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AGMA 913- A98 Reaffirmed May 3, 2016

AMERICAN GEAR MANUFACTURERS ASSOCIATION

AGMA 913- A98

Method for Specifying the Geometry of Spur and Helical Gears

AGMA INFORMATION SHEET (This Information Sheet is NOT an AGMA Standard)

Method for Specifying the Geometry of Spur and Helical Gears American AGMA 913--A98 Gear Manufacturers CAUTION NOTICE: AGMA technical publications are subject to constant improvement, revision, or withdrawal as dictated by experience. Any person who refers to any AGMA Association Technical Publication should be sure that the publication is the latest available from the Association on the subject matter. [Tables or other self--supporting sections may be quoted or extracted. Credit lines should read: Extracted from AGMA 913--A98, Method for Specifying the Geometry of Spur and Helical Gears, with the permission of the publisher, the American Gear Manufacturers Association, 1500 King Street, Suite 201, Alexandria, Virginia 22314.] Approved March 13, 1998

ABSTRACT This information sheet provides information to translate tooth thickness specifications which are expressed in terms of tooth thickness, center distance or diameter into profile shift coefficients, as that term is used in international standards. Published by

American Gear Manufacturers Association 1500 King Street, Suite 201, Alexandria, Virginia 22314 Copyright 1998 by American Gear Manufacturers Association All rights reserved. No part of this publication may be reproduced in any form, in an electronic retrieval system or otherwise, without prior written permission of the publisher.

Printed in the United States of America ISBN: 1--55589--714--2

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AGMA 913--A98

Contents Page

Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv 1

Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2

Terms and symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

3

Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

4

Profile shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

5

Internal gear pair calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Tables 1

Symbols used in equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2

Obsolete terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Figures 1

The basic rack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2

Hypothetical tool . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3

Profile shift of a helical gear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

4

Effect of profile shift on involute tooth profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

5

Distances along the line of action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

6

Root radii cut with rack tool . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

7

Distances along the line of action for an internal gear pair . . . . . . . . . . . . . . . 12

Annexes A

Tool proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

B

Calculation of profile shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

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AGMA 913--A98

AMERICAN GEAR MANUFACTURERS ASSOCIATION

Foreword [The foreword, footnotes and annexes, if any, in this document are provided for informational purposes only and are not to be construed as a part of AGMA Information Sheet 913--A98, Method for Specifying the Geometry of Spur and Helical Gears.] This information sheet is intended to provide sufficient information to allow its users to be able to translate tooth thickness specifications which are expressed in terms of tooth thickness, center distance or diameter into profile shift coefficients, as that term is used in international standards. This AGMA information sheet and related publications are based on typical or average data, conditions or application. AGMA 913--A98 was approved by the AGMA membership on March 13, 1998. Suggestions for improvement of this standard will be welcome. They should be sent to the American Gear Manufacturers Association, 1500 King Street, Suite 201, Alexandria, Virginia 22314.

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AGMA 913--A98

PERSONNEL of the AGMA Nomenclature Committee Chairman: John R. Colbourne . . . . . . . University of Alberta Vice Chairman: D. McCarroll . . . . . . . . . Gleason Works

ACTIVE MEMBERS W.A. Bradley III . . . . . . . . . . . . . . . . . . . . R.L. Errichello . . . . . . . . . . . . . . . . . . . . . . D. Gonnella . . . . . . . . . . . . . . . . . . . . . . . . D.R. McVittie . . . . . . . . . . . . . . . . . . . . . . . O.A. LaBath . . . . . . . . . . . . . . . . . . . . . . . I. Laskin . . . . . . . . . . . . . . . . . . . . . . . . . . . G.W. Nagorny . . . . . . . . . . . . . . . . . . . . . . J.W. Polder . . . . . . . . . . . . . . . . . . . . . . . . L.J. Smith . . . . . . . . . . . . . . . . . . . . . . . . . R.E. Smith . . . . . . . . . . . . . . . . . . . . . . . . .

Consultant GEARTECH Texaco Lubricants Company Gear Engineers, Inc. Cincinnati Gear Company Irving Laskin, P.E. Nagorny & Associates Delft University of Technology Invincible Gear Company R.E. Smith & Co., Inc.

ASSOCIATE MEMBERS K. Acheson . . . . . . . . . . . . . . . . . . . . . . . . M. Allard . . . . . . . . . . . . . . . . . . . . . . . . . . M.R. Chaplin . . . . . . . . . . . . . . . . . . . . . . . A.S. Cohen . . . . . . . . . . . . . . . . . . . . . . . . L. Faure . . . . . . . . . . . . . . . . . . . . . . . . . . . R. Green . . . . . . . . . . . . . . . . . . . . . . . . . . T. Okamoto . . . . . . . . . . . . . . . . . . . . . . . .

The Gear Works -- Seattle, Inc. UNITRAM Contour Hardening, Inc. Engranes y Maquinaria CMD Eaton Corporation Nippon Gear

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AMERICAN GEAR MANUFACTURERS ASSOCIATION

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American Gear Manufacturers Association --

AGMA 913--A98

Annexes A and B provide practical examples on the calculation of tool proportions and profile shift.

Method for Specifying the Geometry of Spur and Helical Gears

2 Terms and symbols 2.1 Terms The terms used, wherever applicable conform to the following standards. ISO 701:1998, International gear notation ---Symbols for geometrical data ANSI/AGMA 1012--F90, Gear Nomenclature, Definitions of Terms with Symbols

1 Scope This information sheet provides a general method for specifying profile shift and rack shift, with gear nomenclature and definitions. This document describes the effect that profile shift has on the geometry and performance of gears, but does not make specific design recommendations.

2.2 Symbols This information sheet uses the ISO symbols in table 1. In cases where there are no ISO symbols, or the definitions are different, other symbols are used.

The equations in the first part of this document (clauses 3 and 4) apply to external gear pairs only. The corresponding equations for internal gear pairs are contained in clause 5.

NOTE: The symbols, definitions and terminology used in this information sheet may differ from other AGMA publications. The user should not assume that familiar symbols can be used without a careful study of these definitions.

Table 1 -- Symbols used in equations ISO Symbols

aw c ci2 d ha0 ha1, ha2 haP0

Other Symbols C1 C2, C3, C4 C5 C6 YJ1, YJ2 aref

Terms Distance to SAP Distances along line of action Distance to EAP Distance between interference points Bending strength geometry factor, pinion and gear Reference center distance Operating center distance Root clearance Required clearance at the tooth root of the internal gear Diameter Addendum of the tool Addendum, pinion and gear Distance on the cutting tool from the reference line to the point near the tooth tip where the straight part of the profile ends and the circular tip begins

Units mm mm mm mm ---mm mm mm mm mm mm mm mm

Where first used Eq 23 Eq 27 Eq 23 Eq 23 Eq 30 Eq 5 Eq 16 Eq 42 Eq 70 Eq 4 Eq 22 Eq 34 Eq 21

(continued)

