Program 60-107 Internal Gear Set Profile Shift Coefficients With Zero Backlash Introduction The purpose of this model is to provide data for a gear set when the tooth thickness and/or the center distance is not standard. The model follows the European convention of gears in mesh without backlash to establish the sum of the profile shift coefficients. (The profile shift coefficient is usually designated as the X factor on European gear drawings.) To provide backlash in the gear set the tooth thickness of one or both gears must be reduced from the values given by the model (or the center distance must be reduced). The AGMA load sharing ratio and the AGMA I-factor are calculated for use in contact stress analysis using AGMA Standard 2001. Data for AGMA Load Sharing Ratio and I Factor Extracted from AGMA Standard 2001-B88, Fundamental Rating Factors and Calculation Methods for Involute Spur and Helical Gear Teeth with the permission of the publisher, American Gear Manufacturers Association, 1500 King Street, Suite 201, Alexandria, VA 22314 The model will check for the following conditions and provide a warning if the geometry is outside the limits: 1. The roll angle at the pinion start of active profile is less than 5 degrees. 2. The roll angle at the gear inside diameter is less than 5 degrees. 3. The width of the tooth top land is less than 0.25/NDP (.25*Normal Module) and the normal diametral pitch is less than 20 (Normal Module is greater than 1.27). 4. The width of the tooth top land is less than 0.275/NDP (.275*Normal Module) and the normal diametral pitch is 20 or greater (Normal Module is 1.27 or less). 5. The teeth are pointed. 6. The normal operating pressure angle is less than 10 degrees or greater than 30 degrees. 7. The total contact ratio is less than one. NOTE: The model does not check for undercut on the pinion and the profile contact ratio is based on contact down to the pinion start of active profile. Undercut must be checked by other means. (See UTS Model 60-450) 8. The specific sliding ratio is greater than three. NOTE: If the specific sliding ratio is greater than three the geometry of the gear set should be checked. However, for gear sets running at higher speeds a specific sliding ratio of three may be too high and noisy operation and scoring may result. 1
UTS Integrated Gear Software 9. The gear inside diameter is less than the base diameter. 10. Trochoidal interference between the gear inside diameter and the pinion tooth tips. Examples Example 1 Example 1 is a spur gear set with a 14 tooth pinion driving a 50 tooth gear. The normal diametral pitch is 10 and the nominal pressure angle is 20 degrees. If this set is made to standard proportions we would use the input data shown in Figures 1-1A and 1-1B, and would have the output data in Report 1-1. Note that the profile shift coefficient for both gears is zero. Fig. 1-1A Fig. 1-1B 2
60-107 Internal Gear Set Profile Shift Coefficients With Zero Backlash Report 1-1 Unit System: US PINION, number of teeth 14 GEAR, number of teeth 50 Gear ratio 3.5714 NORMAL PLANE Normal pitch ( US/British System) Normal module ( SI System) Normal pressure angle Normal base pitch TRANSVERSE PLANE Transverse pitch Transverse pressure angle Transverse module Transverse base pitch COMMON Helix angle Base helix angle Axial pitch Operating center distance Standard center distance 10.000000 1/in 2.540000 mm ` 20.000000 deg 0.295 in 10.0000 1/in 20.0000 deg 2.5400 mm ` 0.295 in 0.000000 deg 0.0000 deg in 1.800 in 1.8000 in 3
UTS Integrated Gear Software Unit System: US Change in Opr CD from "Std" CD 0.0000 in CD modification coefficient 0.00000 Difference of shift coefficients 0.00000 OD, ID coef to maintain clearance 0.00000 Face width TOOTH THICKNESS (Zero Backlash) Pinion 1.000 in Profile shift coefficient 0.00000 TOOTH THICKNESS (Zero Backlash) Pinion At Ref PD TOOTH THICKNESS (Zero Backlash) Pinion At OD TOOTH THICKNESS (Zero Backlash) Gear 0.065 in 0.065 in Profile shift coefficient 0.00000 TOOTH THICKNESS (Zero Backlash) Gear At Ref PD Normal space width Transverse space width TOOTH THICKNESS (Zero Backlash) Gear At ID 0.108 in 4
