program 60-5408—crowned and straight external involute gear ehl film … · 2015. 5. 8. ·...

Program 60-5408—Crowned and Straight External Involute Gear EHL Film Thickness

Introduction Lubrication film thickness is obviously an important factor in the lubrication of gears, because it determines the amount of contact between asperities (high spots) on the surfaces of the working flanks of the teeth. The ability of gears to operate without contact between the teeth has been noted since lubricated metal gears have been used. In many cases the initial machining marks on gears that have been in service for years are still visible. The obvious separation between the teeth cannot be accounted for by hydrodynamic theory alone. Because, for parallel axis external involute gears, the surfaces of the teeth are non-conformal (both convex) the separation predicted by hydrodynamic theory is too small to account for the separation of the teeth. The mode of lubrication between highly loaded gear teeth is not hydrodynamic but elastohydrodynamic (EHL). This theory takes into account not only the hydrodynamic character of the contact but the elastic deformation of the teeth as well. EHL theory successfully predicts the separation (or lack of separation) between gear teeth. The viscosity of lubricating oils increases with pressure. This effect is not very important in contacts such as journal bearings where the pressure in the oil film is relatively low. However, in gear tooth contacts the contact pressure may be in the range of 20,000 to 300,000 psi. At these pressures the viscosity of the lubricant increases dramatically and is responsible for the separation of the surfaces. In order to estimate the film thickness between gear teeth, two sets of data are necessary. First, we must know the relevant design and operating conditions for the gears. Second, we must know how the viscosity of the lubricant changes with both temperature and pressure. The first set of data is available from the design and loading data for the gear set. The second is available in the form of a “lubricant parameter”, developed for each lubricant, that takes into account the pressure and temperature effects on viscosity. These lubricant parameters have been determined by the Mobil Oil Corporation for most of their products and are used in this model. See reference 3 for more information. Once the thickness of the film has been estimated, it must be compared to the roughness of the contacting surfaces to determine the probability of asperity contact. The ratio of the film thickness to the composite roughness of the surfaces is called the specific film thickness and is used, along with speed and shear heating factors, to determine the probability of contact. (Since ALL factors affecting the film thickness are not known with certainty it is necessary to predict the amount of asperity contact using statistical methods.)

UTS Integrated Gear Software

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When the film thickness is too low the gear set may suffer from cold scoring. Cold scoring occurs when asperities on the tooth surfaces contact each other through the lubricant film. Subsequent rolling and sliding of the tooth surfaces produce radial score lines and surface contour is finally destroyed. (Cold scoring must not be confused with hot scoring as they are two different modes of tooth contact failure. See UTS Model 60-560 or reference 2 for information on hot scoring.) Reference 3 uses the EHL film thickness at the operating pitch diameters to assess the probability of cold scoring. This model uses the EHL film thickness at the center of the length of the contact interval. This is done because some gears sets are designed so that the operating pitch point is quite far from the center of mesh. For example, full recess action gears have the outside diameter of the driven gear at the operating pitch point and the operating pitch point is at the very end of the contact interval. Ideal uniform load distribution along gear teeth is seldom obtained due to many factors, including dimensional errors and elasticity in the gear teeth themselves and in supporting elements such as shafts, bearings and housings. Even if parallelism between the teeth is achieved at a specific load condition it cannot be maintained if the load changes because of deflections in the supporting structure. Thermal effects, centrifugal deflections and external shaft bending moments also play a role in causing face mismatch. Of course, any deviation from a perfectly parallel condition between mating teeth causes a non-uniform load distribution along the teeth. The usual design procedure for assessing the compressive stress on teeth with no longitudinal (lead) modification is to use the Hertz equations for parallel cylinders and then to apply a multiplier based on the face mismatch to obtain the maximum compressive stress in the misaligned position. (The use of the parallel cylinder equations is not quite correct as the radii of curvature on involute teeth change across the load “footprint” when under load.) AGMA Standard 218 of the American Gear Manufacturers Association provides a multiplier (designated Face Load Distribution (FLD) Factor, labeled Cmf in the TK Solver model) to account for the increase in compressive stress due to face mismatch above the stress calculated with the Hertz equations. The FLD Factor is defined as the peak load intensity divided by the average load intensity across the face width of the gears. The FLD Factor determined by the “analytical method” in this standard assumes a linear load distribution across the teeth and is based on the stiffness of the teeth along with the load (for helical gears the effect of the profile contact ratio is also included). This method is valid for any design and teeth with crowns must use this method. This model uses the FLD factor to analyze gears with straight teeth.

