addendum modification coefficient

22 GEAR SOLUTIONS • MARCH 2006 • gearsolutionsonline.com

CYLINDRICAL GEAR CONVERSIONS:

AGMA ISO

THE FOLLOWING IS AN IN-DEPTH EXAMINATIONOF THE CONVERSION OF CYLINDRICAL GEARS INTHE AGMA SYSTEM TO ISO STANDARDS WITHTHE ADDENDUM MODIFICATION COEFFICIENT.

TOBy G. González Rey, P. Frechilla Fernández, and R. José García Martín

Therational use of the addendum modification coeffi-cient is one of the topics better known by spe-cialists of gears working in the metric system(ISO). This non-dimensional parameter, also well-known as rack shift coefficient x, allows gear

designs with very good adaptability of the teeth profile in practi-cal applications.

Unfortunately, the gear specialists with designs based on theAGMA system traditionally don’t appreciate the advantages of theaddendum modification coefficient as an adjustment factorbetween ISO and AGMA systems. The publications of specialists[McVittie (1993), Rockwell (2001)] directed toward disclosing thegeneralities of the geometry calculation of gears with the applica-tion of this coefficient are welcome for experts.

The Working Group WG5, under the direction of Henry Deby, approvedin 1981 a Technical Report ISO (ISO/TR 4467) with orientations aboutthe value limits and distribution of the addendum modification of theteeth of external parallel-axis cylindrical involute gears for speed-reducingand speed-increasing external gear pairs. Although the recommendationswere not of a restrictive nature, a general guide was finally published onthe application of the addendum modification coefficient. These recom-mendations were important in order to avoid an incorrect use of thiscoefficient and to prevent designs of gears with very small tooth crestwidth, insufficient value of transverse contact ratio, or cutter interfer-ence. Unquestionably, ISO/TR 4467 showed the advantages of usingthe addendum modification coefficient in the design of gears and givesguided reasonable values of this coefficient for the geometric calculationof gears with more load capacity and better performance.

This article presents some basic definitions and recommenda-tions associated with the correct application of the addendummodification coefficient. Additionally, formulae relating to externalparallel-axis cylindrical involute gears to consider the effect of theaddendum modification coefficient in gear geometry are dis-cussed. The procedure for the solution of two cases—based onthe authors’ experiences in the analysis, recovery, and conversionof helical and spur gears in the AGMA system to ISO standards—show the advantage of the application of the addendum modifica-tion coefficient in the solution of practical problems.

Tooth Profile Reference with No AddendumModification on Cylindrical GearsIt is well known that when external parallel-axis cylindrical gearswith an involute profile are in correct meshing the toothed wheelsshould conjugate with a corresponding rack. The profile of thisrack-tool is denominated basic rack tooth profile. The shape andgeometrical parameters of the basic rack tooth profile for involutegears are set by special standards (see table 2) along with therack shaped tool, such as hobs or rack type cutters, used in thecutting of gears by means of generation methods (see figure 1).

One important definition in the basic rack tooth profile is the datumline corresponding with a straight line drawn parallel to the tip androot lines where the tooth thickness “s” is equal to tooth space width“e.” Most of the dimensions of the basic rack tooth profile are givenwith reference to the datum line and divides the basic rack tooth pro-file in two parts: addendum and dedendum. Figure 2 shows a typicalshape of a basic rack tooth profile and the principal geometricalparameters used to establish the basic dimensions (see table 2 forsome values standards of basic rack tooth profile parameters).

The usable flanks in the basic rack tooth profile are inclined atthe profile angle α to a line normal to the datum line. This angle isthe same as the pressure angle “α” at the reference cylinder of aspur gear or the normal pressure angle “αn” at the reference cylin-der of a helical gear. The relation between pith of the basic racktooth profile “p” and module “m” (in mm) is: p = π • m

The dimensions of the standard basic rack tooth profile aregiven in relation to module and identified by *, such as ha* (factorof addendum), hf* (factor of dedendum), c* (factor of radial clear-ance), and ρf* (factor of root fillet radius). The module m of thestandard basic rack tooth profile is the module in the normal sec-tion mn of the gear teeth. For a helical gear with helix angle β onthe reference cylinder, the transverse module in a transverse sec-tion is mt = mn/cosβ. For a spur gear β = 0 the module is m = mn= mt. As a reference parameter, the module m in ISO standards issimilar to the normal diametral pitch Pnd in AGMA standards andcan be converted by the following equation:

m = 25,4Pnd

Most special standards for basic rack tooth profile—includingISO Standard 57-74—recommend for industrial gears with generalapplication the following values:

• pressure angle: α = 20°• factor of addendum: ha* = ha / m =1• factor of radial clearance: c* = c / m = 0,25

The principal methods for generating the tooth flanks on externalparallel-axis cylindrical involute gears make use of a rack-shapedtool (such as hobs or rack-type cutters) with dimensions similar tothe basic rack tooth profile shown in figure 1. When gears are pro-duced by a generating process, in the case of tooth profile withoutaddendum modification the datum line (MM) of the basic rack toothprofile is tangent to the reference diameter of gear (see figure 3).

Addendum Modification on Cylindrical Gears It’s possible to understand the use of the tooth profile withoutaddendum modification to establish the first standards for thegeometric calculation of cylindrical gears. When gears are generat-ed with a rack-shaped tool and the reference cylinder of gear rollswithout slipping on the imaginary datum line, the standard rela-tionship to set up the tooth shape and geometric calculation arerelatively simple.

For specialists involved with gear design based on ISO standards,it’s often true that the datum line of the basic rack profile need notnecessarily be tangent to the reference diameter on the gear. Thetooth profile and its shape can be modified by shifting the datumline from the tangential position. This displacement makes it possi-ble to use tooth flanks with other parts of the involute curve anddifferent diameters for gears with the same number of teeth, mod-ule, helix angle, and cutting tools.

Explained simply it can be said that the generation of a cylindricalgear with addendum modification, when concluding the genera-tion of tooth flanks, that the reference cylinder is not tangent tothe datum line on basic rack tooth profiles, and there is a radialdisplacement.

gearsolutionsonline.com • MARCH 2006 • GEAR SOLUTIONS 23

The main parameter to evaluate the addendum modification isthe addendum modification coefficient x, also know by Americansas the “profile shift factor,” or the “rack shift coefficient.” Theaddendum modification coefficient quantifies the relationshipbetween distance from the datum line on the tool to the referencediameter of gear ∆abs (radial displacement of the tool) and modulem. This coefficient is defined for pinion x1 and gear x2 as:

x1 = ∆abs1 x2 = ∆abs2m and m

The addendum modification coefficient is positive if the datumline of the tool is displaced from the reference diameter toward thecrest of the teeth (the tool goes away from the center of the gear),and it is negative if the datum line is displaced toward the root ofthe teeth (the tool goes toward the center of gear). To consider theeffect of addendum modifications for a gear pair it is a good prac-tice to define the sum of addendum modification coefficients as:

Σx = x1 + x2

The manufacturing of the gear with addendum modification isnot more complex or expensive than gears without profile shift,because the gears are manufactured in the same cutting machinesand depend solely on the relative position of the gear to be cutand the cutter. The difference can be evident in the blanks with dif-ferent diameters and tooth profiles on gears.

A positive addendum modification (x > 0) results in a greatertooth root width and, thus, in an increase in the tooth root carryingcapacity; more effective in the case of small numbers of teeththan in the case of larger ones, including the possibility of avoidingor reducing undercutting on the pinion. Additionally, an increase ofthe addendum modification coefficient results in a decrease in the tip thickness of the teeth. A negative addendum modification (x < 0) has the reverse effect of positive addendum modificationon tip and root thickness.

The choice of the sum of addendum modifications could be arbi-trary and depends on the center distance or the operational condi-tions applied. Sums that are too high for positive values, or too low

for negative values, may be harmful to the satisfactory performanceof the gear pair. For this reason upper and lower limits are specifiedin the function of the number of teeth (or the virtual number ofteeth for helical gears). In figure 5, recommendations for limits ofsum of addendum modification coefficients are given. For a sum ofaddendum modification coefficients Σx with a correct distribution ofvalues between a pinion and gear, it is possible in a gear pair torealize favorable effects in the ability to balance the bending fatiguelife, pitting resistance, or minimizes the risk of scuffing.