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Table 1 (concluded) ISO Other Terms Symbols Symbols jn Normal operating circular backlash k Tip--shortening coefficient m Module p Circular pitch r1, r2 Reference radius, pinion and gear ra1, ra2 Outside radius, pinion and gear rb1, rb2 Base circle radius, pinion and gear rf1, rf2 Root radius, pinion and gear rfP1 Radius of the pinion fillet circle s Tooth thickness sn1, sn2 Reference normal circular tooth thickness, pinion and gear u Gear ratio ∫ 1.0 xE1, xE2 Generating rack shift coefficient, pinion and gear x1, x2 Profile shift coefficient, pinion and gear x1min Minimum pinion profile shift coefficient to avoid undercut y Profile shift yE Rack shift y1min Minimum pinion profile shift to avoid undercut z1, z2 Number of teeth, pinion and gear ∼n Reference normal pressure angle ∼t Reference transverse pressure angle ∼wt Operating transverse pressure angle ϒ Reference helix angle ∆aref Center distance modification ∆sn Amount of tooth thinning ∆sn1, ∆sn2 Tooth thinning for backlash, pinion and gear ±a0 Radius of the circular tip of the tool ±fP Fillet radius of the basic rack Σx Sum of profile shift coefficients ΣxE Sum of generating rack shift coefficients σF1, σF2 Allowable bending stress, pinion and gear Subscript conversion (none) At reference diameter a At addendum (tip) diameter b At base cylinder diameter f At root diameter n Normal plane t Transverse plane w Operating, running or working y At any (undefined) diameter Tool dimensions 0 1 Pinion 2 Gear or rack

2

Units mm ---mm mm mm mm mm mm mm mm mm ------------mm ---mm ---------------mm mm mm mm mm ------MPa

Where first used Eq 31 Eq 32 Eq 3 Eq 1 Eq 5 Eq 25 Eq 13 Eq 42 Eq 60 Eq 2 Eq 7 Eq 10 Eq 46 Eq 6 Eq 20 Eq 6 Eq 8 Eq 20 Eq 4 Eq 7 Eq 9 Eq 9 Eq 1 Eq 32 Eq 8 Eq 31 Eq 22 Fig 1 Eq 53 Eq 52 Eq 30

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3 Definitions

Table 2 -- Obsolete terms

This information sheet provides definitions of profile shift and rack shift and shows the relations between them and other gear quantities. The terms profile shift and rack shift are used in this information sheet for standardization. The intention is to replace the similar or related terms listed in table 2. 3.1 Basic rack The standard basic rack tooth profile is the tooth profile normal section through the teeth of a basic rack which corresponds to an external gear with number of teeth z =  and diameter d = , see figure 1. A gear with normal module, mn , and normal pressure angle, ∼n , has a basic rack whose normal circular pitch, pn , is πmn and whose normal pressure angle is ∼n . The reference plane of the basic rack is parallel to its tooth tip plane and is the plane on which the normal circular tooth thickness, sn , is equal to one half the normal circular pitch. From this definition it follows that the normal circular tooth thickness on the reference plane is equal to the normal circular space width.

Addendum correction Addendum elongation Addendum increment Addendum modification Addendum ratio Basic rack offset Cutter offset Delta addendum Delta teeth Drop--tooth design Enlargement/reduction Enlarged/reduced center distance Enlarged/reduced number of teeth Half pitch hob pull High/low addendum Hob offset Hob pull Increase/decrease Involute shift Long/short addendum Nonstandard addenda Over/undersize Profile displacement Profile withdrawal Rack withdrawal Tool shift Tool withdrawal Unequal addenda X factor

Normal ∼n pressure angle

π mn 2 Reference line π mn 2 ± fP

hwP

haP0

Dedendum hfP

Addendum haP

Normal base pitch, pbn

Normal circular thickness, sn cP

Normal circular pitch, pn = π mn Figure 1 -- The basic rack

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If the basic rack is oriented so that its teeth make an angle β with the gear axis, see figure 3, the section through the basic rack perpendicular to the gear axis is called the transverse section. On this section, the transverse circular pitch, pt , and the transverse tooth thickness, st , are given by: pn cos β s st = n cos β

pt =

The pitch plane of the basic rack is parallel to the reference plane and is the plane that is tangent to the reference cylinder of the gear. The helix angle of the gear at its reference cylinder is equal to β. 3.3 Reference center distance The reference center distance of an external gear pair is defined as half the sum of the reference diameters.

...(1) ...(2)

and the transverse module, mt , is defined by: m mt = n cos β

a ref =

...(3)

The basic rack represents the theoretical gear tooth form, not the form of the cutting tool. No allowance is made for backlash, finishing stock or manufacturing method. The standard 20 normal pressure angle basic rack of ISO 53 is commonly used. This document is valid for that basic rack and for any other basic rack which meets the criteria of figure 1.

Ꮛd1 + d2Ꮠ 2

= r 1 + r2

...(5)

where r1 and r2 are the radii of the reference cylinders. The reference center distance is not necessarily equal to the operating center distance. It is one of the advantages of involute gears, that the operating center distance can vary from the reference center distance without change in operation. 3.4 Hypothetical tool

3.2 Reference cylinder of the gear (standard pitch cylinder) The reference cylinder of a gear is defined as the pitch cylinder where circular pitch of the gear is equal to circular pitch of the basic rack. If the gear has z teeth and the rack is oriented with its teeth making an angle β with the gear axis, then the diameter, d, of the reference cylinder is given by z mn cos β

The use of the phrase “hypothetical tool” in this document refers to a rack--type cutter. For additional information and an example calculation, see annexes A and B.

π mn 2 Reference line

Hypothetical tool tooth haP0

Addendum ha0

haP

...(4)

Dedendum hf0

d=

The hypothetical tool is the complement of the basic rack as shown in figure 2. The reference line of the hypothetical tool is the line at which its normal circular tooth thickness is equal to π m n . 2

π mn 2

±a0 ∼n

Profile angle

Normal circular pitch, pn Figure 2 -- Hypothetical tool

4

Basic rack

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AGMA 913--A98

3.5 Zero backlash gear pair

3.8 Tooth thickness

A zero backlash gear pair is one which operates in tight mesh (has no backlash) on the operating center distance.

The normal circular tooth thickness, sn , of the zero backlash gear at its reference cylinder is equal to the normal circular space width of the hypothetical tool at its pitch plane when in tight mesh with the zero backlash gear.

3.6 Profile shift The profile shift, y, of the gear is defined as the amount by which the reference plane of the hypothetical tool (conjugate to the basic rack) is displaced from the reference cylinder of the gear. In other words, the gear has profile shift y if the reference plane of the hypothetical tool lies a distance (r + y) from the gear axis, where r is half the diameter, d. Profile shift, y, can be either plus or minus depending on whether the profile shift is to the outside or to the inside of the reference diameter. See figure 3. 3.7 Profile shift coefficient The profile shift coefficient, x, of the gear is defined as the profile shift divided by the normal module. y x=m

...(6)

n

Hypothetical tool

s n = 1 π mn + 2y tan αn 2

...(7)

NOTE: Equations 7, 8 and 9 are for external gears only. The corresponding equations for internal gears are given in 5.1.

