Gear engineering requires professional skills in several action fields such as design, production, operation, maintenance, repair, and recycling. Generally, the main action fields are established by the industry profile. Industries and companies with actions in the maintenance and repair of gears usually ask for skilled professionals in the recovery of these elements.

Typically, the repair of gears implies bigger challenges to the gear engineers, because the problems and solutions involve already-manufactured gears whose geometry is generally unknown. In this situation, the engineer needs to know the previous basic geometry of the gears in order to have a reference for the recovering or remanufacturing.

There are a wide variety of CNC generative gear testers and coordinate measuring machines (CMM) used for inspection and control of spur and helical gears with fully automatic measuring cycles and extremely short measuring times combined with high measuring accuracies. In these advanced gear-measuring machines, the profile of the tooth can be checked and compared with a flank topography reference, and by means of a trial-and-error procedure, it is possible to obtain an approximate geometry of the analyzed gears (Kumar, 2014). Moreover, some advanced measurement machines have incorporated special programs for measuring gears with unknown parameters and determining some important data of the gear basic geometry (Grimsley, 2003). Unfortunately, these machines are costly and often inaccessible to the company or factory involved with gear remanufacturing. Because of this, gear specialists (González Rey, 1999; Innocenti, 2007; Belarifi et al, 2008; and Schultz, 2010) involved with recreating replacement gears are considered alternative procedures to determine the unknown gear geometry using more simple measurement tools.

Consequently, this paper presents a method of reverse engineering to determine the unknown gear geometry in order to have a reference for the design or manufacturing. This method — based on the author’s experiences in the analysis, recovery, and conversion of helical and spur gears — proposes a practical procedure with results not exact, but acceptable, to obtain the fundamental parameters by means of conventional measurement tools. This method is useful for the recreating of new external parallel-axis cylindrical involute gears according to ISO standards by a generation cutting process.

### Basic gear data to determine the unknown external cylindrical involute gear geometry

The question of what data is required to specify an external parallel-axis cylindrical involute gear can be answered by means of the theory associated with the involute helicoids surface of the flank of a helical gear (Maag, 1990). In this case, it is necessary to know the number of teeth, tip diameter, root diameter, base diameter, base helix angle, and base tooth thickness. The first three data can be determined easily by measurement, but the data associated with the base cylinder can be determined only by special gear-measuring equipment. Thus, when only a sample of a gear and not the complete gear data is available initially, the specification for generating the gear can be calculated. Main formulas involved with the theory of the involute helicoids surface of the flank of a helical gear are summarized in Equations 1-8. Some of them are fundamentals in the determination of the gear geometry that fulfills the data requested as a reference for the design or manufacturing.

*Equation 1*

*Equation 2*

*Equation 3*

*Equation 4*

*Equation 5*

*Equation 6*

*Equation 7*

*Equation 8*

Where:

*z *: number of teeth

*m *: normal module (mm)

*x *: addendum modification coefficient

*b *: helix angle at a reference diameter (°)

*d _{a }*: tip diameter (mm)

*d _{f }*: root diameter (mm)

*a _{w }*: center distance (mm)

*d _{b }*: base diameter (mm)

*b _{b }*: base helix angle (°)

*s _{n }*: normal tooth thickness on reference cylinder (mm)

*s _{bn }*: normal base tooth thickness (mm)

*p _{bn }*: normal base pitch (mm)

*a *: pressure angle (°)

*a _{t }*: transverse pressure angle (°)

*h _{a}* *: factor of addendum

*c* *: factor of radial clearance

Standards (Norma NC 02-04-04, 1978; ISO Standard 1340, 1976; and AGMA Standard 910-C90, 1990) with guidelines about the complete information given to the manufacturer in order to obtain the gear required give an example of the proper data to be placed on drawings of the gears for general or special purposes. The previously mentioned information includes details of the gear body, the mounting design, facewidth, and fundamental gear data for manufacturing, inspection, and reference. Usually, the gear data can be efficiently and consistently specified on the gear drawing in a standardized block format. Figure 1 shows the typical gear data block and information required on drawings for standard helical gears according to Cuban Standard NC 02-04-04:1998.