60-107 Internal Gear Set Profile Shift Coefficients With Zero Backlash Unit System: US DIAMETERS Pinion 0.108 in Outside diameter (Optional Input) Roll angle at OD Increased OD to maintain clearance Reference pitch diameter Pointed tooth diameter Base diameter DIAMETERS Gear Inside diameter (Optional Input) Roll angle at ID Decreased ID to maintain clearance Minimum ID (Involute Interference) Reference pitch diameter Pointed tooth diameter Base diameter OPERATING DATA Working depth Normal pressure angle Transverse pressure angle Helix angle Circular pitch Roll angle at pitch point 1.600 in 39.661 deg 1.6000 in 1.400 in 1.689 in 1.316 in 4.857 in 15.0173 deg 4.857 in 4.857 in 5.000 in Below BD in 4.699 in 0.171 in 20.0000 deg 20.0000 deg 0.0000 deg 0.314 in 20.8540 deg 5
UTS Integrated Gear Software Unit System: US OPERATING DATA Pinion Pitch diameter Transverse Tooth Thickness Start of active profile Roll angle at SAP at SAP at SAP OPERATING DATA Gear Pitch diameter Transverse Space Width End of active profile Roll angle at EAP at EAP at EAP Trochoidal clearance: Pin OD/Gear ID Contact Length 1.400 in 1.316 in 0.0086 deg 0.167 in 0.167 in 5.000 in 5.164 in 26.1198 deg 0.231 in 0.231 in 0.035 in Length of contact, transverse plane 0.455 in Approach action (Pinion Driving) 52.57 % Recess action (Pinion Driving) 47.43 % Contact Ratios Profile 1.5420 Helical 0.0000 Total 1.5420 6
60-107 Internal Gear Set Profile Shift Coefficients With Zero Backlash Unit System: US Lines of Contact Across Teeth Max total length 2.000 in Min total length 1.000 in Ratio: (Max length) / (Min length) 2.0000 Specific Sliding Ratios Pinion start of active profile 1747.316 Pinion outside diameter 0.341 Gear start of active profile 0.518 Gear inside diameter 0.999 AGMA Load sharing ratio, mn 1.0000 AGMA I-factor for durability 0.135 Iteration trigger variable for solving CD 1.0 from tooth thickness In solving the model, you receive these warnings: 1. The gear inside diameter has been set (increased) to avoid involute interference 2. The roll angle at pinion the SAP is less than 5 degrees. 3. The specific sliding ratio at the pinion SAP is greater than 3. With 14 teeth in the pinion and standard tooth thickness the pinion will be undercut. The profile contact ratio will not be 1.542 but will be considerably less. Figure 1-2 is the plot of the teeth in mesh, using UTS Gear Program 60-450. The undercut in the driver is obvious. The plot was made using a generating rack with an addendum equal to 1.25, a tooth thickness equal to 1.571 and a tip radius of 0.38 for 1 normal diametral pitch or 1 normal module. Using this rack gives us an actual profile contact ratio of 1.468 because of the undercut on the pinion. The AGMA I- factor is 0.135. 7
UTS Integrated Gear Software Fig. 1-2 If it is not necessary to maintain a standard center distance we may wish to cut the pinion on a 15 tooth blank to eliminate the undercut and decrease the center distance. Figures 1-3A and 1-3B show the additional data inputs; Report 1-2 is the solution for these conditions. Fig. 1-3A 8
60-107 Internal Gear Set Profile Shift Coefficients With Zero Backlash Fig. 1-3B Report 1-2 Unit System: US PINION, number of teeth 14 GEAR, number of teeth 50 Gear ratio 3.5714 NORMAL PLANE Normal pitch ( US/British System) Normal module ( SI System) Normal pressure angle Normal base pitch TRANSVERSE PLANE Transverse pitch Transverse pressure angle Transverse module Transverse base pitch 10.000000 1/in 2.540000 mm ` 20.000000 deg 0.295 in 10.0000 1/in 20.0000 deg 2.5400 mm ` 0.295 in 9
UTS Integrated Gear Software Unit System: US COMMON Helix angle Base helix angle Axial pitch Operating center distance Standard center distance Change in Opr CD from "Std" CD 0.000000 deg 0.0000 deg in 1.742 in 1.8000 in -0.0581 in CD modification coefficient -0.58093 Difference of shift coefficients -0.50000 OD, ID coef to maintain clearance -0.08093 Face width TOOTH THICKNESS (Zero Backlash) Pinion 1.000 in Profile shift coefficient 0.50000 TOOTH THICKNESS (Zero Backlash) Pinion At Ref PD TOOTH THICKNESS (Zero Backlash) Pinion At OD TOOTH THICKNESS (Zero Backlash) Gear 0.194 in 0.194 in 0.035 in 0.035 in Profile shift coefficient 0.00000 10
