Program 60-1111—External Gear Set Tip Relief–Full … · These tolerances are determined by the AGMA quality class of the gear. The tolerances will be calculated by the model

Program 60-1111—External Gear Set Tip Relief–Full Depth Teeth

Introduction At the first point of contact (gear tip and pinion root for a reduction gear) the deflection of the teeth already under load causes the incoming driver tooth to seem to be ahead of where it should be. This insures that the load will be picked up very abruptly at the tip of the driven tooth. (This condition is more serious on spur and LACR helicals than on full helicals.) This applies a heavy shock load at the driven tooth tip where the “cantilever beam” is the longest and produces large root stresses. The same thing happens to the driver tooth at the last point of contact but the effect of dropping the load abruptly is not as severe (mostly noise and vibration along with a rise in compressive stress from the tooth edge breaking the contact “foot print"). From the first point of contact to the pitch point the action is approach action. The driven tooth is approaching the driver tooth and the tip acts much like a sharp edged scraper (or in severe cases, a cutting tool). After contact has passed the pitch point the action is recess action. The driven tooth is retreating from the driver tooth. Recess action is much more conducive to building and maintaining a lubricant film than is approach action, especially at start or end of action. Tooth spacing errors, pitch or profile, add to this effect for those teeth that are “ahead” of where they should be on the driver and “behind” where they should be on the driven. This means that even lightly loaded gears, with little deflection, suffer the same type of problem due to manufacturing error. One method of reducing the deflection problem (or at least not making it worse) is to make the tolerances on profile such that the base pitch of the driver can never be less than theoretical and the base pitch of the driven never more than theoretical. This will reduce the impact at the first point of contact because it will insure that the teeth on the driver are a little “behind” and on the driven a little “ahead". Of course, this will make things worse at the last point of contact but conditions there are much less severe. The practical application of this method means that the profile tolerance on the driver should always be plus at the tip and on the driven always minus. As long as we have to live with these tolerances we may as well put them where they may help us and, at least, not hurt us. Of course, this must not be overdone as we will lose conjugate action. This depends upon the load level, gear quality class, modulus of elasticity, and tooth pressure angle. The model will split the profile tolerance between plus and minus if the gear is AGMA Q9 or less or if the gear is an idler.

UTS Integrated Gear Software

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In addition to tolerance control the application of tip relief on the gears will reduce the engagement shock and the tendency to scrape the driver root. (This will also help the abrupt dropping of load at the last point of contact.) The load can then be picked up smoothly and the full load (plus shock) on the tooth tips eliminated. Since the tip relief can be built into the production tools the advantages cost us very little except getting the specifications right at the design stage. One of the parameters used in determining the proper amount of tip relief is the allowable tolerance on the elements of the gear (pitch tolerance, profile tolerance and lead tolerance). These tolerances are determined by the AGMA quality class of the gear. The tolerances will be calculated by the model if the quality class is entered. If one or more tolerance is not to be according to the quality class the tolerance can be overridden be entering single values in the input column. Examples Example 1 The first example will establish the tip relief specifications for the following steel spur gear set: 22 tooth driver 55 tooth driven 10 normal diametral pitch 20 degree normal pressure angle 2 inch face width Quality class Q12–ANSI/AGMA 2000-A88 Gear Handbook 100 Horsepower at 3600 RPM driver speed Open a new analysis in 60-1111 and follow the sequence of the input form, which is shown in Figure 1. Use the calculated defaults where they appear. Form diameters will not be entered. Neither gear is an idler. Select power instead of driver torque. Select total lead mismatch between teeth instead of AGMA load distribution factor. If the lead mismatch between the teeth is not known it can be calculated using UTS Model 60-102, “Total Lead Mismatch for Gear Pair in Housing". We will use a lead mismatch of 0.0008 inch for these gears. The unmodified profile contact ratio is set to 1.05 to avoid a profile contact less than unity at light loads when the modified portions of the teeth are not in contact. “Base pitches unmodified from mid-point” are set to .525 to split the unmodified portion of