60-5408—Crowned and Straight External Involute Gear EHL Film Thickness

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When straight teeth are misaligned the load at the tooth ends is, of course, higher than the average load. One widely used method of relieving the load on the tooth ends caused by mismatch is called “lead crowning”. With this method the tooth ends of one or both gears are made thinner than the center. The sides of a section parallel to the axis of the gear through the tooth are more or less elliptical or circular rather than straight. The actual shape of the tooth depends on the machining method used and how it is applied. The crown is specified and measured as a lead deviation in transverse planes. In this model a circular arc with a specified rise in the center of the tooth is assumed for the lead deviation. For spur gears, of course, the lead deviation is normal to the tooth and is the actual shape of the tooth. For helical gears, however, the lead deviation is not normal to the tooth. The circular arc deviation for a helical tooth in a direction normal to the tooth is the circular arc obtained by dividing the radius of the deviation by the cosine of the operating helix angle. As we said, for straight teeth, the Hertz equations for parallel cylinders are used for the compressive stress calculation. Here, it is assumed that the shape of the contact zone is a rectangle with the long dimension along the length of the tooth. The width of the rectangle is determined by the load and the curvature of the involute profiles for a given combination of materials. The stress distribution is semi-cylindrical. When the teeth are misaligned the contact zone is no longer a rectangle. The stress distribution is then semi-conical. The multiplier, the FLD Factor, gives us the stress at the wide end of the trapezoidal or triangular load pattern. When we have two surfaces in contact with curvature in all directions (crowned teeth) it is necessary to use the Hertz equations for elliptical contacts instead of the parallel cylinder equations. In this case the zone of contact is assumed to be an ellipse with the long dimension along the tooth (direction of least curvature) and the short dimension in the normal direction of the involute profile. The dimensions of the ellipse are determined by the load, the radius of the crown (or crowns) and the radii of the involute profiles for a given combination of materials. The stress distribution is a semi-ellipsoid. If the contact ellipse (under load) is cut off by the end(s) of the teeth(spur) or by the start or end of the contact zone (helicals) the stress distribution is assumed to be a truncated ellipsoid cut by a plane (or planes) parallel to the minor axis of the contact ellipse. The stress and the minor dimension of the contact pattern are then adjusted to keep the same volume for the truncated ellipsoid and the full contact ellipsoid. (The numerical eccentricities of the elliptical sections are held constant.) This model incorporates both conditions and uses the parallel cylinder equations for straight teeth and the elliptical contact equations for teeth with crown on one or both gears.


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The equations from reference 3 are used for both straight and crowned contacts to obtain a valid comparison of film thickness for the two types of contact. When stress for gears with straight teeth is being calculated, the FLD Factor will also be calculated (or must be input). The stress is then calculated by the methods from reference 3 for straight line contacts. For crowned gears the center of the crown is assumed to be in the center of the face width. (If the teeth are chamfered from the root to the tip at the ends use the mean value for the face widths.) If the initial (no load) contact of the crowned profile(s) falls off the end of the tooth due to the lead mismatch the model will not be solved and the note “INITIAL TOOTH CONTACT OFF END” will appear.

Spur Gears If the contact ellipse (under load) falls off one or both ends of the gear tooth the calculated stress will be increased due to the reduced contact pattern. “CROWNED TOOTH CONTACT OFF END” will appear if the ellipse is off one end of the tooth. If the ellipse falls off both ends of the tooth the comment “CROWN CONTACT OFF BOTH ENDS” will appear.

Helical Gears The contact ellipse is always truncated at the start and end of the contact zone. It is assumed that the initial contact of the CROWN is closest to the start of the contact zone. (In many cases the ellipse is truncated for the entire mesh cycle.) As an example a 12 normal diametral pitch, AGMA quality Q8, helical gear set with 27 teeth in the pinion and 108 teeth in the gear will be used. The gears are steel. The design load is 100 horsepower and the pinion speed is 3600 RPM. The initial temperature will be 200° F. (The initial or inlet temperature is the temperature of the lubricant as it enters the contact zone. This temperature may be somewhat higher than the sump temperature (for splash fed gears) or manifold temperature (for pressure fed gears). NOTE: If the actual value of the surface finish on the gears after break-in is not known, a suggested composite surface finish is available from table “Finish” from the reference. The individual surface finishes should be left blank if the composite finish is used. The face mismatch is .002 inch across a 2 inch face for the gear. The first calculation will be with straight teeth.


