Life Analysis Definitions Note: Variable names in this discussion refer to expressions used in the TK Solver model. Miner's Rule for Duty Cycles Cumulative fatigue damage from a duty cycle where the load and/or speed vary can be assessed using the damage criteria proposed by Miner (Miner's Rule). Miner's Rule assumes that the damage done by each stress repetition at a given stress level is equal, and that the first stress cycle at a uniform stress level is as damaging as the last. Miner's Rule operates on the hypothesis that the portion of useful fatigue life used up by a number of repeated stress cycles at a particular stress is proportional to the total number of cycles in the fatigue life, if that were the only stress level applied to the part. The order in which each stress is applied is not considered significant. Failure is expected when each portion added together reaches 100%. (After solving the model using a duty cycle table bar charts are available on the plot sheet showing the percentage of the life of the material used by each condition in the duty cycle.) To use the procedure, the load spectrum (duty cycle) must be defined. This may done by entering each load condition in the interactive table Miner1 for the pinion and table Miner2 for the gear. The Miner's Rule process requires that a design must exist and stresses must be known for each load level in the load spectrum. The load (power or torque and speed) may be entered instead of gear stress for each load condition and the model will calculate the required stress. The required input items are marked with (i). (Either power or pinion torque may be entered; if both are entered the torque will be calculated from the power and the torque will be overwritten.) The items which may be entered but have calculated default values are marked with (d). In some cases, entries for these factors may take the results outside the standard. In any event, the standard must be consulted to verify that the calculations are valid. The application factors, Ca and Ka, must be equal to one to use Miner's Rule. Any overloads must be included in the duty cycle. In order to apply Miner's Rule the materials selected must have a stress/cycle curve defined. The model contains curve data in the List Functions under AGMA MATERIAL STRESS/CYCLE FUNCTIONS: on the Function sheet for all AGMA materials covered by the Life Factor curves Fig 16-1 and Fig 16-2. Any materials added by you, which are intended to be used in duty cycle calculations, must have stress/cycle data defined. You may use the Material Update tab of the IGS data input form for this model to add user defined materials, including stress/cycle data. 3

UTS Integrated Gear Software General Commercial or Critical Application The Life Factor curves (Fig 16-1 & 16-2 of the standard) provide two different levels at the high cycle portion of the curves. The upper curve may be used for general commercial applications. The lower curves are typically used for critical applications. The model default is set to 'g and the model will use the upper curve with this setting. If you wish to use the lower curve enter 'c to over-ride the default. Geometrical Data Much of the required entry data is geometrical data on the gearset. This data may be obtained from any source. You may find UTS Program 500 or other UTS TK Models useful. Transmission Accuracy Level Number Qv is the transmission accuracy level number. It can be the same value as the AGMA Quality Number for the lowest quality gear in the mesh. See AGMA Standard 2000-A88 for information on quality numbers. Nominal Normal Diametral Pitch The nominal normal diametral pitch is the reference normal diametral pitch for the gearset. Nominal Normal Module The nominal normal module is the reference normal module for the gearset. Nominal Transverse Diametral Pitch The nominal transverse diametral pitch is the number of teeth per inch of reference pitch diameter in the plane of rotation. Nominal Transverse Module The nominal transverse module is the number of millimeters of reference pitch diameter per tooth in the plane of rotation. (The nominal pitch and module should not be confused with the operating pitch and module which is set after the gears are assembled on a particular center distance.) 4

60-540 Gear Load, Stress and Life Analysis Nominal Helix Angle The nominal helix angle is the helix angle of the teeth at the reference pitch diameter. Base Helix Angle The base helix angle is the helix angle on the base cylinders of the gears. Operating Transverse Pressure Angle The operating Transverse pressure angle is the actual running pressure angle of the gearset after the gears have been assembled at the operating center distance. It is the angle between the line of action and a normal to the line of centers in the plane of rotation (transverse plane). Face Width The face width is the net face width of the narrowest of the two gears. (If the gear has no chamfers or tooth end modifications it is the width of the gear blank but if the tooth ends are modified an appropriate correction must be made.) Profile Contact Ratio The profile contact ratio is the length of action in the transverse plane, Z, divided by the transverse base pitch, pb. (It is the time average number of teeth in contact.) Pitting Resistance Geometry Factor The Pitting Resistance Geometry Factor, I, evaluates the radii of curvature of the contacting tooth profiles based on the tooth geometry. Bending Strength Geometry Factor The Bending Strength Geometry Factor, J, evaluates the shape of the tooth, the position at which the most damaging load is applied, and the sharing of the load between oblique lines of contact in helical gears. 5

UTS Integrated Gear Software Load, Stress and Life Factors Application Factors The application factors, Ca and Ka, make allowance for any externally applied loads in excess of the nominal tangential load, Wt. Application factors can only be established after considerable field experience is gained in a particular application. In determining the application factor, consideration should be given to the fact that many prime movers develop momentary peak torques appreciably greater than those determined by the nominal ratings of either the prime mover or the driven equipment. There are many possible sources of overload which should be considered. Some of these are: system vibrations, acceleration torques, overspeeds, variations in system operation, and changes in process load and conditions. When operating near a critical speed of the drive system, a careful analysis of conditions must be made. NOTE: If Miner's Rule is to be used Ca=Ka must be set to one and overloads included in the duty cycle. Service factors Service factors are applied to allow for the type of service the gearset will be put to. Historically, service factors have been used and have included the application factor, and in some cases, the reliability and life factors as well. Where the service factor has included application effects, but has not included reliability and life effects, it should be redefined as Ca or Ka. If a service factor is used it should be defined as: CSF = Ca(CR/CL)^2 for pitting resistance KSF = Ka*KR/KL for bending strength where: Ca,Ka are the Application Factors CR,KR are the Reliability Factors CL,KL are the Life Factors NOTE: If Miner's Rule is to be used with a duty cycle service factors can not be used. 6

60-540 Gear Load, Stress and Life Analysis Temperature factors Temperature factors are used to allow for reduction of gear capacity due to operating temperature. For metal gears the temperature factors, CT and KT, are generally taken as unity when gears operate with temperatures of oil or gear blank not exceeding 250 degrees F (120 degrees C). When gears operate at oil or gear blank temperature above 250 degrees F (120 degrees C), CT and KT are given values greater than one to allow for the effect of temperature on oil film and material properties. (CT is usually set equal to KT.) Consideration must be given to the loss of hardness and strength of some materials due to the tempering effect of temperatures over 350 degrees F (175 degrees C). The effect of temperature on gears made of materials other than metals must be carefully considered. The strength and durability of plastic gears, for example, are very dependent on temperature. Transverse Load Distribution Factor The Transverse Load Distribution Factor, Cmt, accounts for the non-uniform distribution of load among the gear teeth which share the load. It is affected primarily by the correctness of the profiles of mating teeth: i.e., profile modification, and/or profile error. Standard procedures to evaluate Cmt have not been established and Cmt is assumed to be unity. Load Distribution Factors The Load Distribution Factors, Cm & Km, modify the rating equations to reflect the non-uniform distribution of load along the lines of contact. The amount of nonuniformity of load distribution is caused by, and is dependent upon, the following influences: (1) Gear tooth manufacturing accuracy: lead, profile, and spacing. (2) Alignment of the axis of rotation of the pitch cylinders of the mating gear elements. (3) Elastic deflections of gear unit elements: shafts, bearings, housings, and foundations which support the gear elements. (4) Bearing clearances. (5) Hertzian contact and bending deformations at the tooth surface. (6) Thermal expansion and distortion due to operating temperature. (Especially important for wide face gears.) (7) Centrifugal deflections due to operating speed. (8) Tooth crowning and end relief. 7

UTS Integrated Gear Software The load distribution factor is defined as the peak load intensity divided by the average load intensity across the face width. Its magnitude is affected by two components: Cmf face load distribution factor Cmt transverse load distribution factor Cmt is set to unity in the Standard and Cm=Km=Cmf. The model default is the Standard load distribution factor. UTS Model 60-5406 may be used to obtain equivalent Load Distribution Factors if the teeth are crowned. The model may also be used to find the optimum crown to keep the Cm and Km factors to a minimum. Face Load Distribution Factor The Face Load Distribution Factor, Cmf, accounts for the non-uniform distribution of load across the gearing face width. The magnitude of the face load distribution factor is defined as the peak load intensity divided by the average load intensity across the face width. Two different methods of determining the Face Load Distribution Factor, Cmf, may be used, the empirical method and the analytical method. These two methods will result in significantly different results in some cases. Total Lead Mismatch The Total Lead Mismatch, et, is a virtual separation between the tooth profiles at the end of the face width which is composed of the static, no load separation plus a component due to the elastic load deformations. The total lead mismatch between mating teeth is influenced by: (1) Gear tooth manufacturing accuracy: lead, profile, and spacing. (2) Alignment of the axis of rotation of the pitch cylinders of the mating gear elements. (3) Elastic deflections of gear unit elements: shafts, bearings, housings, and foundations which support the gear elements. (4) Bearing clearances. (5) Thermal expansion and distortion due to operating temperature. (Especially important for wide face gears.) (6) Centrifugal deflections due to operating speed. (7) Tooth crowning and end relief. 8