60-1111—External Gear Set Tip Relief—Full Depth Teeth

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the line of action about evenly between the driver and driven. (The actual arc length of the modified portions of the teeth can be changed later if desired.) Select tip relief on both gears. Input and output values are shown in Report 1. Fig. 1


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Report 1

Model Title : Program 60-1111 Unit System: US

MESSAGE Driver

m2 & Driven

m3 Tip

m4 Relief

DRIVER, number of teeth 22

DRIVEN, number of teeth 55

NORMAL PLANE

Diametral pitch 10.000000 1/in `

Pressure angle 20.000000 deg

Module 2.540000 mm `

Base pitch 0.2952 in

TRANSVERSE PLANE



Module 2.5400 mm `


COMMON

Helix angle 0.000000 deg

Base helix angle 0.0000 deg

Operating center distance 3.8500 in


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Model Title : Program 60-1111 Unit System: US Face width 2.0000 in

DIAMETERS

Outside diameter, Driver 2.5000 in

Base Diameter, Driver 2.06732 in

Outside diameter, Driven 5.6000 in

Base Diameter, Driven 5.16831 in

OPERATING DATA

Working depth 0.2000 in

Operating transverse pressure angle 20.0000 deg

Operating PD, Driver 2.2000 in

Form diameter (TIF) 2.1168 in

Driver is an idler n

Operating PD, Driven 5.5000 in


Driven is an idler n

Roll angle at operating PDs 20.854 deg

Length of contact, transverse plane 0.4641 in

Profile 1.5722

Helical 0.0000

Total 1.5722

LOADS

Power 100.00 HP

Driver Torque 1750.69 lbf-in

Driver Speed 3600.0 rpm


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Model Title : Program 60-1111 Unit System: US Tangential pitch line load 1591.54 lbf

Line of action load 1693.68 lbf

Tangential load per unit of face 795.77 lbf/in

Line of action load per unit of face 846.84 lbf/in

K factor 506.40 psi

Unit load 7957.70 psi

MATERIAL

Modulus of elasticity, Driver 30000000.00 psi

Modulus of elasticity, Driven 30000000.00 psi

MESH DEFLECTION (FULL DEPTH TEETH)

Dudley: Tangent to PD, trans plane 0.00027 in

Dudley: Normal to tooth 0.00026 in

AGMA Paper 109.16: Normal to tooth 0.00042 in

AGMA Std 2001: Normal to tooth 0.00048 in

TIP RELIEF AMOUNT DRIVER TOLERANCE

AGMA Q Class 12

Pitch error, Driver 0.00018 in

Profile error, Driver (+ tip) 0.00023 in

Lead error, Driver 0.00034 in

TIP RELIEF AMOUNT DRIVEN TOLERANCE

AGMA Q Class 12

Pitch error, Driven 0.00021 in

Profile error, Driven (- tip) 0.00027 in

Lead error, Driven 0.00034 in


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Model Title : Program 60-1111 Unit System: US TIP RELIEF AMOUNT DUDLEY (GENERAL DESIGN)

AGMA Load Distribution Factor 1.886

Total lead mismatch between teeth 0.00080 in

Tooth stiffness constant 1762000.000 psi

AGMA Face Load Distribution Factor 1.886

AGMA Trans Load Distribution Factor 1.000

Driver tip - last point of contact 0.00042 in

Driven tip - first point of contact 0.00068 in

TIP RELIEF AMOUNT AGMA PAPER 109.16 (PRECISION GEARS)


S_last2 0.00058 in


S_frst2 0.00088 in

TIP RELIEF AMOUNT BASED ON DEFLECTION AND ERRORS

Trans deflection - AGMA Std 2001 0.00048 in





TIP RELIEF LOCATION (NO UNDERCUT)

Tip relief on both gears, 'tb, tb

Unmodified profile contact ratio 1.0500


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Model Title : Program 60-1111 Unit System: US TIP RELIEF LOCATION (NO UNDERCUT) DRIVER

Start of active profile 2.1218 in

Roll angle at SAP 13.2339 deg

Roll angle at TIF 12.6038 deg

Mid-point of length of contact 2.2717 in

Base pitches unmod from mid-point 0.5250

Roll angle at mid-point 26.097 deg

Start tip relief 2.4167 in

Roll angle at start of tip relief 34.688 deg

Arc length of tip relief 0.0495 in

Roll angle - tip relief to OD 4.273 deg

Roll angle at OD 38.961 deg

TIP RELIEF LOCATION (NO UNDERCUT) DRIVEN













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Model Title : Program 60-1111 Unit System: US TIP AND ROOT RELIEF ON DRIVER

Start root relief in

Roll angle at start root relief deg

TIP AND ROOT RELIEF ON DRIVEN



The model offers guidance on tip relief from three different sources.

