Introduction Program 60-146--Center Distance Change When the coefficient of thermal expansion of the material used for gears is different from the coefficient for the mountg or housg material, it is necessary to allow for the size changes caused by this condition. For example, if the housg is made of alumum and the gears of steel, an adjustment must be made the tooth thickness of the gears if a mimum backlash is to be present when the gear unit is below the assembly temperature. As the temperature is lowered, the housg will contract more than the gears and the amount of backlash will become less. If the drive is then put to operation the gears may warm up faster than the housg, and the backlash will be even less than at the equal temperature condition. When both the gears and housg are above the assembly temperature the backlash will usually be creased because of the larger expansion of the center distance the housg compared to the gears. (This is, of course, not the case with plastic gears metal housgs.) UTS Gear Model 60-146 can be used to calculate the amount of these changes and allowance can be made the manufactured tooth thickness of the gears. In addition to thermal effects the center distance tolerance, gear total composite tolerance, bearg runout, bearg radial play, and moisture absorption of plastic gears is taken to account. The model can be used for parallel or crossed axis helical gear sets. Note: Bearg radial play will crease the maximum assembled center distance and the mimum assembled center distance because the gear mesh separatg force will push the gears apart. (The opposite is true for ternal gears.) However, if loads external to the gear mesh push the gears together, the bearg radial play may be entered as a negative value to reflect this condition. The maximum and mimum effective center distance is calculated both on an absolute basis and on a statistical basis. The percentage of assemblies cluded the chosen statistical range is also calculated. The statistical maximum and mimum ECD are based on a normal distribution of part size with the part tolerance. If the part size does not follow a normal distribution, caution should be exercised use of the max and m ECD based on this calculation.
UTS Integrated Gear Software The number of standard deviations will determe the percentage of assemblies that will be the range of the statistical maximum and mimum effective center distance based on a normal distribution of part size with the part tolerance. The value that is usually used engeerg applications is 3 and the default value for the number of standard deviations (+ and -) is set to this value. With a +/- 3 standard deviation range there will be about 1 out of 370 assemblies that will be outside the statistical range. The actual center distance will not reach the maximum or mimum values (absolute or statistical) as the changes the gears are cluded the effective center distance calculations. The effect of runng the gears at the effective center distance on the relative position of the gears will be the same as the actual operatg conditions. The TK Solver model for this module contas a table of the thermal coefficients of expansion (and moisture absorption coefficients) for various materials. The table is shown Figure 1D. The values the model are average handbook values for the materials listed. The thermal coefficient values are per deg F units and should be entered per deg F. They are converted to per deg C if metric units are selected. It is recommended that these values be replaced by (or checked agast) values for the specific materials you are usg. Add or delete materials directly usg the Materials Update tab of the data put form, shown below. To add a new material, click the Add button, make entries each of the fields, then click Save. The Material # and Material Code fields are identifiers required by the TK Solver model. See Figure 1D for examples. Use this procedure also to enter materials already listed the table, but whose values are different from those already given. (Only user-entered materials can be edited and deleted with this procedure.) 2