### Initial data and measurements

In the proposed procedure, to calculate the fundamental gear tooth data of an external parallel-axis cylindrical involute gear, it is necessary to know the following parameters:

- Number of teeth on pinion and gear (
*z*,_{1}*z*)_{2} - Tip diameters on pinion and gear (
*d*,_{a1}*d*) in mm_{a2} - Facewidth on pinion and gear (
*b*,_{1}*b*) in mm_{2} - Base tangent length spanned in k teeth on pinion and gear (
*Wk*,_{1}*Wk*) in mm_{2} - Number of teeth spanned for the base tangent length on pinion and gear (
*k*,_{1}*k*)_{2} - Tooth depth on pinion and gear (
*h*,_{1}*h*) in mm_{2} - Center distance (
*a*) in mm_{w} - Helix angle at tip diameter (
*b*) in degree_{α}

**Number of teeth** (*z*): Special care should be taken when counting the quantity of teeth in the gears. It is good practice to make a mark with chalk in the tooth where the count begins to assure that the number of teeth was correctly determined. An incorrect specification of the number of teeth on gears will be catastrophic in the next calculation.

**Tip diameters** (*d _{a}*): A conventional vernier caliper of suitable size can be used to determine the distance between the two outer extremities of external gear teeth in a position diametrically opposed. The measure will always be more accurate in gears with an even number of teeth, but it is also practically applicable in gears with an odd quantity of teeth, and always better in gears with a large number of teeth.

**Facewidth **(*b*): It is the width over the toothed part of a gear, measured along a generator of the reference cylinder. The measurement can be made using a vernier caliper, as well as a simple ruler with precise millimeters.

**Base tangent length **(*W _{k}*): The measurement is made over a group of teeth using a conventional vernier caliper or plate micrometer. For good results, it is required that the controlled flanks are perfectly clean and without appreciable wear. Moreover, the caliper jaws must penetrate sufficiently into two tooth spaces to make tangent contact with the tooth surfaces without interfering with the teeth adjoining the span measurement. Thus, a measurement is taken of the distance between two parallel planes tangent to the outer flanks of a number of consecutive teeth, along a line tangent to the base cylinder. If not considering the space between nonworking flanks of the mating gears when the working flanks are in contact (zero backlash), the distance measured is equal to the normal thickness of one tooth at the base cylinder

*s*plus the product of the number of teeth spanned less one (

_{bn}*k*– 1) and the normal base pitch

*p*(see Equation 9). Suffixes

_{bn}*k*(for pinion) and

_{1}*k*(for gear) after the letter

_{2}*W*specify the number of teeth between the flanks measured. Figure 2 and Figure 3 illustrate the span measurement applied to spur and helical gears.

*Equation 9*

On an external parallel-axis cylindrical involute gear, the actual base tangent lengths (*Wk _{1}* and

*Wk*) are less than the theoretical dimensions for zero backlash by the necessary amount of the normal backlash allowance, but this doesn’t affect the practical results because standard values of gear backlash (ISO/TR 10064-2, 1996) are relatively small (not bigger than 3 to 7 percent of module) for industrial drives with typical commercial manufacturing tolerances.

_{2}In gears with profile or helix modifications, the span measurement should be carried out on the unmodified part of the tooth flank. In some cases, span measurement cannot be applied when a combination of high helix angle and narrow facewidth prevent the caliper from spanning a sufficient number of teeth (see Equation 10). In this situation, alternative procedures should be considered to determine the unknown gear geometry using conventional measurement tools (Regalado, 2000) or an exhaustive search method with a trial-and-error procedure to obtain an approximate geometry of the analyzed gears.