The first is from Dudley's Handbook of Practical Gear Design and suggests that we need about 0.00042 inch at the driver (pinion) tip and about 0.00068 inch at the driven (gear) tip. This is for general design and Mr. Dudley suggests a more detailed analysis for critical drives.

The second is from AGMA Paper 109.16 by Hans Sigg and recommends 0.00028/0.00058 (inches) at the driver (pinion) tip and about 0.00058/0.00088 at the driven (gear) tip. This data is intended for precision ground gears only (AGMA Q12 and up).

The third set of corrections is based on the actual base pitch errors allowed by the tolerances and on the deflections according to the tooth stiffness from AGMA Standard 2001-B88. The values of 0.00058 inch correction for the driver (pinion) tip and 0.00065 inch for the driven (gear) tip are for the mean average error condition with a normal deflection of 0.00048 inch. This correction would leave about 20% load at the first point of contact.

The fourth set of corrections is also based on the actual base pitch errors allowed by the tolerances and on the deflections according to the tooth stiffness from AGMA Std 2001-B88. The values of 0.00066 inch at the driver tip and 0.00096 inch at the driven tip would be required to remove all load at the first point of contact with the Root-Mean-Square errors plus the deflection. The RMS errors should cover more than 95% of gears.

After the proper amount and location of the tooth modifications have been established it is necessary to transfer this data to the engineering drawings. This is usually done by using “K charts” on the drawing. These charts usually plot the required and allowed deviation from the true involute form for the tooth. Involute checking machines produce a chart which shows the deviation in form plotted against


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the roll angle for the gear. A straight line indicates no deviation from the involute form. This model includes these plots for driver and driven for all four sets of relief calculations included in the model. Go to the TK Solver Plot Sheet, or the pull-down list of plots on the Toolbar, and select the pair of plots you wish to use.

Plot name Tip/root relief method

dr_Dudley Dudley's Handbook-Driver dn_Dudley Dudley's Handbook-Driven

dr_2001_20 AGMA Std 2001 - 20% Load-Driver - Ave Errors dn_2001_20 AGMA Std 2001 - 20% Load-Driven - Ave Errors

dr_2001_0 AGMA Std 2001 - 0% Load-Driver - RMS Errors dn_2001_0 AGMA Std 2001 - 0% Load-Driven - RMS Errors

dr_AGMA_109 AGMA Paper 109-Driver dn_AGMA_109 AGMA Paper 109-Driven

Figures 2 and 3 are the plots from our example for the AGMA Standard 2001 calculations for no load at the first point of contact using the Root-Mean-Square (RMS) values of the allowable tooth errors. (To view plots, toggle to TK Solver and select the plots from the Plots List in the Object Bar.)

Fig. 2


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Fig. 3

NOTE: You may change the vertical scale on these plots by changing the Y-Axis range on the plot subsheet. With the pointer on the plot, right-click the mouse and select “Display Subsheet.” In some cases it may be advantageous to use tip and root relief on a single gear instead of tip relief on both gears. This may occur, for example, with the sun and planet and ring gears in an epicyclic gear set. If tip and root relief is used on the planet gear it will provide relieved contact at the sun and ring gears without any modification to the sun and ring. If root relief is used for any reason it is then necessary to properly locate the diameter at which the root relief would start. Figure 4 and Report 2 show the same model with tip relief on both gears changed to tip and root relief on the driver. The plots for this solution are shown in Figures 5 and 6.