60-146 Center Distance Change Examples Example 1 We will fd the required manufactured tooth thickness to sure a mimum backlash of about 0.0048 ch under the expected operatg temperatures for a 14/55 tooth external parallel axis gear pair. The gears are to be made to the followg specifications: Housg Material Magnesium Gear Material Nickel Steel Spur Gears Pion Teeth 14 Gear Teeth 55 Normal DP 10 Operatg PA 21.7 deg Nomal Gear Tooth Thickness 0.1571 Center Distance 3.4900/3.4940 Mimum Operatg Backlash 0.0048 Inspection and Assembly Temp +68 deg F AGMA Quality Class, Pion 12 AGMA Quality Class, Gear 12 Pion Bearg Runout 0.0005 Gear Bearg Runout 0.0005 Pion bearg radial play.0005/.0010 Gear bearg radial play.0005/.0010 It is estimated that the gears may run 50 degrees F hotter than the housg most of the time. A condition exists when the unit will be quickly exposed to warm air after beg at a low temperature and the housg may be 50 degrees F warmer than the gears. Operatg Temp Conditions: Housg Gears -65 F -15 F Cold Start +100 F +50 F Cold to Hot Transient +160 F +210 F Maximum Operatg Temp Open a new analysis 60-146 and enter the data for the cold start condition. The completed data put form for this condition is shown Figure 1A. 3
UTS Integrated Gear Software Fig. 1A Figure 1B is the completed data entry form for the second temperature condition (cold to hot transient). 4
60-146 Center Distance Change Fig. 1B 5
UTS Integrated Gear Software Figure 1C is the completed data put form for the third condition (maximum operatg temperatures.) Fig. 1C 6
60-146 Center Distance Change Report 1A is a combed report of the solved models for all three operatg conditions. Such a report is created by savg each analysis, openg a new analysis, selectg the saved analyses you want to clude, then clickg the button labeled Compare. See the Introduction for more formation. Report 1A Model: Program 60-146 Unit System: US Description Ex 1A 60-146 Ex 1B 60-146 Ex 1C 60-146 Unit No Group External or Internal Set e e e ASSEMBLY CONDITIONS Normal_Diametral Pitch 10.000000 10.000000 10.000000 1/ ` Normal_Module 2.540000 2.540000 2.540000 mm ` Helix Angle 0.000000 0.000000 0.000000 deg Operatg Transverse Pressure Angle 21.700000 21.700000 21.700000 deg Temperature 68 68 68 F Relative Humidity 50.00 50.00 50.00 % HOUSING Material_Number-See Material Table 8 8 8 Material_Code Magnesium Magnesium Magnesium Thermal Coefficient of expansion 0.00 0.00 0.00 1/degF Moisture Coefficient of expansion 0.00 0.00 0.00 / Basic or nomal center distance 3.4900 3.4900 3.4900 Mimum center distance tolerance 0.0000 0.0000 0.0000 Maximum center distance tolerance 0.0030 0.0030 0.0030 Pion total composite tolerance 0.00099 0.00099 0.00099 Gear total composite tolerance 0.00120 0.00120 0.00120 Pion_bearg runout (TIR) 0.0005 0.0005 0.0005 Gear_bearg runout (TIR) 0.0005 0.0005 0.0005 7
UTS Integrated Gear Software Model: Program 60-146 Unit System: US Description Ex 1A 60-146 Ex 1B 60-146 Ex 1C 60-146 Unit Pion_bearg_radial play, max 0.0010 0.0010 0.0010 Pion_bearg_radial play, m 0.0005 0.0005 0.0005 Gear_bearg_radial play, max 0.0010 0.0010 0.0010 Gear_bearg_radial play, m 0.0005 0.0005 0.0005 Mimum_assembled CD, absolute 3.4900 3.4900 3.4900 Maximum_assembled CD, absolute 3.4967 3.4967 3.4967 Mimum_assembled CD, statistical 3.4914 3.4914 3.4914 Maximum_assembled CD, statistical 3.4953 3.4953 3.4953 PINION Material_Number-See Material Table 4 4 4 Material_Code Nickel St Nickel St Nickel St Number of teeth 