*Equation 10*

Where:

b_{min }: minimum value for facewidth in millimeters. There is an additional value of 1.5 percent to make a stable span measurement.

**Number of teeth spanned for the base tangent length **(*k*): In the case of gears with specified tooth data, the number of teeth spanned for the base tangent length can be calculated (Maag, 1990), but for gears with unknown geometry, the number of teeth between the measuring surfaces can be established so that the points of contact with vernier caliper or plate micrometer are roughly at mid tooth height. The number of teeth to be spanned will be larger for gears with larger numbers of teeth and for gears with a higher helix angle. Recommendations in Table 1, based on the author’s experiences and calculation of the base tangent length, can be used as a guideline for values of the number of teeth for span measurement. More detailed information about values of the number of teeth spanned for the base tangent length from the helix angle, the number of teeth, pressure angle, and the addendum modification coefficient can be obtained in the Maag Gear Book.

**Tooth depth** (*h*): This magnitude is usually specified as the radial distance between the tip and root diameters. Tooth depth may be measured by means of a gear tooth vernier caliper or in a tooth space using a simple vernier caliper with a blade for depth measurements. The caliper blade must penetrate sufficiently and make contact with the surface at the bottom of a tooth space without interfering with adjacent teeth flanks.

**Center distance** (*a _{w}*): Involute gears can operate correctly with a small change of center distance according to the proper tolerances for deviations, but assembled gears with incorrect operating center distance will not operate properly. For this reason, the center distance should be determined with good precision. This magnitude is accepted as the shortest distance between the axes of a gear pair; this is also the distance between the axes of shafts that are carrying the gears.

A common method to determine the gear center distance is the measurement in parallel planes of the center holes distance located in their functional shafts; however, taking into account the accuracy of cylindrical bearing seatings on shafts and in housing bores, a more satisfactory method is to consider the nominal center distance as the sum of the housing bores radii (or outer radii of bearings) plus the distance between them. (See Figure 4 and Equation 11.) Usually, speed reducers and enclosed gear unit boxes have specified the nominal center distance based on a series of preferred numbers (IS0 Standard 3, 1973), and checking it may help determine the nominal value of the center distance.

*Equation 11*

**Helix angle at tip diameter** (*b _{α}*): For spur gears,

*b*=

*b*= 0º, because the helix is a straight line parallel to its rotating axis. However, in the case of helical gears, measurement of the helix angle at the reference diameter is one of the most difficult to specify and should be done with a special helix angle tester. When helix angle measuring is not possible with special equipment, the helix angle at the reference diameter can be approximately determined using a simple method based on measuring the helix angle at tip diameter (

_{α}*b*) with results that are not exact yet acceptable. For this method, it is necessary to apply a marking compound to the tip surface of the external gear teeth and roll the helical gear in a straight line on white paper to collect a generated trace (see Figure 5).

_{α}### Determination of the unknown gear geometry

The output results of the unknown gear have strong relation to the measured values and depend on the uncertainty of the measuring and all manufacturing errors, wear, and deformation on flanks in the gear itself. It is important to understand this concept, because modules, pressure angles, helix angles, addendum modification coefficient, and other gear geometry features are given at calculated values, and they are not necessarily the values used in the initial manufacturing of the gears. However, they are very useful as a reference to establish the fundamental parameters for reproduction of new gears or evaluation of the load capacity of gears.

With the initial data and measurements mentioned earlier, fundamental gear geometry parameters according to ISO standards can be obtained by applying the following calculations.

**Normal module** (*m*): The module m in the normal section of the gear is the same module m of the standard basic rack tooth profile (IS0 Standard 53, 1998) and is defined as the quotient of the pitch *p* (distance measured over the reference circle from a point on one tooth to the corresponding point on the adjacent tooth of the gear), expressed in millimeters, to the number *p*.