60-540 Gear Load, Stress and Life Analysis Usually, the items that contribute the most to et are: (1) Gear accuracy the combined lead mismatch of the pinion and gear. (2) Housing accuracy the lead mismatch due to gear housing machining errors which cause the shafts to be non-parallel and out of plane. (3) Elastic deformation the lead mismatch due to deformations of the gear blanks, shafts, bearings, housing and foundation. The evaluation of et is very difficult, but it is critical to the reliability of the analytical method. UTS Model 60-102 may be helpful in calculating a suitable value for et. Tooth Stiffness Constant The Tooth Stiffness Constant, G, is the average mesh stiffness of a single pair of teeth in the normal direction. The usual range of this value, that is compatible with this analysis, for steel gears is 1.5 to 2.0 x 10^6 psi (1.0 to 1.4 x 10^4 MPa). The model default value for G is obtained by first adjusting the above values based on the modulus of elasticity of the materials. This range is then proportioned according to the operating transverse pressure angle. Reliability Factor The reliability factors, CR and KR, account for the effect of the normal statistical distribution of failures found in materials testing The allowable stress numbers given in Tables 14-1, 14-2, 14-7 and 14-8 are based upon a statistical probability of one failure in 100 at 10^7 cycles. This table (Table 17-1) contains reliability factors which may be used to modify these allowable stress numbers to change that probability: Requirements of Application CR,KR Fewer than one failure in 10,000 1.50 Fewer than one failure in 1,000 1.25 Fewer than one failure in 100 1.00 Fewer than one failure in 10 0.85 Note: At a value in the range of 0.85 plastic flow may occur rather than pitting. The term Fewer than one failure in X must be understood to mean that out of X gearsets one gearset will not reach predicted life and X-1 sets will exceed predicted life. The term does not mean that one set of X sets will immediately fail. 9

UTS Integrated Gear Software Empirical Method Two different methods of determining the Load Distribution Factors, Cm & Km, may be used, the empirical method and the analytical method. These two methods will result in significantly different results in some cases. The empirical method requires a minimum amount of information. This method is recommended for relatively stiff designs which meet the following requirements: (1) Net face width to pinion pitch diameter ratios, F/d, less than or equal to 2.0. (For double helical gears the gap is not included in the face width.) (2) The gear elements are mounted between bearings (not overhung). (3) Face width up to 40 inches. (4) Contact across full face width of narrowest member when loaded. Designs which have high crowns to centralize tooth contact under deflected conditions may not use this method if loaded contact does not extend across the full face width of narrowest member. Analytical Method The analytical method is based on theoretical calculation of the values of elastic tooth deformation under load and lead mismatch. This method requires knowledge of the design, manufacturing, and mounting to evaluate the Load Distribution Factors, Cm and Km. Stresses will be computed for full contact across the face under operating load or for partial contact across the face under the load depending on the tooth stiffness constant, G, and the lead mismatch, et. (G is roughly proportional to the pressure angle.) The analytical method is valid for any gear design and is recommended for the following conditions: (1) Net face to pinion pitch diameter ratio, F/d, greater than 2.0. (For double helical gears the gap is not included in the face width.) (2) Applications with overhung gear elements. (3) Applications with long shafts subject to large deflections or where deflections under load reduce width of contact. (4) Applications where contact does not extend across the full face of narrowest member when loaded. 10

60-540 Gear Load, Stress and Life Analysis For designs which have high crowns to centralize tooth contact under deflected conditions, the Load Distribution Factors, Cm and Km, may be conservatively approximated by this method when equivalent Load Distribution Factors are not used. (Equivalent Load Distribution Factors are not part of the AGMA standard.) UTS Model 60-5406 may be used to obtain equivalent Load Distribution Factors based on the actual crown on one or both members. The model may also be used to find the optimum crown. The equivalent Load Distribution Factors may then be used as input values in this model to determine the effect of the crown. When using the analytical approach, the calculated load capacity of the gears should be compared with past experience since it may be necessary to re-evaluate rating factors to arrive at a rating consistent with past experience. Lead Correction Factor The Lead Correction Factor, Cmc, modifies peak load intensity when crowning or lead modification is applied. Cmc = 1.0 for gears with unmodified leads and 0.8 for gears with leads properly modified by crowning or lead correction. NOTE: Properly modified means that contact extends across the full face width of the narrowest member when under design load. 11

UTS Integrated Gear Software Mesh Alignment Factor The Mesh Alignment Factor, Cma, accounts for the misalignment of the axes of rotation of the pitch cylinders of the mating gear elements from all causes other than elastic deformations. The value of Cma is calculated from the type of unit and the face width. Do NOT enter a value. Mesh Alignment Correction Factor The Mesh Alignment Correction Factor, Ce, is used to modify the mesh alignment factor when the manufacturing or assembly techniques improve the effective mesh alignment. Ce = 0.8 when the gearing is adjusted at assembly or when the compatibility of the gearing is improved by lapping and 1.0 for all other conditions. Pinion Proportion Factor The Pinion Proportion Factor, Cpf, accounts for deflections due to load. The value of Cpf is calculated from the pinion operating pitch diameter and the face width. Do NOT enter a value. Pinion Proportion Modifier The Pinion Proportion Modifier, Cpm, alters Cpf, based on the location of the pinion relative to the bearings: S1/S = (Pinion offset)/(bearing span) Cpm = 1.0 for straddle mounted pinions with (S1/S) < 0.175 Cpm = 1.1 for straddle mounted pinions with (S1/S) > 0.175 If Cpm is not entered it will be calculated from S1 and S. Bearing Span and Offset If Cpm is entered it is not necessary to enter the bearing span, the closest bearing to the pinion CL, or the pinion offset. The span between the bearings and the offset, or distance from the CL of the pinion to the midpoint between the bearings, are used to calculate the pinion proportion modifier, Cpm. In addition to the bearing span you will need to enter either the distance from the closest bearing to the CL of the pinion or the offset. 12

60-540 Gear Load, Stress and Life Analysis Rim Thickness Factor The Rim Thickness Factor, KB, adjusts the calculated bending stress number for thin rimmed gears. It is a function of gear tooth whole depth, ht, and the thickness of the gear rim below the tooth root diameter, tr. If neither the rim thickness factor, nor the whole depth and rim thickness are entered the model will default KB to one. If the whole depth and rim thickness are entered but not the rim thickness factor the model will calculate KB. Surface Finish When surface hardened pinions (48 HRC or above) are run with through hardened gears (180 to 400 BHN), a work hardening effect is achieved which increases the pitting capacity of the gear. As the surface finish of the pinion, fp, is reduced the gear capacity increases. The pinion surface finish is used to calculate the gear hardness ratio factor for pitting, CH. (The model limits the pinion surface finish between 5 and 250 uin [.1 and 6.4 um], rms.) Hardness Ratio Factor The hardness ratio factor, CH, depends on the gear ratio and the hardness of the pinion and the gear. For through hardened gears when the pinion is substantially harder than the gear, the work hardening effect increases the gear capacity. When surface hardened pinions (48 HRC or harder) are run with through hardened gears (180 to 400 BHN), a work hardening effect is achieved. The CH factor varies with the surface finish of the pinion, fp, and the mating gear hardness. (The model limits the surface finish to a range of 5 to 250 microinches [.1 to 6.4 um], rms.) The CH factor is applied only to the gear pitting capacity not to the pinion capacity. Same Flank Contacts Per Revolution The Same Flank Contacts Per Revolution, NC, is the number of contacts made by either the right or left tooth flanks with a mating gear tooth. It does not include opposite flank contact such as occurs on an idler gear meshed with a driver and driven gear. Such an idler would have NC equal to one. 13

UTS Integrated Gear Software Idler If there is reverse bending of the teeth of the gear then the standard requires that the allowable bending stress, Sat, be reduced by 30%. Single Load The relationships between load, stress and life under this heading are for a single load condition. Separate load/stress/life data is listed for pinion pitting data, pinion bending data, gear pitting data and gear bending data. In many cases the load data will be the same for all conditions but this is not required. If you want compatible data for all conditions be sure that the load data reflects the speed and torque differences between pinion and gear because the model does not check for compatibility due to the desirability of back-solving the model. If you have a combination of loads in a duty cycle do not enter data under the Single Load heading. Use the Miner's Rule duty cycle tables. Pitting Resistance The pitting of gear teeth is considered to be a fatigue phenomenon. Initial pitting is non-progressive and is not deemed serious. Initial pitting is characterized by small pits which do not extend over the entire face width or profile height of the affected teeth. The definition of acceptable initial pitting varies widely with gear application. Initial pitting occurs in localized, overstressed areas. It tends to redistribute the load by progressively removing high contact spots. Generally, when the load has been reduced or redistributed, the pitting stops. The ratings for pitting resistance are based on the formulas developed by Hertz for contact pressure between two curved surfaces, modified for the effect of load sharing between adjacent teeth. Bending Strength The bending strength of gear teeth is a fatigue phenomenon related to the resistance to cracking at the tooth root fillet in external gears. The basic theory employed in this analysis assumes the gear tooth to be rigidly fixed at its base, thus the critical stress occurs in the fillet. If the rim supporting the gear tooth is thin relative to the size of the tooth and the gear pitch diameter, another 14