14 14 14 AGMA Quality Class 12 12 12 Thermal Coefficient of expansion 0.00 0.00 0.00 1/degF Moisture Coefficient of expansion 0.00 0.00 0.00 / Nomal_operatg pitch diameter 1.4162 1.4162 1.4162 GEAR Material_Number-See Material Table 4 4 4 Material_Code Nickel St Nickel St Nickel St Number of teeth 55 55 55 AGMA Quality Class 12 12 12 Thermal Coefficient of expansion 0.00 0.00 0.00 1/degF Moisture Coefficient of expansion 0.00 0.00 0.00 / Nomal_operatg pitch diameter 5.5638 5.5638 5.5638 OPERATING CONDITIONS Housg_temperature -65 100 160 F Pion_temperature -15 50 210 F Gear_temperature -15 50 210 F Relative_humidity 50.00 50.00 50.00 % Change_ relative humidity 0.00 0.00 0.00 % 8
60-146 Center Distance Change Model: Program 60-146 Unit System: US Description Ex 1A 60-146 Ex 1B 60-146 Ex 1C 60-146 Unit EFFECTIVE CENTER DISTANCE Change CD (Thermal, Moisture) -0.0048 0.0021 0.0011 Approximate change backlash -0.0038 0.0017 0.0009 EFFECTIVE CD, ABSOLUTE Mimum_effective center distance 3.4852 3.4921 3.4911 Maximum_effective center distance 3.4919 3.4988 3.4978 Mean effective center distance 3.4885 3.4955 3.4945 CD Range 0.00669 0.00669 0.00669 EFFECTIVE CD, STATISTICAL Standard deviations, +/- 3.0000 3.0000 3.0000 Range of deviations 6.0000 6.0000 6.0000 Amount of assemblies cluded the range 99.7300 99.7300 99.7300 % Number_of assemblies cluded range per 370 370 370 assembly outside range Mimum_effective center distance 3.4866 3.4935 3.4926 Maximum_effective center distance 3.4905 3.4974 3.4964 Mean effective center distance 3.4885 3.4955 3.4945 CD Range 0.00388 0.00388 0.00388 For the cold start condition the change effective center distance is -0.0048 ch and the approximate change backlash is -0.0038 ch. (The change backlash is approximate because the operatg pressure angle is averaged for the assembly and effective center distance. It is, however, quite accurate.) For the cold-to-hot transient condition he change effective center distance is +0.0021 ch and the approximate change backlash is +0.0017 ch. At the maximum operatg condition the change effective center distance is +0.0011 ch and the approximate change backlash is +0.0009 ch. The maximum backlash will then occur durg the transient condition and not at maximum operatg condition. 9
UTS Integrated Gear Software Fig. 1D The TK Solver model s table of thermal and moisture coefficients, discussed the Introduction, is shown above. For this analysis, the mimum effective center distance (usg absolute values) is 3.4852 ches and occurs durg the cold start condition. If the mimum operatg backlash is to be 0.0048 ch the tooth thickness of the gears will have to be calculated at this center distance. Figure 1D 10
60-146 Center Distance Change is the solved model from UTS Model 60-104 with the cold start and cold to hot transition conditions entered. (Note that only enough of the model has been solved to obta the tooth thickness.) Report 1B Model: Program 60-104 Unit System: US Description Ex 1E 60-146 Ex 1F 60-146 Unit CAUTION MESSAGE CAUTION MESSAGE m2 m3 m4 DRIVER, number of teeth 14 14 DRIVEN, number of teeth 55 55 Gear ratio 3.9286 3.9286 Gears shaved or ground? ('s,'g,'no) no no NORMAL PLANE Normal pitch 10.000000 10.000000 1/ ` Normal pressure angle 20.000000 20.000000 deg Normal module 2.540000 2.540000 mm` Normal base pitch 0.2952 0.2952 TRANSVERSE PLANE Transverse pitch 10.0000 10.0000 1/ ` Transverse pressure angle 20.0000 20.0000 deg Transverse module 2.5400 2.5400 mm` Transverse base pitch 0.2952 0.2952 COMMON 11