*Equation 12*

The module is a commonly referenced gear parameter in the ISO gear system and important in defining the size of the gear tooth. The module cannot be measured directly from a gear, yet it is a common referenced value. Tooling for commercially available cylindrical gears are stocked in standardized modules (ISO Standard 54, 1996, and ANSI/AGMA 1102-A03, 2003).

Generally, when gear generation is complete, a perfect engagement between gear and its generating hob occurs. Thus, the normal module in the unknown gear geometry may be determined by a simple search of the gear generating hob with a known module, which has a perfect mating with the analyzed gear. However, this procedure requires a complete set of generating hobs to give solutions, and it is not economically desirable, especially when the measurement has to be taken in the field.

Moreover, the normal module could be determined using a more practical procedure based on the difference between values of base tangent lengths over a consecutive number of teeth spanned and their relations with the normal base pitch. Once the base tangent lengths have been measured, the value for reference of the normal module may be calculated applying Equations 13 and 14 for pinion and gear respectively. Because the values *m _{1}* and

*m*do not need to be exactly precise, a value of

_{2}*α*@ 20º can be used for calculation purposes.

*Equation 13*

*Equation 14*

Although mating gears can have different base tangent lengths and number of teeth, mating gears must have the same module and pressure angle. For this reason, the correct normal module for gear *m* should be established equal to the nearest standardized module to the values *m _{1}* and

*m*. Table 2 can be used as a guideline for values of standardized normal modules.

_{2}**Helix angle at reference diameter** (*b*): In spur gears, the helix angle at the reference diameter is *b* = 0º. In the case of helical gears, the helix angle at the reference diameter can be calculated based on the measurement of the helix angle at tip diameter (*b _{α}*) as follows:

*Equation 15*

**Nominal pressure angle** (*a*): This is an important characteristic of the standard basic rack tooth profile for cylindrical involute gears cut by the generating tool and constitutes a geometrical reference for involute gears in order to fix the sizes and profiles of their teeth. In general, gears are generated with a cutter normal profile angle chosen from the range between 14.5º and 25º. Standard values for nominal pressure angle are 14.5º, 17.5º, 20º, 22.5º, and 25º. Some gear manufacturers use nonstandard cutter profile angles to accomplish specific design goals. In these cases, this method of reverse engineering can be used in recreating other new gears with standardized values of pressure angle.

Taking into account the sum of theoretical base tangent lengths of both toothed wheels (*Sw _{tk}* =

*w*+

_{tk1}*w*), the nominal pressure angle can be estimated. By means of mathematical processing of Equations 6, 8, 9, and 16 for pinion and gear, it is possible to determine Equation 17. In particular, Equation 17 is relevant because the numerical values obtained are derived directly from the basic gear data specified previously and can be used as an important factor in the decision-making task.

_{tk2}*Equation 16*

With:

Where:

*Sw _{tk }*: sum of theoretical base tangent lengths of mating pinion and gear

*a _{wk }*: pressure angle at the pitch cylinder

*a _{t }*: transverse pressure angle

To determine the nominal pressure angle in the unknown gear, the sum of the theoretical base tangent lengths (*Sw _{tk}* =

*wt*+

_{k1}*wt*) should be compared with the result of the sum of the measured base tangent lengths (

_{k2}*Sw*=

_{k}*w*+

_{k1}*w*). Thus, the nominal pressure angle a must be estimated equal to the nearest standard value of pressure angle with a smaller difference between the sum of the theoretical (

_{k2}*Sw*) and measured (

_{tk}*Sw*) base tangent lengths of both gears.

_{k}The starting value in the search should be 20°, since the majority of cutting tools use that angle, conforming to worldwide acceptance. Smaller pressure angles can be analyzed for gears with higher transverse contact ratios when lower noise levels are desirable. In this circumstance, gears usually have high numbers of teeth and are lightly loaded. Higher pressure angles are sometimes preferred for gears with lower numbers of teeth and heavily loaded when tooth bending strength is required. Table 3 shows a sample of how to determine a nominal pressure angle.