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Examples Open a new analysis and enter the data shown in Figure 1. Since the gears are steel we can use the default values for Young’s Modulus and Poisson’s Ratio and do not have to enter them. For other materials they must be entered or they will default to the values for steel. Fig. 1

Because we have not chosen a lubricant, we do not know what the lubricant parameter (LP) is. Let’s suppose we wish to hold the cold scoring probability to about 20%. Since TK Solver will solve in any direction, we can set the scoring probability to 20% and solve for a lubricant parameter. However, when solving in this direction the equations cannot be solved directly, so an iterative solution technique must be used. To do this we must toggle to TK Solver and work directly in the Variable Sheet. We will make a guess for the LP of 50. Input 50 for the LP, move the cursor to the status column and type “G”, or double-click the cell and pick “Guess” from the drop-down list of input value types. This will tell TK Solver that 50 is a guess at a starting point and TK's Iterative Solver will be used to solve the equations.


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NOTE: The values that may be used for input for probability of scoring are about 6% to 49% or you may get an error message. If an error occurs with probability set between these values make a different guess for the LP. Your Variable Sheet entries should look like Sheet 1. Sheet 1

Return to the Integrated Gear Software interface and click the toolbar “Solve” button. Report 1 shows the inputs and outputs for this model.

Report 1

Model Title : Program 60-5408 Unit System: US COMMON DATA

Quality Class 8

PINION Speed 3600.0 rpm

Power 100.00 HP


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Model Title : Program 60-5408 Unit System: US Normal pitch 12.000000 1/in `

Normal Pressure Angle 20.000000 deg

Helix Angle 16.000000 deg

Operating Center Distance 5.8667 in

Total Lead Mismatch Between Teeth 0.00200 in

FILM THICKNESS & SCORING PROBABILITY : Crowned Teeth

Actual Width of Contact Ellipse NA in

Length of Full Contact Ellipse NA in

Length of Actual Contact Ellipse NA in

FILM THICKNESS & SCORING PROBABILITY : Straight Teeth

Contact Length 1.0112 in

Maximum Contact Width 0.0047 in

Lubricant Parameter 68.7

Set Lube Parameter Table to y

Initial Lubricant (Inlet) Temp 200 C

EHL Film at Center of Interval 26.63 uin

Composite Surface Finish 35.36 uin

Correction Factor (Inlet Shear) 0.93

Specific Film Thickness 0.75

Probability of Cold Scoring 20 %

PINION DATA

Number of Teeth 27

Outside Diameter 2.5533 in

Face Width 2.2000 in


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Model Title : Program 60-5408 Unit System: US Surface Finish,rms (After break-in) 25.00 uin

GEAR DATA

Number of Teeth 108



Surface Finish,rms (After break-in) 25.00 uin An oil with an LP of about 69 will hold the cold scoring probability to 20%. The next step is to find a lubricant with an LP of about 69 at a temperature of 200° F. The model contains a table of all Mobil lubricants contained in reference 3. When we solved the model this table was updated to 200 degrees F. Sheet 4 is the table in printed form.


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Sheet 4

Mobil Lubricant Parameter AUTOMOTIVE_GEAR_OILS Lube_SHC 38 EP Lube_HD_80W 28 EP Lube_HD_80W_90 50 EP Lube_FE_80W_140 92 EP Lube_HD_85W_140 114 EP Lube_HD_90 65 EP Lube_HD_140 119 EP INDUSTRIAL_GEAR_OILS Cyl_600W 127 Sp_Cyl_600W 143 Sp_Cyl_Minrl_600W 164 X_HECLA_Sp_Cyl 206 X_HECLA_Sp_Cyl_Mnrl 241 DTE_Light 12 DTE_Medium 16 DTE_Hvy_Med 24 DTE_Heavy 34 DTE_Extra_Hvy 45 DTE_BB 67 DTE_AA 102 DTE_HH 140 Glygoyle_11 33 Mild_EP Glygoyle_22 70 Mild_EP Glygoyle_30 96 Mild_EP Glygoyle_80 160 Mild_EP SHC_624 10 Mild_EP SHC_626 21 Mild_EP SHC_629 48 Mild_EP SHC_630 74 Mild_EP