3.9 Rack shift It is customary to first choose the tooth thicknesses in a gear pair, assuming there is no backlash, and to then reduce the tooth thicknesses to allow for backlash. The phrase “profile shift” will be used for the value of y corresponding to the tooth thickness before thinning, and the phrase “rack shift” for the value of y after thinning. Since the rack shift determines the actual tooth thickness at the time of cutting or generating, the symbol yE is used for the rack shift and xE for the rack shift coefficient. If the amount of thinning is ∆sn , the relationship between rack shift and profile shift is:

Normal plane Hypothetical tool reference line Gear reference pitch cylinder Profile shift “y”

Helix angle β

r Basic rack in tight mesh with zero backlash gear

Transverse plane

Base circle

Zero backlash gear without tooth thinning

Figure 3 -- Profile shift of a helical gear

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AGMA 913--A98

yE = y −

∆ sn 2 tan α n

AMERICAN GEAR MANUFACTURERS ASSOCIATION

...(8)

If two external gears are to mesh with no backlash, their profile shift values must satisfy: y1 + y2 =

aref (inv α wt − inv α t) tan α t

...(9)

where aref is the reference center distance; ∼wt

is the operating transverse pressure angle;

∼t

is the reference transverse pressure angle.

3.10 Addendum values The gear addendum, measured from the reference cylinder, is usually chosen as (haP + y). This value depends on the profile shift rather than the rack shift and is therefore independent of the value chosen for backlash. In certain designs, particularly when the center distance is significantly larger than the reference standard center distance, the gear addendum may need to be reduced to allow adequate clearance at the roots of the meshing gear, see 4.10. For internal gear pair equations which replace equations 7 through 9, see 5.1.

4 Profile shift 4.1 Profile shift calculation Profile shift is selected considering the following criteria: --

avoiding undercut;

--

avoiding narrow top lands;

--

balanced specific sliding;

--

balanced flash temperature;

--

balanced bending fatigue life.

The profile shift should be large enough to avoid undercut and small enough to avoid narrow top lands. The profile shifts required for balanced specific sliding, balanced flash temperature and balanced bending fatigue life are usually different. Therefore, the value used should be based on the criterion that is judged to be the most important for the particular application.

6

Figure 4 illustrates how the shape of a gear tooth is influenced by the number of teeth on the gear and the value of the profile shift coefficient. The influence that the number of teeth has on tooth form can be seen by viewing the teeth within any given column of figure 4. With small numbers of teeth, the tooth has larger curvature and the relative thickness of the teeth at the topland and at the form diameter is smaller. As the number of teeth increases, the topland and tooth thicknesses increase and the curvature of the profiles decrease. Tooth thicknesses are maximum for a rack with straight--sided profiles and theoretically infinite number of teeth. Viewing figure 4 horizontally within any given row shows how profile shift changes tooth form. Rows near the top of figure 4 show that gears with few teeth have a tooth form that depends strongly on the value of the profile shift coefficient. For gears with few teeth, the sensitivity to profile shift narrows the choice for profile shift coefficient because too little profile shift results in undercut teeth, whereas too much profile shift gives teeth with toplands that are too narrow. For example, the acceptable values of profile shift coefficient for a 12 tooth gear range from x = 0.4 near undercut, to x = 0.44 for a topland thickness equal to 30% of the module. In contrast, rows near the bottom of figure 4 show that gears with large numbers of teeth are relatively insensitive to profile shift. This means that the gear designer has wider latitude when choosing profile shift for gears with a large number of teeth. As a limiting case, the shape of the teeth of a rack are independent of profile shift. Generally, the performance of a gear is enhanced with increasing numbers of teeth and the optimum value of profile shift. For a fixed gear diameter, with the exception of bending strength, load capacity is increased when the number of teeth increases and the profile shift is designed properly. Resistance to macropitting, adhesive wear and scuffing is improved and the gears usually operate more quietly. The maximum number of teeth is limited by bending strength because a large number of relatively small teeth have high bending stresses. Therefore, the gear designer must limit the number of teeth in the pinion based on maintaining adequate bending strength. Load capacity can be maximized by balancing the pitting resistance and the bending

AMERICAN GEAR MANUFACTURERS ASSOCIATION

strength of the gearset (see AGMA 901--A92). A balanced design has a relatively large number of teeth in the pinion. This makes the gearset relatively insensitive to profile shift, and allows the

AGMA 913--A98

designer to select the profile shift to minimize specific sliding, minimize flash temperature or balance the bending fatigue life of the pinion and gear.

12

Number of teeth

15

20

30

50

100

--0.4

0.0 0.4 Profile shift coefficient

0.8

Figure 4 -- Effect of profile shift on involute tooth profiles

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The minimum profile shift coefficient (to avoid undercut) for the pinion is given by:

4.2 Basic gear geometry z u = z2 , 1

where z2 ≥ z 1

z1 mn 2 cos β z mn = r1 u r2 = 2 2 cos β r1 =

...(10) ...(11) ...(12) ...(13)

r b2 = r2 cos ∼ t = rb1 u

...(14)

Ꮛtancos ∼β Ꮠ n

Ꮛa

∼ wt = arccos

ref

...(15)

Ꮠ

cos ∼ t aw

...(16)

...(21)

haP0 is the distance on the cutting tool tooth from the reference line to the point near the tool tooth tip where the straight part of the profile ends and the circular tip begins. ...(22) h aP0 = ha0 − a0 + a0 sin ∼ n where ha0

is the addendum of the tool;

±a0

is the radius of the circular tip of the tool.

inv ∼ t = tan ∼ t − ∼ t

...(17)

4.5 Avoiding narrow top lands

inv ∼ wt = tan ∼ wt − ∼ wt

...(18)

The maximum permissible profile shift coefficients are obtained by iteratively varying the profile shift coefficients of the pinion and gear until their top land thicknesses are equal to the minimum allowable.

4.3 Sum of profile shift coefficients for zero backlash NOTE: The equations to follow in this section are for external gear pairs only. The corresponding equations for internal gear pairs are given in 5.2.1.

x1 + x2 =

aref (inv ∼ wt − inv ∼ t) mn tan ∼ t

...(19)

4.4 Avoiding involute undercut teeth There are a number of design options to compensate for undercut teeth, including profile shift. Undercut is a condition in generated gear teeth where any part of the fillet curve lies inside a line drawn tangent to the working profile at its point of juncture with the fillet. For such gears, the end of the cutting tool has extended inside of the point of tangency of the base circle and the line of action, and removed an excessive amount of material. This removal of material can weaken the tooth and also may reduce the length of contact, since gear action can only take place on the involute portion of the flank. Should a gear be made by another method that would not undercut the flanks, there may be interference of material and generally the gear would not mesh or roll with another gear. See AGMA 908--B89, Geometry Factors for Determining the Pitting Resistance and Bending Strength of Spur, Helical and Herringbone Gear Teeth.

8

...(20)

where

r b1 = r1 cos ∼ t

∼ t = arctan

y 1 min mn y 1 min = haP0 − r 1 sin2 ∼ t x 1 min =

4.6 Balanced specific sliding Specific sliding is defined as the ratio of the sliding velocity to rolling velocity at a particular point of contact on the gear of interest. Maximum pitting and wear resistance is obtained by balancing the specific sliding at each end of the path of contact. This is done by iteratively varying the profile shift coefficients of the pinion and gear until the following equation is satisfied:

ᏋCC

6 1

− 1

ᏐᏋCC

6 5

Ꮠ

− 1 = u2

...(23)

where C6

is the distance between interference points (see figure 5);

C1

is the distance to SAP (see figure 5);

C5

is the distance to EAP (see figure 5).