UTS Integrated Gear Software Helix angle 0.000000 0.000000 deg Model: Program 60-104 Unit System: US Description Ex 1E 60-146 Ex 1F 60-146 Unit Base helix angle 0.0000 0.0000 deg Axial pitch I n Operatg center distance 3.4852 3.4988 Standard center distance 3.4500 3.4500 Face width 1.0000 1.0000 TOOTH THICKNESS DRIVER AT REF PD Normal tooth thickness 0.1789 0.1769 Transverse tooth thickness 0.1789 0.1769 TOOTH THICKNESS DRIVER AT OD Normal tooth thickness Transverse tooth thickness TOOTH THICKNESS DRIVEN AT REF PD Normal tooth thickness 0.1571 0.1551 Transverse tooth thickness 0.1571 0.1551 TOOTH THICKNESS DRIVEN AT OD Normal tooth thickness Transverse tooth thickness CLEARANCE (FOR UNDERCUT CHECK) Root clearance, Driver (approx) 0.0250 0.0250 Root clearance, Driven (approx) 0.0250 0.0250 DIAMETERS DRIVER Outside diameter Roll angle at OD deg Reference pitch diameter 1.4000 1.4000 Poted tooth diameter 1.7209 1.7180 Base dia 1.3156 1.3156 12
60-146 Center Distance Change Model: Program 60-104 Unit System: US Description Ex 1E 60-146 Ex 1F 60-146 Unit DIAMETERS DRIVEN Outside diameter Roll angle at OD deg Reference pitch diameter 5.5000 5.5000 Poted tooth diameter 5.8584 5.8544 Base dia 5.1683 5.1683 OPERATING DATA Workg depth Basic transverse backlash 0.0010 0.0018 Change Opr CD from "Std" CD 0.0352 0.0488 Normal backlash 0.0048 0.0198 Transverse backlash 0.0048 0.0198 Transverse pressure angle 21.5337 22.0912 deg Helix angle 0.0000 0.0000 deg Roll angle at pitch pot 22.6083 23.2552 deg Circular Pitch 0.3174 0.3186 OPERATING DATA DRIVER Pitch diameter 1.4143 1.4198 Transverse tooth thickness 0.1753 0.1717 Start of active profile Roll angle at SAP deg Normal tooth thickness at SAP Transverse tooth thickness at SAP Mid-pot, length of contact Roll angle at mid-pot deg OPERATING DATA DRIVEN Pitch diameter 5.5561 5.5778 Transverse tooth thickness 0.1373 0.1271 13
UTS Integrated Gear Software In the cold start scenario, if the maximum tooth thickness of the gear is held to 0.1571 ch then the maximum tooth thickness of the pion must be 0.1789 ch. This will result a backlash of 0.0048 ch under these conditions. In the cold to hot transient scenario, if the tooth thickness tolerance is -0.002 ch then the maximum backlash is 0.0198 ch. The tooth thickness values we have been usg are the effective tooth thickness values. Because of allowable tooth element errors that may not be measured by the measurg system beg used the manufactured size may need adjustment. The amount of adjustment depends on the measurg system. (See UTS Gear Model 60-145.) Example 2 Figure 2 and Report 2 model an ternal gear set at normal operatg temperature runng an area with high humidity. The housg is alumum, the pion is nylon 6/6 unfilled and the ternal gear is unfilled acetal. The expansion due to moisture absorption is important here especially for the nylon pion. The large change effective center distance is typical for designs usg plastic gears. The design of plastic gearg, even with small temperature differences, is difficult because of large coefficients of thermal expansion and the expansion due to moisture absorption. We will assume that the parts are measured and assembled at 50% relative humidity and will then operate at 100% humidity. 14
60-146 Center Distance Change Fig. 2 15
UTS Integrated Gear Software Report 2 Model Title : Program 60-146 Unit System: US External or Internal Set i ASSEMBLY CONDITIONS Normal_Diametral Pitch 10.000000 1/ ` Normal_Module 2.540000 mm ` Helix Angle 0.000000 deg Operatg Transverse Pressure Angle 20.000000 deg Temperature 68 F Relative Humidity 50.00 % HOUSING Material_Number-See Material Table 6 Material_Code Alumum Thermal Coefficient of expansion 0.00 1/degF Moisture Coefficient of expansion 0.00 / Basic or nomal center distance 5.4000 Mimum center distance tolerance -0.0030 Maximum center distance tolerance 0.0030 Pion total composite tolerance 0.00540 Gear total composite tolerance 0.00990 Pion_bearg runout (TIR) 0.0000 Gear_bearg runout (TIR) 0.0000 Pion_bearg_radial play, max 0.0015 Pion_bearg_radial play, m 0.0010 16