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Mobil Lubricant Parameter SHC_634 135 Mild_EP Gear_626 25 EP Gear_627 39 EP Gear_629 50 EP Gear_630 71 EP Gear_632 100 EP Gear_634 147 EP Gear_636 198 EP AUTOMATIC_TRANS_FLUID ATF_220 12 AUTOMOTIVE_ENGINE_OILS Mobil_1 18 Delvac_1 17 Delvac_1110 13 Delvac_1120 18 Delvac_1130 34 Delvac_1140 45 Delvac_1150 67 Delvac_1210 14 Delvac_1220 25 Delvac_1230 40 Delvac_1240 51 Delvac_1250 77 Delvac_1310 17 Delvac_1320 26 Delvac_1330 40 Delvac_1340 59 Delvac_1350 78 Oil_10W 12 Oil_20W_20 19


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Mobil Lubricant Parameter Oil_30 34 Oil_40 48 Oil_50 69 Special_10W_30 18 Super_10W_40 18 GAS_ENGINE_OILS Pegasus_360_385_390 30 Pegasus_100_30 30 Pegasus_460_485_490 41 Pegasus_100_40 41 DIESEL_ENGINE_OILS DTE_Oil_#3 33 DTE_Oil_#4 47 DTE_Oil_#5 65 Gard_307 36 Gard_407 51 Gard_507 68 Gard_212 26 Gard_312 41 Gard_412 54 Gard_512 79 Gard_324 44 Gard_424 56 Gard_524 79 Gard_342 47 Gard_442 60 Gard_345 49 Gard_443_445 62 Gard_300 41 Gard_570 101


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Mobil Lubricant Parameter Gard_593 96 Gard_346 41 Gard_444_446 57 Gard_448_450 57 HYDRAULIC_OILS DTE_21 4 DTE_22 7.9 DTE_24 11 DTE_25 15 DTE_26 22 DTE_27 36 Vacuoline_190 58 Vacuoline_1405 11 Vacuoline_1409 25 AIRCRAFT_JET_ENGINE_OILS Jet_254 13 Jet_II 11 TURBINE_OILS DTE_724 11 DTE_797 11 SHC_824 11 SHC_825 16 SPINDLE_OILS Velocite_C 9 Velocite_D 7.3 Velocite_E 3.5 Velocite_S 5.6 Velocite_#3 .9


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Mobil Lubricant Parameter Velocite_#6 3.5 Velocite_#8 5.6 Velocite_#10 7.9 Velocite_#12 9 COMPRESSOR_OILS DTE_103 41 DTE_105 83 Rarus_424 10 Rarus_425 17 Rarus_427_US 36 Rarus_427_ID 25 Rarus_524 11 Rarus_526 25 Rarus_824 12 Rarus_826 23 Rarus_827 26 AIR_TOOL_OILS Almo_325 23 Mild_EP Almo_525 17 Mild_EP Almo_527 35 Mild_EP Almo_529 48 Mild_EP Almo_532 91 Mild_EP MACHINE_OILS Etna_#2 34 Etna_#3 50 Etna_#4 70 Etna_#6 140 Vactra_Light 12 Vactra_Medium 17


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Mobil Lubricant Parameter Vactra_Hvy_Med 24 Vactra_Heavy 35 Vactra_Extra_Hvy 46 Vactra_BB 69 Vactra_AA 103 Vactra_HH 141 Viscolite_Hvy 27 Viscolite_Ex_Hvy 50 Viscolite_BB 77 Viscolite_AA 96 Viscolite_HH 136 Viscolite_SS 294 CIRCULATION_OILS Paper_Mach_J 41 Paper_Mach_K 58 Vacuoline_X28 50 Vacuoline_X33 72 Vacuoline_X37 99 Vacuoline_X46 139 Vacuoline_X48 211 Vacuoline_525 30 MIST_OILS Mist_24 15 EP Mist_27 43 EP Mist_30 59 EP Mist_32 83 EP Mist_34 113 EP REFRIGERATION_OILS Gargoyle_Arctic_Lt 5.9