60-540 Gear Load, Stress and Life Analysis critical stress may occur not at the fillet but in the root area. The rim thickness factor, KB, adjusts the calculated bending stress number for thin rimmed gears. The strength ratings determined by this Standard are based on plate theory modified to consider: (1) The compressive stress at tooth roots caused by the radial component of tooth loading (2) Non-uniform moment distribution resulting from the inclined angle of the load lines on the teeth (3) Stress concentrations at the tooth root fillets (4) The load sharing between adjacent teeth in contact Wear, surface fatigue, or plastic flow may limit bending strength due to stress concentrations around large, sharp cornered pits or wear steps on the tooth surface. Machining steps from mismatched gear production tooling will limit bending strength due to stress concentrations around the steps which are in the most heavily stressed area of the tooth. Neither the standard nor the model will compensate for notches or steps in the tooth profile. If the gear tooling produces steps the strength rating is NOT valid and every effort must be made to correct the tooling. Transmitted Power The transmitted power is the power carried by the gearset. It is not modified to account for efficiency, etc. The transmitted power is a product of the pitch line velocity and the tangential load. Speed This is the speed of rotation of the gear. Unless the gear ratio is one the speed of the pinion is different than the speed of the gear. Torque This is the twisting moment on the gear. Unless the gear ratio is one the torque on the pinion is different than the torque on the gear. 15

UTS Integrated Gear Software Tangential Load This is the load in a direction normal to the line of centers at the operating pitch diameters of the gears. It is the component of the tooth normal load which transmits the power carried by the gearset. Pitch Line Velocity This is the linear component of the velocity of the teeth at the operating pitch diameters. The power transmitted by the gearset is the product of the tangential load and the pitch line velocity. Dynamic Factor Dynamic factors, Cv and Kv, account for internally generated gear tooth loads which are induced by non-conjugate meshing action of the gear teeth. Even if the input torque and speed are constant, significant vibration of the gear masses, and therefore dynamic tooth forces, can exist. These forces result from the relative displacements between the gears as they vibrate in response to an excitation known as transmission error. Transmission error is defined as the departure from uniform relative angular motion of the pair of meshing gears. The dynamic response is influenced by many factors. Refer to Section 8 of the Standard for further information. The model default is the approximate dynamic factor from Fig 8-1 in the standard. The dynamic factor for very accurate gearing (Qv from 12 to 15) is proportioned from the low to the high region of the shaded portion of the figure. When the known dynamic loads (from analysis or experience) are added to the nominal transmitted load, then the dynamic factor can be unity. Contact Stress Number The contact stress number is the value calculated using the pitting resistance rating formulae in the Standard. If the application factor, Ca, is being used the contact stress number is calculated without CT, CR and CH. If the service factor, CSF, is being used the contact stress number includes the effects of CT and CH. 16

60-540 Gear Load, Stress and Life Analysis Adjusted Contact Stress Number The adjusted contact stress number is the contact stress modified by CT and CR when the application factor, Ca, is being used. It is used for life calculations and has no value when the service factor, CSF, is used. Bending Stress Number The bending stress number is the value calculated using the bending strength rating formulae in the standard. If the application factor, Ka, is being used the bending stress number is calculated without KT, and KR. If the service factor, KSF, is being used the bending stress number includes the effects of KT. Adjusted Bending Stress Number The adjusted bending stress number is the bending stress modified by KT and KR when the application factor, Ka, is being used. It is used for life calculations and has no value when the service factor, KSF, is used. Life Factors The Life Factors, CL and KL, are the ratios of the adjusted stress number to the allowable stress number if application factors, Ca and Ka, are used. If service factors, CSF and KSF, are used then the model does not calculate life factors but they would, according to the standard, have a value of one. Contact Life The contact life in cycles is the number of load cycles the gear may be expected to run at the reliability level specified. The contact life in time units is the length of time the gear may be expected to run at the reliability level specified. It has been adjusted for the number of same flank contacts per revolution. Bending Life The bending life in cycles is the number of load cycles the gear may be expected to run at the reliability level specified. It has been adjusted for reverse bending if the gear is an idler. The bending life in time units is the length of time the gear may be expected to run at the reliability level specified. It has been adjusted for the number of same flank contacts per revolution and reverse bending. 17

UTS Integrated Gear Software AGMA Material Factors Elastic Coefficient The Elastic Coefficient, Cp, is a function of the modulus of elasticity and Poisson's ratio for the materials in both gears. The default value for the modulus of elasticity for AGMA materials is from Table 10-1 in the standard. Poisson's ratio is assumed to be 0.30. For NON-AGMA materials the modulus of elasticity must be entered. Poisson's ratio is defaulted to 0.3 but may be entered if you wish to use a different value. Surface Condition Factor The Surface Condition Factor, Cf, is used only in the pitting resistance calculations. It depends on: (1) Surface finish (2) Residual stress (3) Plasticity effects (work hardening) Standard surface condition factors have not been established where there is a detrimental effect. In such cases, a factor greater than unity should be used. Size Factors The size factors, Cs and Ks, reflect the non-uniformity of material properties. It depends primarily on: (1) Tooth size (2) Diameter of parts (3) Ratio of tooth size to diameter of part (4) Face width (5) Area of stress pattern (6) Ratio of case depth to tooth size (7) Hardenability and heat treatment Standard size factors have not been established where there is a detrimental size effect. In such cases, a factor greater than unity should be used. Allowable Contact and Bending Stress Numbers The allowable stress numbers depend on: (1) Material composition and cleanliness (2) Mechanical properties (3) Residual stress (4) Hardness (5) Type of heat treatment (surface or through hardened) 18

60-540 Gear Load, Stress and Life Analysis The material table ( Matl ) in the TK Solver model contains the allowable stress numbers for AGMA materials taken from the standard. These stress numbers are for 10 million cycles of unidirectional load application, application factors Ca and Ka of one and 99% reliability (CR=KR=1). For steel gears, except nitrided, the model is provided with stress/cycle curves so that a life evaluation may be made for AGMA materials. The curves are in accordance with the life factor curves in Fig 16-1 and Fig 16-2 of the standard. If materials that you add to the material table have a defined stress/cycle curve you may add the curve to the model. If your material has only a single point fatigue stress available it may also be added to the table without a curve. The number of cycles at the single point reference stress may be entered in the table if known. You may use the Material Update tab in the IGS data input form for this model to update the material table automatically. If you add materials to the table or change the stress levels in the TK Solver model, be sure to update the table and stress/cycle curves by solving the model with Update material table & stress/cycle curves? set to 'y. Top Land Normal Tooth Thickness The top land normal tooth thickness is the circular thickness of the tooth at the outside diameter in a direction normal to the tooth. For carburized and induction hardened teeth the suggested maximum effective case depth is the lesser of 0.4/Pd (0.4*m) and 56% of the top land normal tooth thickness. Min Recommended Eff Case Depth This value is for carburized, flame hardened and induction hardened teeth. Surface hardened gear teeth require adequate case depth to resist the sub-surface shear stresses developed by tooth contact loads and the tooth root fillet tensile stresses. The Min Recommended Eff Case Depth is the least depth required, at the pitch line, to resist the stresses produced by the specified load. The effective case depth for carburized and hardened gears is defined as the depth below the surface at which the Rockwell 'C' hardness, HRC, has dropped to HRC 50. The effective case depth for induction or flame hardened gears is defined as the depth below the surface at which the Rockwell 'C' hardness, HRC, has dropped to 10 points below the specified minimum surface hardness. 19

UTS Integrated Gear Software Max Recommended Eff Case Depth This value is for carburized, flame hardened and induction hardened teeth. Surface hardened gear teeth require adequate case depth to resist the sub-surface shear stresses developed by tooth contact loads and the tooth root fillet tensile stresses. However, depths must not be so great as to result in brittle tooth tips and high residual tensile stress in the core. For carburized and induction hardened teeth the maximum recommended effective case depth, at the pitch line, is the lesser of 0.4/Pd (0.4*m) and 56% of the top land tooth thickness. The effective case depth for carburized and hardened gears is defined as the depth below the surface at which the Rockwell 'C' hardness, HRC, has dropped to HRC 50. The effective case depth for induction and flame hardened gears is defined as the depth below the surface at which the Rockwell 'C' hardness, HRC, has dropped to 10 points below the specified minimum surface hardness. Minimum Total Case Depth This value is for nitrided teeth. Surface hardened gear teeth require adequate case depth to resist the sub-surface shear stresses developed by tooth contact loads and the tooth root fillet tensile stresses. The Minimum Total Case Depth is the least depth required, at the pitch line, to resist the stresses produced by the specified load. The case depth for nitrided gears is specified as the total case depth, and is defined as the depth below the surface at which the hardness has dropped to 110% of the core hardness. 20