60-146 Center Distance Change Model Title : Program 60-146 Unit System: US Gear_bearg_radial play, max 0.0017 Gear_bearg_radial play, m 0.0012 Mimum_assembled CD, absolute 5.3801 Maximum_assembled CD, absolute 5.4019 Mimum_assembled CD, statistical 5.3828 Maximum_assembled CD, statistical 5.3992 PINION Material_Number-See Material Table 10 Material_Code Nyl 6/6UF Number of teeth 17 AGMA Quality Class 7 Thermal Coefficient of expansion 0.00 1/degF Moisture Coefficient of expansion 0.00 / Nomal_operatg pitch diameter 1.7000 GEAR Material_Number-See Material Table 9 Material_Code Acetal UF Number of teeth 125 AGMA Quality Class 6 Thermal Coefficient of expansion 0.00 1/degF Moisture Coefficient of expansion 0.00 / Nomal_operatg pitch diameter 12.5000 OPERATING CONDITIONS Housg_temperature 90 F Pion_temperature 110 F Gear_temperature 100 F 17
UTS Integrated Gear Software Model Title : Program 60-146 Unit System: US Relative_humidity 100.00 % Change_ relative humidity 50.00 % EFFECTIVE CENTER DISTANCE Change CD (Thermal, Moisture) -0.0085 Approximate change backlash 0.0062 EFFECTIVE CD, ABSOLUTE Mimum_effective center distance 5.3716 Maximum_effective center distance 5.3934 Mean effective center distance 5.3825 CD Range 0.02180 EFFECTIVE CD, STATISTICAL Standard deviations, +/- 3.0000 Range of deviations 6.0000 Amount of assemblies cluded the 99.7300 % range Number_of assemblies cluded range 370 per assembly outside range Mimum_effective center distance 5.3743 Maximum_effective center distance 5.3907 Mean effective center distance 5.3825 CD Range 0.01644 The absolute values of the tolerances and changes produced by the temperature and humidity conditions produces a center distance range of 0.0218 ch. This is a considerable amount of change and when combed with the variation produced at possible cold and dry conditions design of an acceptable gear set may be very difficult. It is very unlikely that any gear set would be at these extremes sce all dimensions would have to be at the max tolerance limit and such a direction as to all add up the same way. Because of this, many designs use a statistical center distance variation. In this case a range of +/- 3 sigma produced a center distance range of 18
60-146 Center Distance Change 0.0164 ch. This reduction may make design much easier. The number of units that could be expected to be outside this range is about 1 370. NOTE: In this example we only looked at one condition of humidity and temperature. In actual design work it is necessary to run the model under the followg conditions: 1. Max Temp & Max Humidity 2. Max Temp & M Humidity 3. M Temp & Max Humidity 4. M Temp & M Humidity The maximum and mimum values of the effective center distances from all 4 conditions are then used for design purposes. (The effect of different humidity levels can be modeled by changg the operatg relative humidity.) Application Note: All versions of this model prior to Version 4.1 adjusted the maximum and mimum effective center distance effect of the total composite tolerance as follows: Maximum center distance with master gear = tight mesh CD with gear at maximum tooth thickness + 1/2 of total composite tolerance Mimum center distance with master gear = tight mesh CD with gear at mimum tooth thickness - 1/2 of total composite tolerance Ver 4.1 (and later) adjusts the maximum and mimum effective center distance effect of the total composite tolerance as follows: Maximum center distance with master gear = tight mesh CD with gear at maximum tooth thickness Mimum center distance with master gear = tight mesh CD with gear at mimum tooth thickness - total composite tolerance This method is accordance with ANSI/AGMA 2002 B88. Reference: Paul M. Dean, Center Distance Gear Handbook, Chapter 6 Darle W. Dudley, Editor McGraw-Hill Book Company, New York, NY (1st Edition 1962) 19