Addendum modification coefficient (*x _{1}*,

*x*): The profile shift is the amount that is added to, or subtracted from, the gear teeth addendum to enhance the operational performance of the gear mating or meet fixed design criteria. For specialists involved with gear design based on ISO standards, it’s known that the datum line of the basic rack profile does not necessarily need to be tangent to the reference diameter on the gear, thus the tooth profile and its shape can be modified by shifting the datum line from the tangential position (González Rey, G. et al, 2006). The main parameter to evaluate the addendum modification is the addendum modification coefficient

_{2}*x*, also known as profile shift factor or rack shift coefficient.

The addendum modification coefficients for pinion (*x _{1}*) and gear (

*x*) can be estimated by Equations 18 and 19 obtained with consideration of normal backlash and mathematical processing of Equations 6, 8, 9, and 15.

_{2}*Equation 18*

*Equation 19*

Where:

*j _{bn}* = normal backlash (mm)

Normal backlash is the shortest distance between nonworking flanks of two gears when the working flanks are in contact. Some backlash should be present in all gear meshes. It is required to assure that the nondriving flanks of the teeth do not make contact. Backlash in a given mesh varies during operation as a result of changes in speed, temperature, and load. The amount of backlash required depends on the size of the gears, their accuracy, mounting, and the application. For the purpose of this procedure, normal backlash is preferably measured with feeler gauges when gears are mounted in the housing under static conditions. When normal backlash can not be measured, Table 4 can be used as a guideline for values of minimum backlash (ISO/TR 10064-2, 1996) recommended for industrial drives with ferrous gears in ferrous housings, working at pitchline speeds less than 15 m/s, and with typical commercial manufacturing tolerances for housings, shafts, and bearings.

Equations 20 and 21, derived from Equation 5, give a possible cross-check for the estimated values of addendum modification coefficients if the normal tooth thicknesses on the reference cylinder for pinion and gear (*s _{n1}* and

*s*) are known.

_{n2}*Equation 20*

*Equation 21*

**Factor of radial clearance** (*c**) **and factor of addendum** (*h _{a}**): Shape and geometrical parameters of the basic rack tooth profile for involute gears are set by special standards (see Table 5) corresponding with the rack-shaped tool (such as hobs or rack-type cutters) used in the cutting of gears by means of generation methods. The dimensions of the standard basic rack tooth profile give information about standardized values of radial clearance and addendum as a multiple of the normal module.

The factor of radial clearance is the distance, along the line of centers, between the root surface of a gear and the tip surface of its mating gear given in relation to normal module. Radial clearance is the same between the root surface and the tip surface for pinion and gear with the same tooth depth (see Figure 6).

Equations 22 and 23 can be used to determine the factor of radial clearances. For the purpose of this procedure, radial clearances are preferably measured with gauges when gears are mounted in the housing under static conditions.

*Equation 22*

*Equation 23*

Equations 24 and 25 derived directly from the basic gear data are given to estimate values of the factor of addendum.

*Equation 24*

*Equation 25*

Since the majority of cutting tools use values of *h _{a}* *= 1 and

*c**= 0.25, conforming to worldwide acceptance, these values should be analyzed first in the search method. It is possible to find another non-standard cutter to accomplish this specific purpose as

*h*= 0.75 for stub gears or

_{a}**h*= 1.25 for gears with deep teeth. In the case of a non-standard system of basic rack tooth profile, Equations 22-25 can be used in recreating other new gears with standardized values.

_{a}*### Conclusion

The theory of the involute surface of the flank of a cylindrical gear can give information about basic gear tooth data needed to determine the unknown gear geometry. Based in the mentioned theory, a procedure of reverse engineering to determine the basic geometry of external parallel-axis cylindrical involute gears has been presented. The proposed method can be used as an alternative procedure to determine the unknown gear geometry using conventional measurement tools.

### References

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