60-540 Gear Load, Stress and Life Analysis Core Hardness The core hardness for nitrided gears has an important effect on the allowable bending stress number. The model default for this hardness is the minimum requirement from the standard for AGMA material. If you override the default for AGMA material you will be outside the standard. Non-AGMA Material Factors: Modulus of Elasticity Young's modulus of elasticity must be entered for materials which are not specified in the standard. It is used in the calculation of the elastic coefficient, Cp. Poisson's Ratio Poisson's ratio must be entered for materials which are not specified in the standard. It is used in the calculation of the elastic coefficient, Cp. It is defaulted to 0.3 but you may use a different value. Material and Heat Treatment The material and heat treatment must be specified for steel materials which are not specified in the standard. The Brinell hardness must be entered for through hardened steels. The model will process these materials in the same manner as steel materials contained in the standard. If your material is not steel the entry should be made for Other materials. 21

UTS Integrated Gear Software Examples Example 1 Example 1 is a helical gearset with a 22 tooth pinion driving a 55 tooth gear for use in a commercial enclosed gear unit. The Empirical Method will be used to rate the set. Both gears are made of carburized and hardened steel conforming to AGMA Grade 1 (Tables 14-1 & 14-7 in the standard). No lead crowning or correction is to be used. The gears will not be adjusted at assembly or lapped. Neither gear has a thin rim under the tooth roots. The pinion is straddle mounted between bearings 5 inches apart and the pinion centerline is 2 inches from the closest bearing centerline. (Note that if the pinion was not straddle mounted we could not use the Empirical Method.) The geometry and load details needed as input data are: AGMA Quality Number 10 Nominal NDP 10/in Nominal Helix 23 deg Base Helix 22.541 deg Opr Trans PA 22.5704 deg Net Face Width 2" Profile CR 1.4328 AGMA I Factor.191 Pinion Teeth 22 Opr Pitch Dia 2.4069" Rim Thick Factor 1 Same Flank Contacts per Revolution 1 Pinion is not an idler AGMA J Factor.479 Pinion Material # 19 Top Land Normal Tooth Thickness.0659" Gear Teeth 55 Rim Thick Factor 1 Same Flank Contacts per Revolution 1 Gear is not an idler AGMA J Factor.481 Gear Material # 19 Top Land Normal Tooth Thickness.0765" 22

60-540 Gear Load, Stress and Life Analysis Transmitted Power 100 HP Pinion Speed 2100 RPM Gear Speed 840 RPM Service Factors 1.5 Opening a new analysis in 60-540 brings up the data input form. Enter the data as shown above and in Figure 1-1. A report for the solved model is shown in Report 1-1. We will assume that the gearset is for a general commercial application. (See the standard for differences between general commercial and critical applications.) The pinion does not have a thin rim under the tooth roots so we will enter a thickness factor of one. Neither of our gears is an idler. 23

UTS Integrated Gear Software Fig. 1-1 24

60-540 Gear Load, Stress and Life Analysis Report 1-1 Gear Set Stress/Life - B88 (Program 60-540) Unit System: US Description Value Unit Comment Miner's Rule for Pinion and Gear Life from Duty Cycle? General commercial or critical application COMMON n g Transmission accuracy level number 10 Nominal normal diametral pitch 10.000000 1/in Nominal normal module 2.540000 mm ` Nominal transverse diametral pitch 9.205049 1/in Nominal transverse module 2.759355 mm ` Nominal helix angle 23.0000 deg Base helix angle 22.5410 deg Operating transverse pressure angle 22.5704 deg Face width, net 2.000 in Profile contact ratio 1.4328 AGMA I-Factor 0.1910000 LOAD FACTORS Application factor - pitting Application factor - bending Service factor - pitting 1.500 25

UTS Integrated Gear Software Gear Set Stress/Life - B88 (Program 60-540) Unit System: US Description Value Unit Comment Service factor - bending 1.500 Reliability factor -pitting Sets per failure Reliability factor -bending Sets per failure Temperature fact - pitting 1.000 Temperature fact - bending 1.000 Transverse load distrib fact 1.000 MATERIAL FACTORS Elastic coefficient 2290 Surface condition factor 1.000 Size factor - pitting 1.000 Size factor - bending 1.000 ANALYTICAL METHOD Total lead mismatch Tooth_stiffness constant EMPIRICAL METHOD in psi Lead correction factor 1 Mesh_alignment factor 0.158 Type of Unit: 2 Mesh alignment correction factor 1 Pinion_proportion_factor 0.071 Pinion_proportion_modifier 1.000 26

60-540 Gear Load, Stress and Life Analysis Gear Set Stress/Life - B88 (Program 60-540) Unit System: US Description Value Unit Comment Bearing span 5.000 in Closest Bearing to CL pinion 2.000 in Pinion offset 0.500 in PINION Number of teeth 22 Operating pitch diameter 2.4069 in Aspect ratio 0.831 Rim thickness factor 1.000 Whole depth in Rim thickness in Surface_finish, rms uin Same flank contacts per rev 1.00 Idler n AGMA J-Factor 0.479 No Group Pinion material # 19 Pinion material name CBG1 Allowable Contact Stress Number 180000 psi Cycles at Allowable Stress Numbers 10000000.00 cy Allowable Bending Stress Number 55000 psi Cycles at Allowable Stress Numbers 10000000.00 cy PINION CARB, FLAME OR INDUCTION HARDENED Top land normal tooth thickness 0.0659 in 27

UTS Integrated Gear Software Gear Set Stress/Life - B88 (Program 60-540) Unit System: US Description Value Unit Comment Min_recommended eff case depth 0.019 in Hardening process factor 6400000.00 psi Max recommended eff case depth 0.037 in PINION NITRIDED Minimum total case depth Core hardness Core hardness coefficient PINION FOR NON-AGMA MATERIALS Modulus of elasticity Poisson`s ratio Material and Heat Treatment? Brinell hardness SINGLE LOAD PITTING (PINION) in BHN psi Power 100.00 HP Speed 2100.00 rpm Torque 3001.19 lbf-in Tangential load 2493.82 lbf Pitch line velocity 1323.3 ft/min Dynamic factor 0.867 Face_load dist fact 1.229 Load_dist fact 1.229 K factor 725.3 psi Contact stress number 173894 psi 28

60-540 Gear Load, Stress and Life Analysis Gear Set Stress/Life - B88 (Program 60-540) Unit System: US Description Value Unit Comment Adjusted_contact stress- CT,CR psi Life_factor Contact_life: Cycles cy Contact_life: Time SINGLE LOAD BENDING (PINION) hr Power 100.00 HP Speed 2100.00 rpm Torque 3001.19 lbf-in Tangential load 2493.82 lbf Pitch line velocity 1323.3 ft/min Dynamic factor 0.867 Load_dist fact 1.229 Bending_stress number 50943 psi Adjusted_bending stress (KT,KR) psi Life_factor Bending_life: Cycles cy Bending_life: Time hr GEAR Number of teeth 55 Rim thickness factor 1.000 Whole depth Rim thickness Hardness ratio factor 1.000 in in 29

UTS Integrated Gear Software Gear Set Stress/Life - B88 (Program 60-540) Unit System: US Description Value Unit Comment Same flank contacts per rev 1.00 Idler n AGMA J-Factor 0.481 Gear material # 19 Gear material name CBG1 Allowable Contact Stress Number 180000 psi Cycles at Allowable Stress Numbers 10000000.00 cy Allowable Bending Stress Number 55000 psi Cycles at Allowable Stress Numbers 10000000.00 cy GEAR CARB, FLAME OR INDUCTION HARDENED Top land normal tooth thickness Min_recommended eff case depth Hardening process factor Max recommended eff case depth GEAR NITRIDED Minimum total case depth Core hardness Core hardness coefficient GEAR FOR NON-AGMA MATERIALS Modulus of elasticity Poisson`s ratio Material and Heat Treatment? 0.0765 in 0.019 in 6400000.00 psi 0.040 in in BHN psi 30

60-540 Gear Load, Stress and Life Analysis Gear Set Stress/Life - B88 (Program 60-540) Unit System: US Description Value Unit Comment Brinell hardness SINGLE LOAD PITTING (GEAR) Power 100.00 HP Speed 840.00 rpm Torque 7502.98 lbf-in Tangential load 2493.82 lbf Pitch line velocity 1323.3 ft/min Dynamic factor 0.867 Face_load dist fact 1.229 Load_dist fact 1.229 K factor 725.3 psi Contact stress number 173894 psi Adjusted_contact stress- CT,CR,CH psi Life_factor Contact_life: Cycles cy Contact_life: Time hr SINGLE LOAD BENDING (GEAR) Power Speed Torque Tangential load 100.00 HP 840.00 rpm 7502.98 lbf-in 2493.82 lbf Pitch line velocity Dynamic factor 0.867 1323.3 ft/min 31