60-107 Internal Gear Set Profile Shift Coefficients With Zero Backlash Unit System: US TOOTH THICKNESS (Zero Backlash) Gear At Ref PD Normal space width Transverse space width TOOTH THICKNESS (Zero Backlash) Gear At ID DIAMETERS Pinion Outside diameter (Optional Input) Roll angle at OD Increased OD to maintain clearance Reference pitch diameter Pointed tooth diameter Base diameter DIAMETERS Gear Inside diameter (Optional Input) Roll angle at ID Decreased ID to maintain clearance Minimum ID (Involute Interference) Reference pitch diameter Pointed tooth diameter Base diameter 0.094 in 0.094 in 1.700 in 46.892 deg 1.7162 in 1.400 in 1.742 in 1.316 in 4.800 in 11.9758 deg 4.784 in 4.772 in 5.000 in Below BD in 4.699 in 11
UTS Integrated Gear Software Unit System: US OPERATING DATA Working depth Normal pressure angle Transverse pressure angle Helix angle Circular pitch Roll angle at pitch point OPERATING DATA Pinion Pitch diameter Transverse Tooth Thickness Start of active profile Roll angle at SAP at SAP at SAP OPERATING DATA Gear Pitch diameter Transverse Space Width End of active profile Roll angle at EAP at EAP at EAP Trochoidal clearance: Pin OD/Gear ID 0.192 in 13.8246 deg 13.8246 deg 0.0000 deg 0.304 in 14.0993 deg 1.355 in 0.201 in 1.324 in 6.5154 deg 0.202 in 0.202 in 4.839 in 0.201 in 5.072 in 23.2814 deg 0.187 in 0.187 in 0.013 in 12
60-107 Internal Gear Set Profile Shift Coefficients With Zero Backlash Unit System: US Contact Length Length of contact, transverse plane 0.464 in Approach action (Pinion Driving) 18.78 % Recess action (Pinion Driving) 81.22 % Contact Ratios Profile 1.5702 Helical 0.0000 Total 1.5702 Lines of Contact Across Teeth Max total length 2.000 in Min total length 1.000 in Ratio: (Max length) / (Min length) 2.0000 Specific Sliding Ratios Pinion start of active profile 0.838 Pinion outside diameter 0.504 Gear start of active profile 1.014 Gear inside diameter 0.456 AGMA Load sharing ratio, mn 1.0000 AGMA I-factor for durability 0.276 Note that the CD modification coefficient is NOT equal to the Difference of shift coefficients. The operating center distance is 1.7419 mm. 13
UTS Integrated Gear Software Since the OD coefficient to maintain clearance is no longer zero it would be necessary to increase the pinion outside diameter (and, under certain conditions, to decrease the gear inside diameter) to keep the same root clearance. This is seldom done with internal gear sets since the clearance is a little larger than standard and we will stay with the standard outside diameter. Note the increase in AGMA I-factor and the reduction in the specific sliding ratio at the pinion SAP. Figure 1-4 shows these gears in mesh. The plot was generated in UTS Gear Program 60-450. Fig. 1-4 14
60-107 Internal Gear Set Profile Shift Coefficients With Zero Backlash Example 2 Example 2 is a helical gear set with a 20 tooth pinion driving a 85 tooth internal gear. The normal module is 4 mm, the nominal pressure angle is 20 degrees, the nominal helix angle is 25 degrees and the face is 30 mm. We are required to run on a 140 mm center distance, which is less than the standard center distance. Figures 2-1A and 2-1B are the data inputs and Report 2-1 is the solution for the given data. (We need to solve this part of the model to obtain the Difference of profile shift coefficients.) Fig. 2-1A Fig. 2-1A 15
UTS Integrated Gear Software Report 2-1 Unit System: Metric PINION, number of teeth 20 GEAR, number of teeth 85 Gear ratio 4.2500 NORMAL PLANE Normal pitch ( US/British System) Normal module ( SI System) Normal pressure angle Normal base pitch TRANSVERSE PLANE Transverse pitch Transverse pressure angle Transverse module Transverse base pitch COMMON Helix angle Base helix angle Axial pitch Operating center distance Standard center distance 6.350000 1/in 4.000000 mm ` 20.000000 deg 11.809 mm 5.7551 1/in 21.8802 deg 4.4135 mm ` 12.867 mm 25.000000 deg 23.3990 deg 29.735 mm 140.000 mm 143.4391 mm 16