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Mobil Lubricant Parameter Gargoyle_Arctic_C 15 Gargoyle_Arctic_C_Hvy 18 Gargoyle_Arctic_Ex_Hvy 23 Gargoyle_Arctic_155 12 Gargoyle_Arctic_300 23 METAL_CUTTING_FLUIDS Mobilmet_Omicron 11 Mobilmet_Upsilon 4.2 WAY_OILS Vactra_#1 12 Mild_EP Vactra_#2 23 Mild_EP Vactra_#3 32 Mild_EP Vactra_#4 69 Mild_EP INITIAL_TEMP___degF 167 INITIAL_TEMP___degC 75 Note that the temperature to which the LP’s are set appears at the bottom of the table. (The listing of lubricants in the table does not imply that they are suitable for the temperature listed. Lubricants must be selected on the basis of high and low temperatures to which they will be exposed. Check your selection with a qualified lubricant supplier.) Since we need an LP of about 69 we will choose Mobil Gear 634. Blank the scoring probability, input 65 for the LP on the data input form and solve. The completed data input form is shown in Figure 2, the report for the solved model in Report 2.


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Fig. 2

Report 2

Model Title : Program 60-5408 Unit System: US Message Field STRAIGHT Message Field TEETH Message Field TEMP Message Field HIGH Message Field _

COMMON DATA

Quality Class 8


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Model Title : Program 60-5408 Unit System: US PINION Speed 3600.0 rpm

PINION Torque 1750.6944 lbf-in

Power 100.00 HP

Tangential Load at Operating PD 1492.0607 lbf

Pitch Line Velocity 2211.7 ft/min

Normal pitch 12.000000 1/in `

Normal Module 2.116667 mm `



Transverse Pitch 11.535140 1/in `

Transverse Module 2.201967 mm `

Transverse Pressure Angle 20.738600 deg

Base Helix Angle 15.0116 deg


Operating Trans Pressure Angle 21.1225 deg

Operating Helix Angle 16.0389 deg

Effective Face Width 2.0000 in


Face Load Distribution Factor 4.116

Tooth Stiffness Constant 1815000.000 psi

Distance from Center of Tooth to Initial 1.0405 in Tooth Contact

Gear Ratio 4.0000

Profile Contact Ratio (Theoretical) 1.5824

Helical Contact Ratio (Theoretical) 2.1057


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Model Title : Program 60-5408 Unit System: US FILM THICKNESS & SCORING PROBABILITY : Crowned Teeth

Actual Width of Contact Ellipse NA in

Length of Full Contact Ellipse NA in

Length of Actual Contact Ellipse NA in


Contact Length 1.0112 in

Maximum Contact Width 0.0047 in









PINION DATA

Number of Teeth 27


Transverse Crown Rise (Default=0) 0.000 in


Tooth End to Initial Contact Point 0.1041 in

Tooth End to Crown Contact Ellipse With in Crown Centered On Tooth

Surface Finish,rms (After break-in) 25.00 uin


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Model Title : Program 60-5408 Unit System: US Young's Modulus (Default=steel) 30000000.000 psi

Poisson's Ratio (Default=.3) 0.300

Operating Pitch Diameter 2.3467 in

Start of Active Profile-No Undercut 2.2473 in

Base Diameter 2.1890 in

GEAR DATA

Number of Teeth 108





Tooth End to Crown Contact Ellipse With in Crown Centered On Tooth


Young's Modulus (Default=steel) 30000000.000 psi





ROLL ANGLES : Pinion

Start of Active Profile 13.304 deg

Low Single Tooth Contact _ deg

Pitch Diameter, Operating 22.134 deg

Center of Contact Interval 23.853 deg

High Single Tooth Contact _ deg


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Model Title : Program 60-5408 Unit System: US Outside Diameter, Effective 34.403 deg

ROLL ANGLES : Gear






Outside Diameter, Effective 24.342 deg

With Mobil DTE HH we have about a 23% chance of scoring. (If the scoring probability is 20% or higher a message will be given in the message field suggesting the use of an extreme pressure lubricant instead of a non-reactive lubricant.) The film thickness is about 25.64 microinch and the specific film thickness about .73. This is probably a good choice of lubricant for this application, as the table shows that Mobil Gear 634 is classed as an extreme pressure (EP) oil. The pour point and viscosity should be checked to make sure that they are “OK” for this application. EP additives are meant to react with the tooth surfaces at high temperatures and form a film to prevent welding. EP lubricants were developed primarily for prevention of hot scoring due to high temperatures at the tooth contact from high speed and load but have been found to help in cold scoring also. (This is probably due to the hot scoring characteristics of the asperity contact.)