UTS Integrated Gear Software Gear Set Stress/Life - B88 (Program 60-540) Unit System: US Description Value Unit Comment Load_dist fact 1.229 Bending stress number Adjusted_bending stress (KT,KR) Life_factor Bending_life: Cycles Bending_life: Time 50731 psi We see that in no case do we exceed the allowable stress values with the loads imposed. The model constructs plots of contact stress vs cycles for materials that have a stress/cycle curve. Figures 1-2 through 1-5 are the plots for our example. To view them, toggle to TK Solver and select them from the drop-down list on the Object Bar, or from the Plot Sheet. The plot names are: pinc for pinion allowable contact stress pinb for pinion allowable bending stress gearc for gear allowable contact stress gearb for gear allowable bending stress psi cy hr 32

60-540 Gear Load, Stress and Life Analysis Fig. 1-2 Fig. 1-3 33

UTS Integrated Gear Software Fig. 1-4 Fig. 1-5 Of course the pinion and gear plots are the same for this example, as we used the same material for the pinion and the gear. For this example we are pretty close to the allowable stresses with the loads and service factors specified. However, if we want to know how much power we can carry for pinion and gear pitting and bending to utilize all the allowable stress it is only 34

60-540 Gear Load, Stress and Life Analysis necessary to blank out the power data and enter the allowable contact stress numbers for the pinion and gear. The model will then backsolve and give us the maximum allowable power ratings. For such backsolving we can use either the Power User form or, as here, the TK Solver Variable Sheet. (See your TK Solver documentation if you need further information.) Sheet 1 is a portion of the Variable Sheet after solving. Sheet 1 *Single Load: *Pitting (Pinion): Pcp 107.15 HP *Power 2100 ncp rpm *Speed Tcp 3215.65 lbf-in *Torque Wtcp 2672.03 lbf *Tangential load vtcp 1323.3 ft/min *Pitch line velocity Cvp.867 *Dynamic factor (Def=Std Cv) Cmfp 1.229 *Face_load dist fact (Def=Std Cmf) Cmp 1.229 *Load_dist fact (Def=Std Cm) Kp 777.1 psi *K factor 180000 scp psi *Contact stress number scp` psi *Adjusted_contact stress- CT,CR CLp *Life_factor Ncp cy *Contact_life: Cycles Lcp hr *Contact_life: Time *Bending (Pinion): Pbp 107.96 HP *Power 2100 nbp rpm *Speed Tbp 3240.23 lbf-in *Torque Wtbp 2692.45 lbf *Tangential load vtbp 1323.3 ft/min *Pitch line velocity Kvp.867 *Dynamic factor (Def=Std Cv) Kmp 1.229 *Load_dist fact (Def=Std Km) 55000 stp psi *Bending_stress number stp` psi *Adjusted_bending stress (KT,KR) KLp *Life_factor Nbp cy *Bending_life: Cycles Lbp hr *Bending_life: Time * * * *Single Load: 35

UTS Integrated Gear Software *Pitting (Gear): Pcg 107.15 HP *Power 840 ncg rpm *Speed Tcg 8039.13 lbf-in *Torque Wtcg 2672.03 lbf *Tangential load vtcg 1323.3 ft/min *Pitch line velocity Cvg.867 *Dynamic factor (Def=Std Cv) Cmfg 1.229 *Face_load dist fact (Def=Std Cmf) Cmg 1.229 *Load_dist fact (Def=Std Cm) Kg 777.1 psi *K factor 180000 scg psi *Contact stress number scg` psi *Adjusted_contact stress- CT,CR,CH CLg *Life_factor Ncg cy *Contact_life: Cycles Lcg hr *Contact_life: Time *Bending (Gear): Pbg 108.42 HP *Power 840 nbg rpm *Speed Tbg 8134.39 lbf-in *Torque Wtbg 2703.69 lbf *Tangential load vtbg 1323.3 ft/min *Pitch line velocity Kvg.867 *Dynamic factor (Def=Std Cv) Kmg 1.229 *Load_dist fact (Def=Std Km) 55000 stg psi Bending stress number stg` psi *Adjusted_bending stress (KT,KR) KLg *Life_factor Nbg cy *Bending_life: Cycles Lbg hr *Bending_life: Time We did not obtain an estimated life for our gears because we used service factors and it is not known how much of the service factor is due to load and how much due to reliability. If we want a life prediction, we would need to enter application factors, which are set by overloads and reliability factors which are a function of the number of gearsets we expect to reach the predicted life. For this example let's assume that of the service factor of 1.5; it is all due to overload (of 50%) and not due to reliability. In this case we should set the application factors to 1.5 and the reliability factors to 1.0. Figure 1-7 is the input form and Report 1-2 shows the solved model after making these changes. 36

60-540 Gear Load, Stress and Life Analysis Fig. 1-7 37

UTS Integrated Gear Software Report 1-2 Gear Set Stress/Life - B88 (Program 60-540) Unit System: US Description Value Unit Comment Miner's Rule for Pinion and Gear Life from Duty Cycle? General commercial or critical application COMMON n g Transmission accuracy level number 10 Nominal normal diametral pitch 10.000000 1/in Nominal normal module 2.540000 mm ` Nominal transverse diametral pitch 9.205049 1/in Nominal transverse module 2.759355 mm ` Nominal helix angle 23.0000 deg Base helix angle 22.5410 deg Operating transverse pressure angle 22.5704 deg Face width, net 2.000 in Profile contact ratio 1.4328 AGMA I-Factor 0.1910000 LOAD FACTORS Application factor - pitting 1.500 Application factor - bending 1.500 Service factor - pitting 38

60-540 Gear Load, Stress and Life Analysis Gear Set Stress/Life - B88 (Program 60-540) Unit System: US Description Value Unit Comment Service factor - bending Reliability factor -pitting 1.000 Sets per failure 100.0000 Reliability factor -bending 1.000 Sets per failure 100.0000 Temperature fact - pitting 1.000 Temperature fact - bending 1.000 Transverse load distrib fact 1.000 MATERIAL FACTORS Elastic coefficient 2290 Surface condition factor 1.000 Size factor - pitting 1.000 Size factor - bending 1.000 ANALYTICAL METHOD Total lead mismatch Tooth_stiffness constant EMPIRICAL METHOD in psi Lead correction factor 1 Mesh_alignment factor 0.158 Type of Unit: 2 Mesh alignment correction factor 1 Pinion_proportion_factor 0.071 Pinion_proportion_modifier 1.000 39

UTS Integrated Gear Software Gear Set Stress/Life - B88 (Program 60-540) Unit System: US Description Value Unit Comment Bearing span 5.000 in Closest Bearing to CL pinion 2.000 in Pinion offset 0.500 in PINION Number of teeth 22 Operating pitch diameter 2.4069 in Aspect ratio 0.831 Rim thickness factor 1.000 Whole depth in Rim thickness in Surface_finish, rms uin Same flank contacts per rev 1.00 Idler n AGMA J-Factor 0.479 Pinion material # 19 Pinion material name CBG1 Allowable Contact Stress Number 180000 psi Cycles at Allowable Stress Numbers 10000000.00 cy Allowable Bending Stress Number 55000 psi Cycles at Allowable Stress Numbers 10000000.00 cy PINION CARB, FLAME OR INDUCTION HARDENED Top land normal tooth thickness Min_recommended eff case depth 0.0659 in 0.019 in 40

60-540 Gear Load, Stress and Life Analysis Gear Set Stress/Life - B88 (Program 60-540) Unit System: US Description Value Unit Comment Hardening process factor 6400000.00 psi Max recommended eff case depth 0.037 in PINION NITRIDED Minimum total case depth Core hardness Core hardness coefficient PINION FOR NON-AGMA MATERIALS Modulus of elasticity Poisson`s ratio Material and Heat Treatment? Brinell hardness SINGLE LOAD PITTING (PINION) in BHN psi Power 100.00 HP Speed 2100.00 rpm Torque 3001.19 lbf-in Tangential load 2493.82 lbf Pitch line velocity 1323.3 ft/min Dynamic factor 0.867 Face_load dist fact 1.229 Load_dist fact 1.229 K factor 725.3 psi Contact stress number 173929 psi Adjusted_contact stress- CT,CR 173929 psi 41