C 6 = Ꮛrb1 + rb2Ꮠ tan ∼ wt = a w sin ∼ wt C 1 = C6 −

Ꭹr2a2

C 5 = Ꭹr2a1 − r2b1

− r2b2

...(24) ...(25) ...(26)

C 2 = C5 − pbt

...(27)

C 3 = rb1 tan ∼ wt

...(28)

C 4 = C1 + pbt

...(29)

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∼wt

AGMA 913--A98

rb2

ra2 Line of action

aw

pbt EAP

pbt

HPSTC P

LPSTC SAP

ra1

C1 C2 C3 rb1

C4 C5

C6

Figure 5 -- Distances along the line of action for external gear pair 4.7 Balanced flash temperature According to Blok’s theory, the maximum scuffing resistance is obtained by minimizing the contact temperature. This is done by iteratively varying the profile shift coefficients of the pinion and gear, while calculating the flash temperature by Blok’s equation (see annex A of ANSI/AGMA 2101--C95, Fundamental Rating Factors and Calculation Methods for Involute Spur and Helical Gear Teeth), until the flash temperature peaks in the approach and recess portions of the line of action are equal. The flash temperature should be calculated at the points SAP, LPSTC, HPSTC, EAP and at several points in the

two pair zones (between points SAP and LPSTC and between points HPSTC and EAP, see figure 5). 4.8 Balanced bending strength Maximum bending resistance is obtained by iteratively varying the profile shift coefficients of the pinion and gear until the ratio of the bending strength geometry factors equals the ratio of allowable bending stresses, i.e., Y J1 σ = σF2 Y J2 F1

...(30)

See ANSI/AGMA 2101--C95, clause 5.2 through 5.2.3, for an explanation of YJ1, YJ2, σF1 and σF2.

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4.9 Tooth thinning for backlash

4.10.1.2 Full working depth -- option 2

The small adjustments of the position of the cutting tool to thin the gear teeth for backlash are considered independently of the profile shift coefficients (x1 and x2) by specifying the amount the pinion and gear teeth are thinned for backlash, ∆sn1 and ∆sn2. This way, the outside diameters are independent of tooth thinning for backlash. The total thinning coefficients are selected such that:

Ꮛ Ꮠ

a ∆ s n1 + ∆ s n2 = j n aref w

...(31)

where jn





 

h a1 = 1 + x 1 − 1 k m n 2 h a2 = 1 + x 2 − 1 k m n 2

...(36) ...(37)

CAUTION: Option 2 (full working depth) may give insufficient tip--to--root clearance if aw & aref. Check clearances or use option 3 (full tip--to--root clearance) to be safe.

4.10.1.3 Full tip--to--root clearance -- option 3 h a1 = Ꮛ1 + x 1 − kᏐ m n

...(38)

h a2 = Ꮛ1 + x 2 − kᏐ m n

...(39)

4.10.2 Root radius and clearance is normal operating circular backlash

A common convention among gear manufacturers is to reduce the normal tooth thickness of each member by the same amount, which may be a value in m or a function of the normal module, such as 0.024mn . This maintains the same whole depth for both members. However, for other directions of tooth thickness measurement, see ANSI/AGMA 2002--B88.

Root radii (cut with rack tool). See figure 6. r f1 = r1 − ha01 + x E1 m n

...(40)

r f2 = r2 − ha02 + x E2 m n

...(41)

The root clearances are: c 1 = aw − rf1 − ra2 c 2 = aw − rf2 − ra1

k = x1 + x2 −

∆ a ref mn

...(32)

...(43)

tool reference line

4.10 Tip--shortening coefficient for external gearsets For gears operating on extended centers (aw > aref), the outside radii of the gears may be shortened to maintain adequate tip--to--root clearance. The amount of adjustment of the outside radii is proportional to the tip--shortening coefficient, k:

...(42)

yE

ha0

tool pitch line

r

rf

yE = xE mn

where ∆ a ref = aw − aref

...(33)

For internal gear sets, see 5.2.3.

4.11 Addendum circle radii

4.10.1 Tip--shortening options Three of the tip shortening options are as follows: 4.10.1.1 Full length teeth -- option 1

r a1 = r1 + ha1

...(44)

r a2 = r2 + ha2

...(45)

For internal gear sets, see 5.3.

h a1 = Ꮛ1 + x 1Ꮠ mn

...(34)

h a2 = Ꮛ1 + x 2Ꮠ mn

...(35)

CAUTION: Option 1 (full length teeth) may give insufficient tip--to--root clearance if aw & aref. Check clearances or use option 3 (full tip--to--root clearance).

10

Figure 6 -- Root radii cut with rack tool (refer to annex A for additional information)

4.12 Generating rack shift coefficients ∆ s n1 2 mn tan ∼ n ∆ s n2 x E2 = x 2 − 2 mn tan ∼ n x E1 = x 1 −

...(46) ...(47)

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For internal gear sets, see 5.4.

AGMA 913--A98

5 Internal gear pair calculations

4.13 Normal circular tooth thickness





 

s n1 = 1 π + 2 x E1 tan ∼ n m n 2 s n2 = 1 π + 2 x E2 tan ∼ n m n 2 For internal gears, see 5.5.

5.1 Internal gear rack shift ...(48) ...(49)

4.14 Determining profile shift coefficients of existing gear pairs If the normal circular tooth thicknesses are known, the generating rack shift coefficients are found from equations 50 and 51. sn1 π mn − 2 ...(50) x E1 = 2 tan ∼ n sn2 π mn − 2 ...(51) x E2 = 2 tan ∼ n For internal gear sets, see 5.6. 4.14.1 Sum of generating rack shift coefficients Σ x E = x E2 + x E1

4.14.2 Normal operating circular backlash

Ꮛ2 a

w

s n = 1 π mn − 2 y tan ∼ n ...(57) 2 ∆ sn yE = y + ...(58) 2 tan ∼ n a (inv ∼ wt − inv ∼ t) ...(59) y 2 − y 1 = ref tan ∼ t In equation 59, the subscripts 1 and 2 refer to the pinion and the gear. The addendum of an external gear is generally chosen equal to (1.0 mn + y). The corresponding value for an internal gear would be (1.0 mn - y), but this value often leads to interference at the pinion tooth fillets. It is common to choose the largest addendum possible, consistent with no interference. The tip circle radius of the internal gear is then given by:

...(52)

For internal gear sets, see 5.6.1.

jn =

Equations 57 through 59 are equations 7 through 9 altered for the case of an internal gear.1)

Ꮠ

m n tan ∼ n Ꮛ Σ x − Σ x EᏐ aref

...(53)

r 2a2 = r2b2 + n ∼ wt +

For internal gear sets, see 5.6.2.

The tooth thinning coefficients must satisfy equation 31. However, it is usually impossible to determine the ratio ∆sn1/∆sn2 that was used for existing gears. The following analysis is based on common practice where ∆sn1 Ζ ∆sn2Ι in which case:

Ꮛ Ꮠ

0.025 mnᏐ − r 2b1

Ꮖ

fP1 +

2

...(60)

rfP1 is the radius of the pinion fillet circle, i.e., the radius at which the involute tooth profile meets the tooth fillet.