Let’s put .0015 inch crown on the pinion and see what change this will make in the cold scoring probability. (Crown should not be added without consideration of the change in compressive and bending stress. See UTS TK Solver model 60-5406.) Figure 3 shows a portion of the data input form with the crown added. Click the toolbar “Refresh Values” button, then the “Solve” button. The solved model is shown in Report 3.


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Fig. 3

Report 3

Model Title : Program 60-5408 Unit System: US

Message Field CROWNED

Message Field TEETH

Message Field TEMP

Message Field HIGH

Message Field _

COMMON DATA

Quality Class 8

PINION Speed 3600.0 rpm


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Model Title : Program 60-5408 Unit System: US PINION Torque 1750.6944 lbf-in

Power 100.00 HP

Tangential Load at Operating PD 1492.0607 lbf

Pitch Line Velocity 2211.7 ft/min

Normal pitch 12.000000 1/in `

Normal Module 2.116667 mm `



Transverse Pitch 11.535140 1/in `

Transverse Module 2.201967 mm `

Transverse Pressure Angle 20.738600 deg

Base Helix Angle 15.0116 deg


Operating Trans Pressure Angle 21.1225 deg

Operating Helix Angle 16.0389 deg

Effective Face Width 2.0000 in


Face Load Distribution Factor NA

Tooth Stiffness Constant 1815000.000 psi

Distance from Center of Tooth to Initial 0.4197 in Tooth Contact

Gear Ratio 4.0000

Profile Contact Ratio (Theoretical) 1.5824

Helical Contact Ratio (Theoretical) 2.1057


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Model Title : Program 60-5408 Unit System: US FILM THICKNESS & SCORING PROBABILITY : Crowned Teeth

Actual Width of Contact Ellipse 0.0153 in

Length of Full Contact Ellipse 1.1630 in

Length of Actual Contact Ellipse 1.1630 in


Contact Length NA in

Maximum Contact Width NA in









PINION DATA

Number of Teeth 27





Tooth End to Crown Contact Ellipse With 0.1461 in Crown Centered On Tooth



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Model Title : Program 60-5408 Unit System: US Young's Modulus (Default=steel) 30000000.000 psi





GEAR DATA

Number of Teeth 108





Tooth End to Crown Contact Ellipse With 0.0420 in Crown Centered On Tooth


Young's Modulus (Default=steel) 30000000.000 psi





ROLL ANGLES : Pinion







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Model Title : Program 60-5408 Unit System: US Outside Diameter, Effective 34.403 deg

ROLL ANGLES : Gear






Outside Diameter, Effective 24.342 deg

With the crown on the pinion and Mobil DTE HH we have about a 21% chance of scoring. The film thickness is about 26.48 microinch and the specific film thickness about .75. The addition of crown helped the scoring probability somewhat. This will not always be the case. In some gear sets the addition of crown makes matters worse. (See UTS Model 60-5406.)

This example is not meant to imply that a 20% chance of scoring is suitable for gear sets. This choice must be made by the gear engineer for each application.

References:

1. AGMA 218.01, AGMA Standard for Rating the Pitting Resistance and Bending Strength of Spur and Helical Involute Gear Teeth

2. AGMA 217.01, AGMA Information Sheet - Gear Scoring Design Guide for Aerospace Spur and Helical Power Gears

Data extracted from reference 1 and 2 with the permission of the publisher, the American Gear Manufacturers Association, 1500 King Street, Suite 201, Alexandria, VA 22314

3. Mobil EHL Guidebook, Third Edition Mobil Oil Corporation Commercial Marketing Technical Publications 3225 Gallows Road Fairfax, VA 22037

program 60-5408—crowned and straight external involute gear ehl film … · 2015. 5. 8. ·...

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