UTS Integrated Gear Software Gear Set Stress/Life - B88 (Program 60-540) Unit System: US Description Value Unit Comment Life_factor 0.966 Contact_life: Cycles 44500000.00 cy Contact_life: Time SINGLE LOAD BENDING (PINION) 352.8 hr Power 100.00 HP Speed 2100.00 rpm Torque 3001.19 lbf-in Tangential load 2493.82 lbf Pitch line velocity 1323.3 ft/min Dynamic factor 0.867 Load_dist fact 1.229 Bending_stress number 50963 psi Adjusted_bending stress (KT,KR) 50963 psi Life_factor 0.927 Bending_life: Cycles 726000000.00 cy Bending_life: Time 5763.7 hr GEAR Number of teeth 55 Rim thickness factor 1.000 Whole depth Rim thickness Hardness ratio factor 1.000 Same flank contacts per rev 1.00 in in 42

60-540 Gear Load, Stress and Life Analysis Gear Set Stress/Life - B88 (Program 60-540) Unit System: US Description Value Unit Comment Idler n AGMA J-Factor 0.481 Gear material # 19 Gear material name CBG1 Allowable Contact Stress Number 180000 psi Cycles at Allowable Stress Numbers 10000000.00 cy Allowable Bending Stress Number 55000 psi Cycles at Allowable Stress Numbers 10000000.00 cy GEAR CARB, FLAME OR INDUCTION HARDENED Top land normal tooth thickness Min_recommended eff case depth Hardening process factor Max recommended eff case depth GEAR NITRIDED Minimum total case depth Core hardness Core hardness coefficient GEAR FOR NON-AGMA MATERIALS Modulus of elasticity Poisson`s ratio Material and Heat Treatment? Brinell hardness 0.0765 in 0.019 in 6400000.00 psi 0.040 in in BHN psi 43

UTS Integrated Gear Software Gear Set Stress/Life - B88 (Program 60-540) Unit System: US Description Value Unit Comment SINGLE LOAD PITTING (GEAR) Power 100.00 HP Speed 840.00 rpm Torque 7502.98 lbf-in Tangential load 2493.82 lbf Pitch line velocity 1323.3 ft/min Dynamic factor 0.867 Face_load dist fact 1.229 Load_dist fact 1.229 K factor 725.3 psi Contact stress number 173929 psi Adjusted_contact stress- CT,CR,CH 173929 psi Life_factor 0.966 Contact_life: Cycles 44500000.00 cy Contact_life: Time 882.1 hr SINGLE LOAD BENDING (GEAR) Power Speed Torque Tangential load 100.00 HP 840.00 rpm 7502.98 lbf-in 2493.82 lbf Pitch line velocity Dynamic factor 0.867 1323.3 ft/min 44

60-540 Gear Load, Stress and Life Analysis Gear Set Stress/Life - B88 (Program 60-540) Unit System: US Description Value Unit Comment Load_dist fact 1.229 Bending stress number Adjusted_bending stress (KT,KR) 50751 psi 50751 psi Life_factor 0.923 Bending_life: Cycles Bending_life: Time 918000000.00 cy 18212.5 hr The predicted life for pinion pitting is about 353 hours and for gear pitting about 882 hours. The predicted bending lives are much longer, about 5,764 hours and 18,213 hours, respectively. This is an indication that the diametral pitch is too coarse and the use of a finer pitch should be considered. Before too much reliance is put upon the predicted life using the Empirical Method, it is useful to check the same gears using the Analytical Method. The Analytical Method requires more information about the accuracy of both the gears and the housing in which they will run, but it will give a more accurate picture of the operation of the system. 45

UTS Integrated Gear Software Example 2 Example 2 is an 8 normal diametral pitch helical gearset with a 21 tooth pinion driving a 61 tooth gear for use in an enclosed gear unit for critical service. A repeating duty cycle of 355 hours has been determined. We wish to obtain an estimate of the life of the gears when subjected to the duty cycle. The Analytical Method will be used. The estimated life is to be made at a reliability factor of one which sets the reliability at less than one failure per 100 units. (See the standard for an explanation of the meaning of failure rate.) Both gears are made of carburized and hardened steel. The pinion will conform to AGMA Grade 3 specifications and the gear to AGMA Grade 2. (Tables 14-1 and 14-7 in the standard). These materials are #21 and #20, respectively, in the material table in the model. When specifying AGMA materials the standard should be consulted to ensure that all requirements for the materials will be met. No lead crowning or correction is to be used. Neither gear has a thin rim under the tooth roots. The gears are precision ground to AGMA Quality 12 and the total lead mismatch is 0.0011 inch. (UTS Model 60-102 may be of help in assessing the total lead mismatch or consult the methods in the standard.) The gear data will be imported to the model from a data access file from Program 500 with file name 540Temp1. (This data could, of course, be typed in from the keyboard.) Report 2-1 is an output data sheet from Program 500 for this gearset. It is included for reference as the data is already in the file 540Temp1. Report 2-1 Job Name : 540Temp1 Normal Diam Pitch 8.0000 Opr Trans Diam Pitch 7.2913 Normal Pressure Angle 20.0000 Opr Trans Press 22.5192 Angle Helix Angle 23.5000 Opr Helix Angle 23.6296 Trans Diam Pitch 7.3365 Line of Action 0.5625 Trans Pressure Angle 21.6475 % Approach Action 46.49 46

60-540 Gear Load, Stress and Life Analysis Base Helix Angle 22.0059 % Recess Action 53.51 Opr Center Distance 5.6231 Profile C.R. 1.4132 Helical C.R. 2.0308 Face Width 2.0000 Total C.R. 3.4440 Basic Trans Backlash 0.0005 Contact Lines: 1.0108 Max/Min Total Opr Trans BL 0.0059 DRIVER (Deg Roll) DRIVEN (Deg Roll) Number of Teeth 21 61 Outside Diameter 3.1600 (36.72) 8.5800 (27.63) Cut Transverse Backlash 0.0027 0.0027 Delta Addendum 0.0257 0.0089 Total Normal Finish Stock 0.0124 0.0135 HOB FORM DATA NON-TOPPING NON-TOPPING DriverOff Lead: Hob Press Ang 14.5000 20.0000 Hob Tip to Ref Line 0.1750 0.1750 Hob Tooth Thickness at Ref 0.1692 0.1828 Both: Full Rad-Hob Tip Radius 0.0597 0.0506 Hob Protuberance 0.0067 0.0073 Hob SAP from Ref Line 0.0577 0.1163 Hob Space Width at Hob SAP 0.1821 0.1253 Normal Tooth Thickness at OD 0.0844 0.0979 Normal Tooth Thickness-(Hobbed) 0.2250 0.2139 Normal Tooth Thickness-(Ground) 0.2126 0.2004 47

UTS Integrated Gear Software DRIVER (Deg Roll) DRIVEN (Deg Roll) Dia @ Mid-point of Line of Action 2.8955 (24.61) 8.3510 (23.46) Pitch Diameter- (Ref) 2.8624 (22.74) 8.3146 (22.74) Operating Pitch Diameter 2.8801 (23.75) 8.3661 (23.75) Base Diameter 2.6605 7.7282 Dia- (Start of Active Profile) 2.7230 (12.49) 8.1545 (19.29) Form Diameter 2.7167 (11.84) 8.1482 (19.14) Root Diameter 2.5567 7.9755 Root Clearance 0.0548 0.0553 Helical Lead 20.6814 60.0745 Max Undercut 0.0067 0.0076 Diameter at Max Undercut 2.6777 ( 6.53) 8.0712 (17.26) Finished Grind Diameter 2.6777 ( 6.52) 8.0712 (17.26) Minimum Fillet Radius 0.0608 0.0534 Helical Factor- C(h) 1.387 1.387 Y Factor 0.499 0.557 Almen-Straub Strength Factor 1.85E+2 1.66E+2 Almen-Straub Pitting Factor 3.51E+3 Load Sharing Ratio- m(n) 0.659 0.659 AGMA Stress Corr Fact- K(f) 1.442 1.542 J-Factor 0.525 0.548 I Factor 0.209 Hob Tool Number Steel Gears- Finish Ground Case Carburized 48

60-540 Gear Load, Stress and Life Analysis Open a new analysis in 60-540 and bring in the data from Program 500 analysis 540Temp1 (See the Integrated Gear Software Introduction for more information on these procedures.). Figure 2-1 is the 60-540 data entry form with the Program 500 data. Fig. 2-1 At this point we enter the duty cycle data for the pinion. Click the tab labeled Miner s Table and the radio button Miner1. Enter the first column of data as shown in Figure 2-2. 49

UTS Integrated Gear Software Fig. 2-2 This completes the data input for the first case. There will be six cases in all. Click Add to add a column for each case, and an extra column to complete the data input. The completed duty cycle table for the pinion, Miner1 is shown in Figure 2-3. The gear duty cycle table receives data from the pinion duty cycle input. Check the gear table by clicking the radio button for Miner2. 50

60-540 Gear Load, Stress and Life Analysis Fig. 2-3 Now we are ready to solve the model to complete the duty cycle tables. Click the Solve button in the Miner s Table tab. The solved Miner1 and Miner2 tables are shown in Figures 2-4 and 2-5. 51