...(54)

...(55)

5.2 Internal gear pair profile shift calculation

From equations 46 and 47: ∆ s n1 2 mn tan ∼ n ∆ s n2 x 2 = x E2 + 2 mn tan ∼ n For internal gear sets, see 5.6.3.

fP1 +

2

ᎩᏋr

Equation 60 insures that no interference occurs at the pinion root fillet. A similar equation can be used to avoid the possibility of interference with the root fillets of the cutter, which would cause the tooth tips of the internal gear to be undercut. Other considerations affecting tip shortening of internal gears are discussed in 5.2.3.

4.14.4 Profile shift coefficients

x 1 = x E1 +

Ꮛrb2 − rb1Ꮠ tan ∼ wt +

where

4.14.3 Tooth thinning for backlash

a ∆ s n1 = ∆ sn2 = 1 j n aref w 2

ᎩᏋr

Ꮑ

The basic gear geometry equations are given in 4.2. ...(56)

These equations are valid for both external and internal gearsets.

_______________________ 1)

CAUTION: There are different conventions for the sign of x used in internal gear calculations.

11

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5.2.1 Sum of profile shift coefficients for zero backlash

where C 6 = Ꮛrb2 − rb1Ꮠ tan ∼ wt = a w sin ∼ wt ...(64)

C 1 = Ꭹr2a2 − r2b2 − C6

For internal gearsets: a ref = r2 − r1 x2 − x1 =

...(61)

aref (inv ∼ wt − inv ∼ t) mn tan ∼ t

...(62)

5.2.2 Balanced specific sliding

...(65)

C 5 = Ꭹr2a1 − r2b1

...(66)

C 2 = C5 − pbt

...(67)

C 3 = rb1 tan ∼ wt

...(68)

C 4 = C1 + pbt

...(69)

5.2.3 Tip shortening for internal gearsets To balance the specific sliding at each end of the path of contact for an external gear pair see 4.6. For an internal gear pair, this is done by iteratively varying the profile shift coefficients of the pinion and gear, see figure 7, until the following equation is satisfied:

ᏋCC + 1 ᏐᏋCC + 1 Ꮠ = u 6

6

1

5

2

For internal gear pairs operating at extended centers (aw > aref), the addendum values are generally reduced for a number of reasons. The tip shortening coefficient given by equation 32 is valid for an external gear pair, in which both gears are cut by a rack--type cutter. Since an internal gear is usually cut by a pinion cutter, the tip shortening coefficient is not particularly useful for calculating the addendum circle radii of an internal gear pair.

...(63)

EAP HPSTC

C4 C3

C5 P

C2 LPSTC C1 C6

SAP

ra1

∼wt

rb1 rb2

rfP1

ra2 O1 aw O2 Figure 7 -- Distances along the line of action for an internal gear pair

12

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To choose the addendum circle radius of the pinion, the root circle radius, rf2, of the internal gear must first be calculated. This will depend on the required tooth thickness and the diameter and tooth thickness of the pinion cutter. The addendum circle radius of the pinion should then be given by: r a1 = rf2 − aw − c i2 ...(70)

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5.5 Internal gear pair normal circular tooth thickness


 s n2 =
1 π − 2 x E2 tan ∼ n m n 2 s n1 = 1 π + 2 x E1 tan ∼ n m n 2

...(75) ...(76)

where ci2

is the required clearance at the tooth roots of the internal gear.

An expression for the addendum circle radius, ra2, of the internal gear was given in equation 60. This value was chosen to ensure that there would be no interference between the internal gear tooth tips and the pinion tooth fillets. There are several other requirements that should also be considered. There must be no interference between the internal gear tooth tips and the tooth fillets of the pinion cutter. The base circle of the pinion cutter should lie inside the tooth tips of the internal gear (c1 > 0). In addition, there must be no rubbing between the cutter and gear during the return strokes of the cutter. Finally, there must be no interference between the tooth tips of the internal gear and those of the pinion, which can occur when the difference between the tooth numbers is small. Reference [4] describes a design procedure for internal gear pairs which includes all of the above considerations.

5.6 Determining profile shift coefficients of existing internal gear pairs If the normal circular tooth thicknesses are known, the rack shift coefficients are found from equations 76 and 77. sn1 π mn − 2 x E1 = 2 tan ∼ n s n2 π mn − 2 x E2 = − 2 tan ∼ n

  

...(77)

  

...(78)

5.6.1 Sum of rack shift coefficients Σ x E = x E2 − x E1

...(79)

5.6.2 Normal operating circular backlash jn = −

Ꮛ2 a

w

Ꮠ

mn tan ∼ n Ꮛ Σ x − Σ x EᏐ ...(80) a ref

5.3 Internal gear pair addendum circle radii r a1 = r1 + ha1

...(71)

r a2 = r2 − ha2

...(72)

5.4 Internal gear pair rack shift coefficients ∆ s n1 x E1 = x 1 − 2 mn tan ∼ n ∆ s n2 x E2 = x 2 + 2 mn tan ∼ n

5.6.3 Internal coefficients

gear

pairs

profile

shift

From equations 72 and 73: ∆ s n1 2 mn tan ∼ n

...(73)

x 1 = x E1 +

...(74)

x 2 = x E2 −

∆ s n2 2 mn tan ∼ n

...(81) ...(82)

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(This page is intentionally left blank.)

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Annex A (informative) Tool proportions [The foreword, footnotes and annexes, if any, are provided for informational purposes only and should not be construed as a part of AGMA 913--A98, Method for Specifying the Geometry of Spur and Helical Gears.]

A.1 Purpose

A.3 Equations for calculation of tool proportions

This annex defines a method for deriving the geometry of the hypothetical tool basic rack. The tool geometry used by ISO 6336 for the purpose of calculating gear ratings is hypothetical rather than actual tool geometry. The tool proportions required are those of a hypothetical tool basic rack that is conjugate to (the complement of) the basic rack, zero backlash tooth form. The hypothetical tool contains the combined effects of the roughing tool, which produces the root, and the finishing tools for the flank. Changing the stock allowance for finishing will change the tool addendum.

This annex uses the geometry of the measured roughing tool and the finishing stock allowance. See ISO 53 for tooth form basic rack definitions and detailed information. In ISO 6336--3, figure 2, specific tooth form and hypothetical tool basic racks are defined which include the effects of tool protuberance and stock allowance for finishing, but not tooth thinning for backlash. Figure A.1 illustrates the relations between the basic racks of the tooth form, the hypothetical tool and the roughing tool.