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Fig. 4


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Report 2


MESSAGE Driver

m2 Tip &

m3 Root

Model Title : Program 60-1111 Unit System: US m4 Relief



NORMAL PLANE





TRANSVERSE PLANE



Module 2.5400 mm `


COMMON



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Model Title : Program 60-1111 Unit System: US Base helix angle 0.0000 deg


Face width 2.0000 in

DIAMETERS





OPERATING DATA











Profile 1.5722

Helical 0.0000

Total 1.5722

LOADS

Power 100.00 HP


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Model Title : Program 60-1111 Unit System: US Driver Torque 1750.69 lbf-in


Tangential pitch line load 1591.54 lbf




K factor 506.40 psi


MATERIAL






AGMA Paper 109.16: Normal to tooth 0.00042 in



AGMA Q Class 12





AGMA Q Class 12



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Model Title : Program 60-1111 Unit System: US Profile error, Driven (- tip) 0.00027 in


TIP RELIEF AMOUNT DUDLEY (GENERAL DESIGN)










S_last2 0.00058 in


S_frst2 0.00088 in








Tip relief on both gears, 'tb, trdr



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18


Start root relief 2.1617 in

Roll angle at start root relief 18 deg



Roll angle at start root relief deg Fig. 5


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Fig. 6

NOTE: The deflection given by AGMA 109.16 is for steel only and if the modulus of elasticity for either gear is other than for steel the deflection will not be calculated. (The tip relief amounts also will not be calculated.)


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Example 2 The second example is a 23-degree helical gear set made of powdered metal with a modulus of elasticity of 20,000,000 psi. The pinion has 19 teeth and the gear 45. The normal diametral pitch is 26 with a 20-degree pressure angle. We will establish the tip relief specifications for these gears with average errors and with 20% load left at the first point of contact. Figure 7 and Report 3 are the inputs and outputs for this model. Fig. 7


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Report 3


MESSAGE Driver

m2 & Driven

m3 Tip

m4 Relief



NORMAL PLANE





TRANSVERSE PLANE



Module 1.0613 mm `


COMMON


Base helix angle 21.5410 deg



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Model Title : Program 60-1111 Unit System: US Face width 0.3100 in

DIAMETERS





OPERATING DATA











Profile 1.4844

Helical 1.0025

Total 2.4868

LOADS

Power 0.62 HP

Driver Torque 22.33 lbf-in



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Model Title : Program 60-1111 Unit System: US Tangential pitch line load 56.25 lbf




K factor 325.06 psi


MATERIAL






AGMA Paper 109.16: Normal to tooth Q12 Steel in



AGMA Q Class 9





AGMA Q Class 9


Profile error, Driven (- tip) 0.00041 in



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Model Title : Program 60-1111 Unit System: US TIP RELIEF AMOUNT DUDLEY (GENERAL DESIGN)









Driver tip - last point of contact NA in

S_last2 NA in

Driven tip - first point of contact NA in

S_frst2 NA in








Tip relief on both gears, 'tb, tb



25


























26






Roll angle at start root relief deg Figures 8 and 9 are the plots from our example for the AGMA Std 2001 deflection for 20% load at the first point of contact using the average values of the allowable tooth errors. Note that at this quality class the profile error is + or - instead of + tip for the driver and - tip for the driven. Fig. 8


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Fig. 9

The model checks various items and, if any problem is found, it will appear in the “MESSAGE” section of the report, along with advisory messages along the way. All equations and methods used are on the Rule Sheet and in the functions of the TK Solver model. References:

Data extracted from AGMA Standard 2001-B88, Fundamental Rating Factors and Calculation Methods for Involute Spur and Helical Gear Teeth, and from AGMA Paper 109.16, Profile and Longitudinal Corrections on Involute Gears, with the permission of the Publisher, the American Gear Manufacturers Association, 1500 King Street, Suite 201, Alexandria, VA 22314. Handbook of Practical Gear Design, 2nd Edition, by Darle W. Dudley, Dudley Engineering Co., Published by McGraw-Hill, Inc, 1984.

Documents

Program 60-1111—External Gear Set Tip Relief–Full … · These tolerances are determined by the AGMA quality class of the gear. The tolerances will be calculated by the model