UTS Integrated Gear Software Fig. 2-4 52

60-540 Gear Load, Stress and Life Analysis Fig. 2-5 Now we are ready to complete the data entry form and solve the model. The completed data entry form is shown in Figure 2-6. 53

UTS Integrated Gear Software Fig. 2-6 After you complete data entry, a box will appear indicating that the model is solving. At this point you must toggle to the TK Solver model to respond to four prompts, beginning with this: 54

60-540 Gear Load, Stress and Life Analysis If we respond with Yes to this prompt the model will calculate new Cm and Km factors (for straight teeth) and place them in the table regardless of wether or not there are already factors in the table. If you have typed Cm and Km factors into the table (for example to account for crowned teeth) then respond with No so that the values will not be re-calculated and replaced. Next, this prompt appears: If we respond with Yes to this prompt the model will calculate new Cv and Kv factors and place them in the table regardless of wether or not there are already factors in the table. If you have typed Cv and Kv factors into the table that you wish to use instead of the standard factors then respond with No so that the values will not be replaced. Finally, this prompt will appear: If we respond with Yes to this prompt the model will use the CT and KT factors from the Variable Sheet and place them in the table regardless of whether or not there are already temperature factors in the table. If you have typed CT and KT factors into the table that you wish to use instead of the standard factors (for example to account for different temperature effects on pinion and gear) then respond with No so that your values will not be replaced. After solving, the Variable Sheet should look like Report 2-2. The plots show the percentage of the life of the gearset that is used up by each load and speed level in the duty cycle, in the form of the % life used vs the load condition number. For pitting conditions #1 and #2 use most of the life and for bending conditions #1 and #5 are the most significant This data is also available in the TK Solver model. The names are %LifePinC for pinion pitting, %LifePinB for pinion bending, %LifeGearC for gear pitting, and %LifeGearB for gear bending. 55

UTS Integrated Gear Software Report 2-2 Gear Set Stress/Life - B88 (Program 60-540) Unit System: US Description Value Unit Comment Miner's Rule for Pinion and Gear Life from Duty Cycle? General commercial or critical application COMMON y c Transmission accuracy level number 12 Nominal normal diametral pitch 8.000000 1/in Nominal normal module 3.175000 mm ` Nominal transverse diametral pitch 7.336481 1/in Nominal transverse module 3.462151 mm ` Nominal helix angle 23.5000 deg Base helix angle 0.3841 deg Operating transverse pressure angle 22.5192 deg Face width, net 2.000 in Profile contact ratio 1.4132 AGMA I-Factor 0.2092035 LOAD FACTORS Application factor - pitting 1.000 Application factor - bending 1.000 Service factor - pitting 56

60-540 Gear Load, Stress and Life Analysis Service factor - bending Gear Set Stress/Life - B88 (Program 60-540) Unit System: US Description Value Unit Comment Reliability factor -pitting 1.000 Sets per failure 100.0000 Reliability factor -bending 1.000 Sets per failure 100.0000 Temperature fact - pitting 1.000 Temperature fact - bending 1.000 Transverse load distrib fact 1.000 MATERIAL FACTORS Elastic coefficient 2290 Surface condition factor 1.000 Size factor - pitting 1.000 Size factor - bending 1.000 ANALYTICAL METHOD Total lead mismatch Tooth_stiffness constant PINION 0.00110 in 1880000.00 psi Number of teeth 21 Operating pitch diameter 2.8801 in Aspect ratio 0.694 Rim thickness factor 1.000 Whole depth in 57

UTS Integrated Gear Software Rim thickness Gear Set Stress/Life - B88 (Program 60-540) in Unit System: US Description Value Unit Comment Surface_finish, rms Same flank contacts per rev 1.00 Idler n AGMA J-Factor 0.525 uin Pinion material # 21 Pinion material name CBG3 Allowable Contact Stress Number 275000 psi Cycles at Allowable Stress Numbers 10000000.00 cy Allowable Bending Stress Number 75000 psi Cycles at Allowable Stress Numbers 10000000.00 cy PINION CARB, FLAME OR INDUCTION HARDENED Top land normal tooth thickness GEAR 0.0844 in Number of teeth 61 Rim thickness factor 1.000 Whole depth Rim thickness Hardness ratio factor 1.000 Same flank contacts per rev 1.00 Idler n AGMA J-Factor 0.548 in in 58

60-540 Gear Load, Stress and Life Analysis Gear material # 20 Gear Set Stress/Life - B88 (Program 60-540) Unit System: US Description Value Unit Comment Gear material name CBG2 Allowable Contact Stress Number 225000 psi Cycles at Allowable Stress Numbers 10000000.00 cy Allowable Bending Stress Number 65000 psi Cycles at Allowable Stress Numbers 10000000.00 cy GEAR CARB, FLAME OR INDUCTION HARDENED Top land normal tooth thickness 0.0979 in 59

UTS Integrated Gear Software Pinion Pitting - % Life Used By Duty Cycle Conditions 60

60-540 Gear Load, Stress and Life Analysis Pinion Bending - % Life Used By Duty Cycle Conditions 61

UTS Integrated Gear Software Gear Pitting - % Life Used By Duty Cycle Conditions 62

60-540 Gear Load, Stress and Life Analysis Gear Bending - % Life Used By Duty Cycle Conditions 63

UTS Integrated Gear Software The gear and not the pinion would control the useful life of this gearset, as the gear pitting life is about 3300 hours and the bending life is about 3900 hours. The life of the pinion is much higher. The reason for this is undoubtedly the premium material being used for the pinion. 64

60-540 Gear Load, Stress and Life Analysis Example 3 If you have not gone through the first two examples please do so before proceeding with Example 3. In this example it is assumed that you know how to enter the data in the data input forms and the duty cycle tables. Example 3 is a 5 normal diametral pitch spur gearset with a 17 tooth pinion driving a 34 tooth gear for use in an enclosed gear unit for general service. A repeating duty cycle of 17 hours has been established. We will obtain an estimate of the life of the gears when subjected to the duty cycle. The estimated life is to be made at a reliability factor of one which sets the reliability at less than one failure per 100 units. (See the standard for an explanation of the meaning of failure rate.) Both gears are made of carburized and hardened steel conforming to AGMA Grade 1 specifications. (Tables 14-1 & 14-7 in the standard). This material is #19 the material table in the model. No lead crowning or correction is used. Neither gear has a thin rim under the tooth roots. The gears are hobbed and after heat treatment are to conform to AGMA Quality 8. The the total lead mismatch is 0.0018 inch. (UTS Model 60-102 may be of help in assessing the total lead mismatch or consult the methods in the standard.) The gear data will be imported to the model from a data access file from Program 500 with file name 540TEMP2. (This data could, of course, be typed in from the keyboard.) The duty cycle data could be typed in but it has been stored on a file also named 540TEMP2, so we will import this data also. Report 3-1 is a Program 500 report for this gearset. It is included for reference; the data is already in the file 540TEMP2. 65

UTS Integrated Gear Software Report 3-1 Job Name : 540 Example 3 Normal Diam Pitch 5.0000 Opr Diam Pitch 4.9103 Normal Pressure Angle 25.0000 Opr Pressure Angle 27.1214 Helix Angle 0.0000 Trans Diam Pitch 5.0000 Line of Action 0.7683 Trans Pressure Angle 25.0000 % Approach Action 44.58 % Recess Action 55.42 Opr Center Distance 5.1932 Profile C.R. 1.3492 Face Width 2.5000 Basic Backlash 0.0036 Total Operating BL 0.0077 DRIVER (Deg Roll) DRIVEN (Deg Roll) Number of Teeth 17 34 Outside Diameter 3.9242 (45.18) 7.2622 (35.72) Cut Transverse Backlash 0.0021 0.0020 Delta Addendum 0.0621 0.0311 Total Normal Finish Stock 0.0074 0.0074 HOB FORM DATA NON-TOPPING NON-TOPPING Hob Pressure Angle 25.0000 25.0000 Hob Tip to Ref Line 0.2700 0.2700 Hob Tooth Thickness at Ref 0.3068 0.3068 Both: Full Rad-Hob Tip Radius 0.0504 0.0504 Hob Protuberance 0.0042 0.0042 Hob SAP from Ref Line 0.1499 0.1755 66