A.2 Definitions The basic rack is an imaginary rack having the standard basic rack tooth profile in the normal section. It corresponds to a zero backlash gear with an infinite pitch radius. A gear with a normal module, mn , and a normal pressure angle, αn , at its reference circle, has a basic rack whose normal pitch is π mn and whose pressure angle is ∼n . The basic rack reference line is defined as the datum line where the normal tooth thickness is equal to (π mn )/2. Therefore, the tooth thickness equals the space between the teeth on the reference line. The tool measurement line is an arbitrary datum line on the actual tool where the tool tooth thickness is measured and from which the tool addendum is measured. The hypothetical tool is by definition conjugate to the basic rack, so the hypothetical tool reference line is coincident with the basic rack reference line for the zero backlash gear. The tool reference line is defined as the line where the actual tool normal tooth thickness is equal to (π mn )/2 or π/(2 Pnd ). See ISO 6336--3, figure 2. The reference circle of a gear is that pitch circle whose diameter is equal to (z mn )/cosβ, where z is the number of teeth and β is the helix angle. For additional and more formal definitions see ISO 1122--1 and ANSI/AGMA 1012--F90.

A.3.1 Normalized dimensions (coefficients) Some of the tool geometry requires coefficients, which are also known as normalized dimensions. To normalize SI dimensions, divide the dimension (mm) by the module (mm). For the example used, the addendum of the 4.233 module hob basic rack is 6.248 mm, so the normalized addendum is 6.248/4.233 = 1.476. To normalize an English dimension, multiply the dimension (inches) by the diametral pitch (in --1). In English units, since the example 6 Pnd hob has an addendum of 0.246 inch, the normalized addendum is 6 ¢ 0.246 = 1.476. As can be seen, the resulting coefficient is the same because it is dimensionless. A.3.2 Calculation from the finishing stock allowance and tool dimensions mn

is the normal module (for SI units, mm);

Pnd

is the normal diametral pitch, in --1;

q

is the finish stock allowance per flank, in (mm);

∼n

is the normal pressure angle at the reference diameter;

ha

is the measured tool addendum (from the tool tip to the tool measurement line), in (mm);

t

is the normal circular tooth thickness of roughing tool at the measurement line, in (mm);

15

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AMERICAN GEAR MANUFACTURERS ASSOCIATION Normal circular pitch

π mn

Note the value for ∆s is shown positive. It may have a negative value, which would change the relationships shown accordingly.

π mn 2

π mn 2

Hypothetical tool Hypothetical tool dedendum,

hf0

Basic rack addendum,

Material allowance, q, for finish machining

Tool reference line

haP

t

∆s 2

∆s 2 tan(∼ n)

Hypothetical tool and basic rack reference line Tool measurement line

Hypothetical tool addendum, ha0 Gear basic rack dedendum, hfP

Measured tool (hob) addendum,

q sin(∼ n)

ha

Tool

Basic rack “Zero Backlash” gear tooth

Profile angle,

∼n

Figure A.1 -- Basic rack and hypothetical tool represented in rack form

∆s

is the adjustment if the roughing tool measurement line is not coincident with the tool reference line. When the measured tool addendum is taken from the tool reference line, then ∆s = 0. Otherwise, it may be calculated as:

∆s = π − t 2P nd

A.3.3 Tool basic rack addendum coefficient (normalized) Tool basic rack addendum coefficient (normalized) is: h a0n = P nd ha0 h h a0n = ma0 n

...(A.1)

...(A.3) ...(A.3M)

A.3.4 Additional tool data required ∆s =

ha0

π mn − t 2

is the hypothetical tool basic rack addendum, in (mm). It is measured from the tip to the reference plane, where the hypothetical tool normal tooth thickness is (π mn )/2 or π/(2 Pnd ). The hypothetical tool basic rack addendum is equal to hfP, the dedendum of the zero backlash tooth form basic rack. See figure A.1. Either SI or English units may be used in the following equation:

h fP = h a0 = h a +

16

...(A.1M)

q ∆s − 2 tan ∼ n sin ∼ n

...(A.2)

pr

is the protuberance, as measured on tool, in (mm). Note that the 1996 release of ISO 6336--1, page 9, lists the symbol for protuberance as qpr, yet ISO 6336--3, figure 2, uses the symbol pr, which is also used here.

spr

is net protuberance spr = pr -- q

±a0

is the tip radius, in (mm). This is assumed to be equal to the tooth form basic rack root radius, ±fP. The maximum value that ±a0 can have is that for a full fillet radius. An equation for a full fillet radius may be derived from ISO 6336--3, equation 12, by setting E = 0 and solving for ±a0 max:

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π cos ∼ + s − h sin ∼ n pr n a0 4 P nd ± a0 max = 1 − sin ∼ n ...(A.4) π m cos ∼ + s − h sin ∼ n n pr n a0 ± a0 max = 4 1 − sin ∼ n ...(A.4M)

A.3.5 Basic rack fillet radius coefficient (normalized)

Pnd q ∼n ha t

Pinion Wheel Unit 6.0 in--1 Normal diametral pitch. 0.0053 0.0053 in Finish stock allowance per flank. 20 deg Normal pressure angle at the reference diameter. 0.2460 in Measured tool addendum (to reference line). 0.2618 in Normal circular tooth thickness of roughing tool.

Basic rack fillet radius coefficient (normalized): ± a0n = P nd ± a0 ± ± a0n = ma0 n

...(A.5)

Calculated values:

...(A.5M) ∆s =

A.4 Example calculation of tool proportions The following example shows how the tool proportions are calculated in English units for the sample problem shown. This sample problem is based on AGMA 918--A93, example 3.1.3, see figure A.2. In this example, the tool measurement line is coincident with the tool reference line. A.4.1 Addendum of tool basic rack (inch units)

π −t 2 P nd

= π − 0.2618 = 0.0000 2(6)

h fP = h a0 = h a + = 0.246 +

∆s − q 2 tan ∼ n sin ∼ n

0 − 0.0053 = 0.2305 2 tan
20 sin
20

h a0n = P nd h a0 = 6(0.2305) = 1.383 Tool basic rack addendum, normalized.

Data needed for calculation:

0.5236 R =0.0682 Full

0.006

0.2618

0.2460

20.00 20.00 Ref Protuberance normal plane

Figure A.2 -- Hob basic rack (linear dimensions in inches)

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A.4.2 Addendum of tool basic rack (SI units)

Calculated values: Net protuberance.

Data needed for calculation:

mn q ∼n ha t

spr

Pinion Wheel Unit 4.23333 mm Normal module 0.13462 0.13462 mm Finish stock allowance per flank. 20 deg Normal pressure angle at the reference diameter. 6.24840 mm Measured tool addendum (to reference line). 6.6497 mm Normal circular tooth thickness of roughing tool.

= pr -- q = 0.006 -- 0.0053 = 0.0007

Full fillet radius check.

π cos(∼ ) + s − h sin(∼ ) n pr n a0 4 P nd ± a0 max = 1 − sin(∼ n) π cos(20) + 0.0007 − 0.2305 sin(20) 4
6 = 1 − sin(20) = 0.0682

Normalized basic rack fillet radius ± a0n = P nd ± a0

Calculated values:

= 6(0.0682) = 0.4092

π mn ∆s = −t 2 =

π
4.23333 − 6.6497 = 0.0000 2

h fP = h a0 = h a + = 6.24840 +

∆s − q 2 tan ∼ n sin ∼ n

0 − 0.13462 = 5.8548 2 tan
20 sin
20

h h a0n = ma0 = 5.8548 = 1.383 n 4.2333

A.4.4 Fillet radius (SI units) Data needed for calculation: mn ∼n ha0 pr q

Pinion Wheel 4.23333 20

Unit mm Normal module deg Normal pressure angle at the reference diameter. 5.85470 5.85470 mm Hypothetical tool basic rack addendum. 0.15240 0.15240 mm Protuberance. 0.13462 0.13462 mm Finish stock allowance per flank.