60-107 Internal Gear Set Profile Shift Coefficients With Zero Backlash Unit System: Metric Change in Opr CD from "Std" CD -3.4391 mm CD modification coefficient -0.85978 Difference of shift coefficients -0.79050 OD, ID coef to maintain clearance -0.06928 Face width 30.000 mm The required operating center distance of 140 mm is 3.439 mm less than standard. Therefore the Difference of shift coefficients is -.79050. We will apply all of the difference to the pinion as positive profile shift and leave the gear standard. Figure 2-2 shows the additional data input and Report 2-2 is the solution for the model with this condition. Fig. 2-2 17
UTS Integrated Gear Software Report 2-2 Unit System: Metric PINION, number of teeth 20 GEAR, number of teeth 85 Gear ratio 4.2500 NORMAL PLANE Normal pitch ( US/British System) Normal module ( SI System) Normal pressure angle Normal base pitch TRANSVERSE PLANE Transverse pitch Transverse pressure angle Transverse module Transverse base pitch COMMON Helix angle Base helix angle Axial pitch Operating center distance Standard center distance 6.350000 1/in 4.000000 mm ` 20.000000 deg 11.809 mm 5.7551 1/in 21.8802 deg 4.4135 mm ` 12.867 mm 25.000000 deg 23.3990 deg 29.735 mm 140.000 mm 143.4391 mm 18
60-107 Internal Gear Set Profile Shift Coefficients With Zero Backlash Unit System: Metric Change in Opr CD from "Std" CD -3.4391 mm CD modification coefficient -0.85978 Difference of shift coefficients -0.79050 OD, ID coef to maintain clearance -0.06928 Face width TOOTH THICKNESS (Zero Backlash) Pinion 30.000 mm Profile shift coefficient 0.79050 TOOTH THICKNESS (Zero Backlash) Pinion At Ref PD TOOTH THICKNESS (Zero Backlash) Pinion At OD TOOTH THICKNESS (Zero Backlash) Gear 8.585 mm 9.472 mm 1.716 mm 1.952 mm Profile shift coefficient 0.00000 TOOTH THICKNESS (Zero Backlash) Gear At Ref PD Normal space width Transverse space width TOOTH THICKNESS (Zero Backlash) Gear At ID 6.283 mm 6.933 mm 6.283 mm 6.933 mm 3.515 mm 19
UTS Integrated Gear Software Unit System: Metric DIAMETERS Pinion 3.864 mm Outside diameter (Optional Input) Roll angle at OD Increased OD to maintain clearance Reference pitch diameter Pointed tooth diameter Base diameter DIAMETERS Gear Inside diameter (Optional Input) Roll angle at ID Decreased ID to maintain clearance Minimum ID (Involute Interference) Reference pitch diameter Pointed tooth diameter Base diameter OPERATING DATA Working depth Normal pressure angle Transverse pressure angle Helix angle Circular pitch Roll angle at pitch point 102.594 mm 43.210 deg 103.1485 mm 88.270 mm 105.130 mm 81.912 mm 367.149 mm 19.1988 deg 366.594 mm 358.779 mm 375.149 mm 352.307 mm 348.125 mm 7.723 mm 16.5253 deg 18.0548 deg 24.4716 deg 13.533 mm 18.6771 deg 20
60-107 Internal Gear Set Profile Shift Coefficients With Zero Backlash Unit System: Metric OPERATING DATA Pinion Pitch diameter Transverse Tooth Thickness Start of active profile Roll angle at SAP at SAP at SAP OPERATING DATA Gear Pitch diameter Transverse Space Width End of active profile Roll angle at EAP at EAP at EAP Trochoidal clearance: Pin OD/Gear ID Contact Length 86.154 mm 10.008 mm 87.188 mm 20.8941 deg 8.873 mm 9.769 mm 366.154 mm 10.008 mm 378.496 mm 24.4496 deg 7.589 mm 8.387 mm 0.906 mm Length of contact, transverse plane 15.952 mm Approach action (Pinion Driving) 0.00 % Recess action (Pinion Driving) 100.00 % Contact Ratios Profile 1.2398 Helical 1.0089 Total 2.2487 21
UTS Integrated Gear Software Unit System: Metric Lines of Contact Across Teeth Max total length 40.746 mm Min total length 40.457 mm Ratio: (Max length) / (Min length) 1.0071 Specific Sliding Ratios Pinion start of active profile 0.081 Pinion outside diameter 0.434 Gear start of active profile 0.767 Gear inside diameter 0.088 AGMA Load sharing ratio, mn 0.7415 AGMA I-factor for durability 0.571 Note that this gear set is full recess action with the pinion driving and all contact takes place after the pitch point on the line of centers. (This contact geometry would probably not be satisfactory with the gear driving.) Figure 2-3 shows the shape of the teeth in mesh in the normal plane for these gears generated with the same rack used in Example 1. The plot was generated in UTS Gear Program 60-450. 22
60-107 Internal Gear Set Profile Shift Coefficients With Zero Backlash Fig. 2-3 23