For a given center distance and gear ratio, the assembly of thegears can be adjusted with a convenient calculating of the adden-dum modifications. There are different criteria for distributing andapplying the addendum modification coefficient depending on theeffect required in the gear transmission.

• Addendum modification coefficient for gears with a small num-ber of teeth to avoid undercutting on the pinion.

x ≥ ha* -

• Total addendum modification coefficient.If 0 ≤ xΣ ≤ 0,5 then x1 = xΣ y x2 = 0If - 0,5 ≤ xΣ ≤ 0 then x1 = 0 y x2 = xΣ

• Value of addendum modification coefficient inversely propor-tional to the number of teeth.

If 0 ≤ xΣ ≤ 0,8 then x1 =

If - 0,8 ≥ xΣ ≤ 0 then x1 = xΣ • 1 -

• Value of addendum modification coefficient according to MAAG:

Where:


FIGURE 1 Cutting a gear by means of generation methods and a rack-shaped tool. In this case, generated tooth profile with severe cutter inter-ference. This case is very typical in gears with a small number of teethwithout adequate addendum modification.

FIGURE 2 Profiles conjugation between rack shaped tool (1) and gear (2).Line MM is identified as the datum line.

z • sen2α2 • cos3β

z2

z1 + z2

xΣ • z2

z1 + z2

( )

A = 0,50 in case of α = 20,0°A = 0,38 in case of α = 22,5°A = 0,23 in case of α = 25,0°

The basic geometry of the external parallel-axis cylindrical invo-lute gears, taking into account the addendum modification coeffi-cients that can be calculated with the formulas found in table 3.

Sample Practical CasesTo illustrate the application of the addendum modification coeffi-cient in the solution of practical problems in the adjusting gearsfrom the AGMA system to ISO standards, two samples of practicalcases are shown. These cases cover the most frequent geometri-cal problems faced by gear experts during the conversion to theISO Standard of helical and spur gears manufactured with AGMAstandards. Cases with spur gear are more difficult to adjust due tothe fact that the helix angle is fixed and can’t be modified. Othercases can be taken into account by a similar approach.

Case 1: Setting gears from AGMA to ISO with determination of thesum of addendum modification coefficients for a given center dis-tance and gear ratio.

• Statement of the problem—In the gearbox of the power transmission system of a heavy truck it was necessary torecover a spur gear transmission cut originally with toolsbased on AGMA standards. For the new manufacture cuttingtools based on ISO standards should be used. The basicsolution is as follows:

• Initial dataCenter distance; aw = 203,2 mm (8 inch).Gear ration (approximate), u = 4,13Normal diametral pitch (original), Pd = 4 Pressure angle for the rack shaped tool, α = 20°

• Basic solution

a) Proposal of module (ISO).

Standard module by ISO 54-77, m = 6 (mm)

b) Transverse pressure angle:

c) Number of teeth for pinion and gear.

Approximation is preferable by defect.

d) Pressure angle at the pitch cylinder.

e) Sum of addendum modification coefficients.


FIGURE 3 Normal position without addendum modification of the rack-shaped tool and gear when cutting is completed. The reference diameterof the gear is tangent to the datum line on basic rack tooth profile.

TABLE 1 Conversion of values of module m and normal diametral pitch Pnd. Note: Values in bold are preferred.

TABLE 2 Some standard values of basic rack tooth profile parameters.

f) Addendum modification coefficients for pinion and gear (sever-al criterions to distribute the coefficients can be used, but alwayskeep the calculated sum).

x1 = 0,482x2 = 0,463

g) Other geometrical parameters.

Tip diameters: da1 = 94,84 mm ; da2 = 334,62 mm

Root diameters: df1 = 68,78 mm ; df2 = 308,56 mm

Case 2: Setting gears from AGMA to ISO with determination of the sumof addendum modification coefficients for a given center distance,


FIGURE 4 Different positions of the datum line of the tool (MM) in relation to the reference diameter of the gear “d” when the generation has finished.a) Gear generated without addendum modification: x = 0; b) Gear generated with negative addendum modification: x< 0; c) Gear generated with positiveaddendum modification: x > 0.

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keeping the same number of teeth and geargeneration using a cutting tool with a smallerpressure angle than the original.