Calculated values: A.4.3 Basic rack fillet radius (inch units)

Net protuberance

The basic rack fillet radius is normally given on the tool drawing. However, if a full fillet radius is specified without a dimension being given, it may be calculated as follows. Further, a given fillet radius may be checked to be sure that it does not exceed the value for a full fillet radius.

s pr = pr − q = 0.1524 − 0.13462 = 0.01778

Data needed for calculation:

Pnd ∼n ha 0 pr q

18

Pinion Wheel Unit 6.0 in--1 Normal diametral pitch. 20 deg Normal pressure angle at the reference diameter. 0.2305 0.2305 in Hypothetical tool basic rack addendum. 0.0060 0.0060 in Protuberance. 0.0053 0.0053 in Finish stock allowance per flank.

Full fillet radius check π m cos(∼ ) + s − h sin(∼ ) n n pr n a0 ± a0 max = 4 1 − sin(∼ n) π

4.2333 cos 20 + 0.01778 − 5.8547 sin 20 =4 1 − sin 20 = 1.7321

Normalized basic rack fillet radius ± ± a0n = ma0 n = 1.7321 = 0.4092 4.233

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Annex B (informative) Calculation of profile shift [The foreword, footnotes and annexes, are provided for informational purposes only and should not be construed as a part of AGMA 913--A98, Method for Specifying the Geometry of Spur and Helical Gears.]

B.1 Purpose This annex gives a procedure for calculating profile shift coefficients from the finished normal circular tooth thickness. It is based on the assumption that backlash is split equally between the gear and the pinion. Profile shift coefficients for the gear and pinion affect the calculation of tooth stiffness, which affects load distribution across the face of the teeth (KHβ), internal dynamic factor (Kv ) and root bending strength. The nominal (zero backlash) values of profile shift coefficient must be used for consistent results. The profile shift coefficient is the distance (expressed as a coefficient, i.e., normalized) between the basic rack reference line and the reference circle on the gear when the basic rack is positioned for zero backlash teeth. See annex A for definitions. Figure B.1 shows the hypothetical rack tool cutting the zero backlash tooth form on the left, and cutting the finished tooth form on the right. For a positive value of the x factor, the tool is “held out” from the part to produce thicker teeth than “standard”. Regardless

x mn

Hypothetical tool

Tangent line to gear reference circle

Profile shift (zero backlash)

∆x m 2 n

The generating rack shift coefficient for the pinion, xE1, and gear, xE2, may be calculated from the finished normal circular tooth thickness at the reference circle (equations B.1 and B.2). The zero backlash profile shift coefficients are determined, with this method, by adding equal amounts to the generating rack shift coefficients, such that they would produce a zero backlash gear pair. Figure B.2 shows, in rack form, the hypothetical tool, the basic rack and a finished gear tooth. In rack form the simple trigonometric relationships may be seen. A tool (hob) is also shown in rack form to illustrate the difference between an actual tool and the hypothetical tool. For gearing produced to final shape without finishing stock, there is no difference between the actual tool and the hypothetical tool.

For backlash tooth thinning

Hypothetical tool and basic rack reference line

Zero backlash gear

of the sign of x, the tool is “fed in” a little further to thin the teeth for backlash. When the tool is positioned to produce these finished teeth, the distance between the hypothetical tool reference line and the gear reference circle is known as the generating rack shift coefficient, xE .

xE mn

Generating rack shift

Hypothetical tool reference line

Hypothetical tool

Gear reference circle

Finished gear

Figure B.1 -- Hypothetical tool with zero backlash and finished gear

19

π mn 2

π mn 2

π mn 2 t

Hypothetical tool

Material allowance, q, for finish machining Tool reference line Basic rack reference line

∆s 2 tan (∼n)

Finished gear tooth

∆x mn x mn 2 For backlash Profile shift (zero backlash) tooth thinning

Reference line of hypothetical tool positioned for finished (thinned) tooth

Tool measurement line Hypothetical tool Measured addendum, ha0 tool (hob) Gear basic rack addendum, dedendum, hfP ha

q sin (∼n )

Finished gear tooth normal circular thickness, sn , at reference circle

Tool

xE mn Generating rack shift

Tangent line to gear reference circle Gear dedendum

Basic rack “Zero Backlash” gear tooth

Zero backlash tooth thickness

Profile angle, ∼n

Basic rack reference line

}

Hypothetical tool dedendum, Basic rack hf0 addendum, haP

αs/2

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Figure B.2 -- Basic rack, hypothetical tool and finished gear tooth represented in rack form

AGMA 913--A98

20 Normal circular pitch π mn

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In figures B.1 and B.2:

The generating rack shift coefficients for the pinion and the wheel are:

-- The tooth thinning for backlash is greatly exaggerated; -- The profile shift coefficient, x, is multiplied by the normal module, mn , to obtain the amount of profile shift, because x is a coefficient; -- Note that the values for ∆s, xE and x are shown positive. They may have negative values, which would change the relationships shown in the figures accordingly.

x E1 =

s n1 P nd − π 2 2 tan ∼ n

x E1 =

x E2 =

s n1 π mn − 2

...(B.1)

...(B.1M)

2 tan ∼ n

s n2 P nd − π 2 2 tan ∼ n s n2 π mn − 2

...(B.2)

The root diameter of the part may be calculated from the hypothetical tool addendum, ha0, and the generating rack shift, xE . Using the zero backlash profile shift coefficient, x, rather than xE to calculate the root diameter will result in a larger calculated root diameter. The actual root diameter is process dependent, but it is generally smaller than that calculated with x.

Calculate the sum of the nominal “zero--backlash” profile shift factors for this gear pair as:

B.2 Equations for calculation of profile shift coefficients

inv( ) is the involute function of the angle. inv(∼) = tan(∼) -- ∼ (where ∼ in radians)

Pnd

is the normal diametral pitch;

mn

is the normal module;

z1, z2

is the number of teeth on pinion and wheel, respectively;

sn1, sn2 is the maximum finished normal circular tooth thickness (after all finishing operations, including tooth thinning for backlash) of pinion and wheel respectively, measured at the reference (standard pitch) diameter; ∼wt

is the operating pressure angle (transverse);

∼t

is the pressure angle at the reference (standard pitch) diameter (transverse);

∼n

is the pressure angle at the reference (standard pitch) diameter (normal);

xE1, xE2 is the generating rack shift coefficients of pinion and wheel, respectively; x1, x2

is the nominal zero backlash profile shift coefficient of pinion and wheel, respectively.

x E2 =

...(B.2M)

2 tan ∼ n

ᒑ x = z1 +2 z2

inv∼ wt − inv∼ t tan ∼ n

...(B.3)

Determine the adjustment needed to bring the generating rack shift coefficients, xE1 and xE2, to their zero--backlash values. ∆x =

ᒑ x − ᏋxE1 + xE2Ꮠ

...(B.4)

If the value of ∆x is negative, you should review the geometry of the gear pair, since interference between the finished gear flanks is indicated. The nominal zero backlash profile shift coefficients, x1 and x2, of the pinion and wheel are: x 1 = x E1 + ∆x 2

(B.5)

x 2 = x E2 + ∆x 2

...(B.6)

B.3 Example calculation of profile shift coefficients (inch units) The following example shows how the profile shift coefficients are calculated (with English units) for the sample problem shown in the other appendices. This sample problem is based on AGMA 918--A93 example 3.1.3.