60-540 Gear Load, Stress and Life Analysis DRIVER (Deg Roll) DRIVEN (Deg Roll) Hob Space Width at Hob SAP 0.1817 0.1579 Normal Tooth Thickness at OD 0.0706 0.1032 Normal Tooth Thickness-(Hobbed) 0.3774 0.3486 Normal Tooth Thickness-(Ground) 0.3700 0.3412 Dia @ Mid-point of Line of Action 3.5009 (30.89) 6.8867 (28.57) Pitch Diameter- (Ref) 3.4000 (26.72) 6.8000 (26.72) Operating Pitch Diameter 3.4621 (29.35) 6.9243 (29.35) Base Diameter 3.0814 6.1629 Dia- (Start of Active Profile) 3.2083 (16.61) 6.5799 (21.43) Form Diameter 3.1983 (15.93) 6.5699 (21.16) Root Diameter 2.9799 6.3181 Root Clearance 0.0722 0.0721 Max Undercut 0.0043 0.0044 Diameter at Max Undercut 3.1357 (10.79) 6.4407 (17.40) Finished Grind Diameter 3.1357 (10.80) 6.4407 (17.40) Roll- radians- (1 tooth load) 0.659 (37.79) 0.559 (32.02) Minimum Fillet Radius 0.0641 0.0605 Helical Factor- C(h) 1.000 1.000 Y Factor 0.750 0.771 Load Sharing Ratio- m(n) 1.000 1.000 AGMA Stress Corr Fact- K(f) 1.736 1.762 J-Factor 0.432 0.437 I Factor 0.121 Steel Gears- Finish Ground Soft or Quenched and Tempered Figures 3-1a through 3-1c show the Power User data input form with the required inputs, including the Program 500 data. 67

UTS Integrated Gear Software Fig. 3-1a 68

60-540 Gear Load, Stress and Life Analysis Fig. 3-1b 69

UTS Integrated Gear Software Fig. 3-1c To load duty cycle data, click the Miner s Table tab and click the Load Duty Cycle Data button. A standard Windows load file form will appear. Simply double-click the icon of the desired file. (Use the Save Duty Cycle Data button to save duty cycle data that you want to reuse.) Figure 3-2 shows the data for Miner s table 1 from 540TEMP2. 70

60-540 Gear Load, Stress and Life Analysis Fig. 3-2 Report 3-2 shows the data for the solved model. The complete duty cycle data is shown in Figures 3-3a and 3-3b for Miner s table 1 and in Figures 3-4a and 3-4b for Miner s table 2. 71

UTS Integrated Gear Software Report 3-2 Gear Set Stress/Life - B88 (Program 60-540) Unit System: US Description Value Unit Comment Miner's Rule for Pinion and Gear Life from Duty Cycle? General commercial or critical application COMMON Transmission accuracy level number 8 Nominal normal diametral pitch Nominal normal module Nominal transverse diametral pitch Nominal transverse module Nominal helix angle Base helix angle Operating transverse pressure angle Face width, net 5.000000 1/in 5.080000 mm ` 5.0000 1/in 5.0800 mm ` 0.000000 deg 0.0000 deg 27.2673 deg 2.500 in Profile contact ratio 1.3276 AGMA I-Factor 0.1212080 LOAD FACTORS Application factor - pitting 1.000 Application factor - bending 1.000 Service factor - pitting 72

60-540 Gear Load, Stress and Life Analysis Gear Set Stress/Life - B88 (Program 60-540) Unit System: US Description Value Unit Comment Service factor - bending Reliability factor -pitting 1.000 Sets per failure 100.0000 Reliability factor -bending 1.000 Sets per failure 100.0000 Temperature fact - pitting 1.000 Temperature fact - bending 1.000 Transverse load distrib fact 1.000 MATERIAL FACTORS Elastic coefficient 2290 Surface condition factor 1.000 Size factor - pitting 1.000 Size factor - bending 1.000 ANALYTICAL METHOD Total lead mismatch Tooth_stiffness constant PINION 0.00180 in 2000000.00 psi Number of teeth 17 Operating pitch diameter 3.4667 in Aspect ratio 0.721 Rim thickness factor 1.000 Whole depth in 73

UTS Integrated Gear Software Gear Set Stress/Life - B88 (Program 60-540) Unit System: US Description Value Unit Comment Rim thickness in Surface_finish, rms uin Same flank contacts per rev 1.00 Idler AGMA J-Factor 0.356 Pinion material # 19 (CBG1) Pinion material name Allowable Contact Stress Number 180000 psi Cycles at Allowable Stress Numbers 10000000 cy Allowable Bending Stress Number 55000 psi Cycles at Allowable Stress Numbers 10000000 cy PINION CARB, FLAME OR INDUCTION HARDENED Top land normal tooth thickness GEAR 0.0711 in Number of teeth 34 Rim thickness factor 1.000 Whole depth Rim thickness Hardness ratio factor 1.000 Same flank contacts per rev 1.00 Idler AGMA J-Factor 0.359 Gear material # 19 (CBG1) in in 74

60-540 Gear Load, Stress and Life Analysis Gear Set Stress/Life - B88 (Program 60-540) Unit System: US Description Value Unit Comment Gear material name Allowable Contact Stress Number 180000 psi Cycles at Allowable Stress Numbers 10000000 cy Allowable Bending Stress Number 55000 psi Cycles at Allowable Stress Numbers 10000000 cy GEAR CARB, FLAME OR INDUCTION HARDENED Top land normal tooth thickness 0.1068 in 75

UTS Integrated Gear Software Fig. 3-3a Fig. 3-3b 76

60-540 Gear Load, Stress and Life Analysis Fig. 3-4a Fig. 3-4b After solving you will get a message advising that load distribution factors are greater than two for all conditions, so contact does not extend across the face for any condition. 77

UTS Integrated Gear Software The pitting life for the pinion is about 460 hours and for the gear about 920 hours. The bending life is much larger for both gears, indicating that a finer pitch (more teeth on this center distance) would increase the pitting life and reduce the bending life. We will assume that the life balance between pitting and bending is satisfactory and see if an improvement in life is possible by adding crown to the pinion. UTS Model 60-5406 was used to obtain the following Cm factors for the gearset with 0.001 inch crown on the pinion (Condition #4 was used to find the optimum crown): Straight teeth Crowned pinion Condition #1 Cm = 3.406 Cm = 2.628 #2 Cm = 3.447 Cm = 2.634 #3 Cm = 4.005 Cm = 2.832 #4 Cm = 4.128 Cm = 2.868 The duty cycle tables, after entering the new Cm values and solving, are shown in Figures 3-5a and 3-5b (Miner s table 1) and in Figures 3-6a and 3-6b (Miner s table 2). Note: When solving this time be sure to tell the model NOT to calculate new Cm and Km factors for the table. 78

60-540 Gear Load, Stress and Life Analysis Fig. 3-5a Fig. 3-5b 79

UTS Integrated Gear Software Fig. 3-6a Fig. 3-6b 80

60-540 Gear Load, Stress and Life Analysis The predicted life for pinion pitting went from about 460 hours to about 290,000 hours due to the reduction in contact stress. The gear life went from about 920 hours to about 580,000 hours. The stress reduction from the addition of crown can be considerable in cases where the total lead mismatch is large. This gearset was designed as an example only and is not necessarily the best answer to a design problem. All circumstances must be considered in the design of a best gearset and this was not done in the example. 81

UTS Integrated Gear Software Example 4 If you have not gone through the first two examples, please do so before proceeding with Example 4. In this example we will enter data into the material table that is not for an AGMA material. The new material will be an acetal plastic for which we have gear test data when a plastic gear is meshed with a steel pinion. Open a new analysis in 60-540 and select the Material Update tab of the data input form. Click the Add button. A blank form for data entry appears. We will select a material name of Ace#1wS. The material designation will be Acetal #1 with Steel. We will set the AGMA values for general commercial applications. At this point we must choose between stress and cycle data and allowable stress data. Here the choice is stress and cycle data, to enable us to calculate life data for this material. If we choose allowable stress data, life calculations could not be made. It is necessary to enter stress and life cycle data for at least two points in order to establish a curve. The first point for which we must enter data is the low cycle end of the compressive stress/cycle curve. These values are 10,000 cycles at 6,000 psi. The second point is the high end, 10 million cycles at 3,000 psi. The bending stress points are 10,000 cycles at 6,000 psi and 6 million cycles at 6,000 psi. Enter the compressive stress and bending stress in US units and cycles in the appropriate columns. Click the asterisk button (*) to the left of the table row to create a new row. When you are finished entering the data, the form should look like Figure 4-1a. After the application updates the form, the alternate units column in this case, metric will be calculated automatically, a message appears saying the new material has been added, and the form looks like Figure 4-1b. 82

60-540 Gear Load, Stress and Life Analysis Fig. 4-1a 83

UTS Integrated Gear Software Fig. 4-1b 84

60-540 Gear Load, Stress and Life Analysis Now that we have our new material in the table, we will calculate the life of a.5 module spur gearset with a steel pinion of 27 teeth driving an acetal plastic gear of 65 teeth. This time we will work in the metric system. Go to the Input Parameters tab of the data input form and click the radio button to select metric units. Figure 4-2 is the completed input form and Report 4-1 a report of the model after solving for an AGMA Grade 1 steel pinion driving a gear made of our new material. The gear is an idler gear subjected to reverse bending. The ambient temperature is 38 deg C and the gearset will have initial lubrication only. (Plastic gears are very sensitive to lubrication conditions.) The application factor, Ka, was set to 1.25 to allow a little extra safety in beam fatigue breakage. 85

UTS Integrated Gear Software Fig. 4-2 86