• Statement of the problem—In thefinal drive of earth-moving equipmentit was necessary to manufacture apair of spur gears originally manufac-tured under AGMA standards. Geargeneration should be done by meansof a cutting tool based on the ISO sys-tem and a pressure angle of 20°. Theoperator of the hobbing machine pro-ceeded to cut the gears and observedan undercutting on the pinion notpresent in the original. During thestudy of the original pinion it wasestablished that the original genera-tion of the gear was made with a cut-ting tool with a 25° pressure angle.The challenge in the new design sup-posed engineering calculations with acutting tool with a 20° pressure angleto guarantee the original center dis-tance and number of teeth withoutundercutting on the pinion. The basicsolution is as follows:

• Initial dataCenter distance, aw = 11 inch (279,4 mm)Number of teeth on pinion, z1 = 14Number of teeth on gear, z2 = 41Module in new design of gear, m = 10Normal diametral pitch in original gear,Pd = 2,5Pressure angle for cutting tool (original),α = 25°Pressure angle for cutting tool (new), α = 20°Factor of addendum, ha* = 1

• Basic solution

a) Proposal of module (ISO).

Standard module by ISO 54-77, m = 10 (mm)

b) Transverse pressure angle:

c) Pressure angle at the pitch cylinder.

d) Sum of addendum modificationcoefficients.

e) Addendum modification coefficientfor pinion to avoid undercutting


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f) Addendum modification coefficients for gear.x1 = 0,462 > 0,298x2 = xΣ - x1 = 0,462 - 0,462 = 0

g) Other geometrical parameters.Tip diameters: da1 = 168,8 mm ; da2 = 429,56 mm

Root diameters: df1 = 124,24 mm ; df2 = 385,0 mm

SummaryBased on experiences in the analysis, recovery, and conversionof helical and spur gears in the AGMA system to ISO standards,the main definitions and recommendations associated to theaddendum modification coefficient have been presented.Examples with the application of the addendum modificationcoefficient in the solution to practical problems in relation tothe geometry and manufacture of gears shows the applicationof the addendum modification coefficient as an adjustment fac-tor between ISO and AGMA systems. Moreover, the calculationutilized in the two practical cases and formulae relating tobasic gear geometry given in table 3 can be taken as a basefor solution in similar cases of geometrical conversion from theAGMA to the ISO system.

REFERENCES• McVittie, D.; The European Rack Shift Coefficient “X” for

Americans. Gear Technology, Jul-Aug 1993, Pages. 34-36.• Rockwell, P. D.; Profile Shift in External Parallel-Axis Cylindrical

Involute Gears. Gear Technology, Nov-Dec 2001, Pages. 18-25.• Technical Report ISO/TR 4467-1982; Addendum modification

of the teeth of cylindrical gears for speed-reducing and speed-increasing gear pairs. ISO 1982.


FIGURE 5 Conventional and recommended limits for the sum of theaddendum modifications and zones for special cases, according to ISO/TR4467-1982. In the graphic, ΣZv is the sum of virtual number of teeth.

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• ISO 53:1998, Cylindrical gearsfor general and heavy engineer-ing. Standard basic rack toothprofile.

• MAAG Gear Corp., “MAAG GearBook.” Zurich. 1990.

• González Rey, G. Apuntes parael cálculo de engranajes cilíndri-cos según Normas ISO delComité Técnico 60. EUP deZamora. Universidad deSalamanca. 2001


ABOUT THE AUTHORS:Dr. Gonzalo González Rey is principal professor in the machine elements division of the faculty of Mechanical Engineering at the InstitutoSuperior Politécnico José A. Echeverría (CUJAE) in Havana, Cuba. He is also an AGMA member with expertise in the area of ISO/TC60/WG6-13.He can be reached at (537) 266-3607 or via e-mail at [email protected]. Prof. Pablo Frechilla Fernández is titular professorof mechanical engineering at the Salamanca University in Spain. His background is in machine design, with more than 30 years of experi-ence as an advisor. He can be reached at [email protected]. Eng. D. Roberto José García Martín is collaborator professor of mechanical engi-neering at University of Salamanca-Spain at the campus of E.S.P. of Zamora. His background is in the area of design and machine control,and he can be reached at [email protected].

TABLE 3 Formulae relating to basic gear geometry.

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