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ᒑ x = z1 +2 z2

Data needed for calculation: Pinion

Pnd β ∼n sn

z1, z2 a

Wheel

Unit in--1 Normal diametral pitch. 15 deg Standard helix angle. 20 deg Normal pressure angle at the reference diameter. 0.32262 0.25780 in Actual normal circular tooth thickness at reference diameter. 21 86 -Number of teeth. 9.3175 in Center distance.

= 21 + 86 2 = 0.53425

6.0

inv(α) = tan(α) -- α

Adjustment to bring the generating rack shift coefficients to their zero--backlash values: ∆x =

ᒑ x − ᏋxE1 + xE2Ꮠ

= 0.06591 The values of x1 and x2 are:

Ꮛtancos∼β Ꮠ

x 1 = x E1 + ∆x 2

n

Ꮛ

= 0.50131 + 0.06591 = 0.5343 2

Ꮠ

= tan −1 tan 20 = 20.6469 degrees cos 15

x 2 = x E2 + ∆x 2

Operating pressure angle (transverse):

ᏁᏋ

Ꮛ

z 1 + z 2Ꮠ

Ꮑ

cos ∼ t 2 P nd a cos β

ᏐᏆ

cos(20.6469) Ꮛ2(6)(9.3175) ᏐᏆ cos(15)

= cos −1 (21 + 86) = 22.0120 degrees

Generating rack shift coefficients: s n1 P nd − π 2 x E1 = 2 tan ∼ n 0.32262(6) − π 2 = 0.50131 = 2 tan(20) s n2 P nd − π 2 x E2 = 2 tan ∼ n 0.25780(6) − π 2 = − 0.03296 = 2 tan(20) Sum of the nominal “zero--backlash” profile shift coefficients:

22

(where α in radians)

= 0.53425 − (0.50131 + (− 0.03296))

Pressure angle at the reference (standard pitch) diameter (transverse):

∼ wt = cos −1

inv(22.012) − inv(20.6469) tan(20)

inv( ) is the involute function of the angle.

Calculated values:

∼ t = tan −1

inv ∼ wt − inv ∼ t tan ∼ n

= − 0.03296 + 0.06591 = 0.0000 2 B.3 Example calculation of profile shift coefficients (SI units) Data needed for calculation: mn β ∼n sn

z1, z2 a

Pinion Wheel 4.23333 15

Unit mm Normal module deg Standard helix angle. 20 deg Normal pressure angle at the reference diameter. 8.19455 6.54812 mm Actual normal circular tooth thickness at reference diameter. 21 86 -- -Number of teeth. 236.66450 mm Center distance.

Calculated values: Pressure angle at the reference (standard pitch) diameter (transverse):

Ꮛtancos∼β Ꮠ

∼ t = tan −1

Ꮛ

n

Ꮠ

= tan −1 tan 20 = 20.6469 degrees cos 15

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Operating pressure angle (transverse):

Ꮑ

Ꮛ

ᏐᏆ

m n cos ∼ t ∼ wt = cos−1 Ꮛz 1 + z2Ꮠ 2 a cos β

Ꮑ

cos (20.6469) Ꮛ(4.23333) ᏐᏆ 2(236.6645) cos (15)

= cos −1 (21 + 86) = 22.0120 degrees

ᒑ x = z1 +2 z2 = 21 + 86 2

8.19455 − π 2 = 0.50131 = 4.23333 2 tan(20) s n2 π mn − 2 x E2 = − 2 tan α n 6.54812 − π 2 = − 0.03296 = 4.23333 2 tan(20) Sum of the nominal “zero--backlash” profile shift coefficients:

inv(22.012) − inv(20.6469) tan(20)

= 0.53425 Adjustment to bring the generating rack shift coefficients to their zero--backlash values:

Generating rack shift coefficients: s n1 π mn – 2 x E1 = 2 tan α n

inv ∼ wt − inv ∼ t tan ∼ n

∆x =

ᒑ x − ᏋxE1 + xE2Ꮠ

= 0.53425 − (0.50131 + (− 0.03296)) = 0.06591 The values of x1 and x2 are: x 1 = x E1 + ∆x 2 = 0.50131 + 0.06591 = 0.5343 2 x 2 = x E2 + ∆x 2 = − 0.03296 + 0.06591 = 0.0000 2

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AGMA 913--A98

AMERICAN GEAR MANUFACTURERS ASSOCIATION

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AMERICAN GEAR MANUFACTURERS ASSOCIATION

AGMA 913--A98

Bibliography The following documents are either referenced in the text of AGMA 913--A98, Method for Specifying the Geometry of Spur and Helical Gears, or indicated for additional information. 1. AGMA 908--B89, Information Sheet -- Geometry Factors for Determining the Pitting Resistance and Bending Strength of Spur, Helical and Herringbone Gear Teeth, 1989 2. ANSI/AGMA 2101--C95, Fundamental Rating Factors and Calculation Methods for Involute Spur and Helical Gear Teeth, 1995 3. AGMA 901--A92, Information Sheet -- A Rational Procedure for the Preliminary Design of Minimum Volume Gears, 1992 4. Colbourne, J.R., The Geometric Design of Internal Gear Pairs, AGMA Paper No. 87FTM2 5. McVittie, D.R., Describing Nonstandard Gears -- An Alternative to the Rack Shift Coefficient, AGMA Paper No. 86FTM1 6. DIN 3992, Profilverschiebung bei Stirnrädern mit Aussenverzahnung 7. ISO 53:1998, Cylindrical gears for general and heavy engineering -- Basic rack 8. ISO 1122--1:1998, Glossary of gear terms -- Part 1: Geometrical definitions 9. ISO/TR 4467:1982, Addendum modification of the teeth of cylindrical gears for speed--reducing and speed--increasing gear pairs 10. ISO/TR 10064--2:1996, Cylindrical gears -- Code of inspection practice -- Part 2: Inspection related to radial composite deviations, runout, tooth thickness and backlash 11. Lorenz Gear Cutting Tools -- 1961 12. MAAG Gear Book, MAAG Gear Company, Ltd., Zurich, Switzerland, 1990 13. DIN 3960:1980, Begriffe und Bestimmungsgrößen für Stirnräder (Zylinderräder) und Stimradpaare (Zylinderradpaare) mit Evolventenverzahnung 14. ISO 6336--3:1992, Calculation of load capacity of spur and helical gears -- Part 3: Calculation of tooth strength 15. AGMA 918--A93, Information Sheet -- A Summary of Numerical Examples Demonstrating the Procedures for Calculating Geometry Factors for Spur and Helical Gears 16. ANSI/AGMA 2002--B88 (R95), Tooth Thickness Specification and Measurement, 1988

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