**By Prasmit Kumar Nayak, A. Velayudham and C. Chandrasekaran**

Evaluation of geometry of used and broken gear is a seldom practice followed in the fields such as maintenance repair and recycling. Although gear calculations and main parameters are standardized, this task can be complicated, especially when there is no information about mating gear and gearbox assembly, change in the accuracy of used gears due to heavy wear, and sometimes the gear geometry probably is non-standard. Such situation calls for effective and accurate methods of gear profile geometry evaluation. Different methods like reverse engineering, iterative process, and analytical methods are being followed in gear industries.

Apart from these, a diverse number of CNC gear generative testing equipment and coordinate measuring machines are available in the industry to inspect the gears using fully automated gear measurement cycles. But 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 basic gear geometry [Grimsley, 2003]. Unfortunately, these machines are costly and often inaccessible to the company or factory involved with gear remanufacturing. Because of this, several researchers [Innocenti, 2007; Belarifi et al, 2008; and Schultz, 2010] have proposed alternative procedures to determine the unknown gear geometry.

Gonzalez et al, 2016, have proposed a procedure to obtain the fundamental gear parameters using conventional measurement tools. In their approach, it is assumed that the involute surface of the flank of a cylindrical gear can give information about the basic gear tooth data needed to determine the unknown gear geometry. Jadhav and Sandooja, 2012, have adopted a step wise analytical approach to find out the basic gear parameters for an unknown gear pair, which eliminated frequent trial and errors, iterations, and complex measurements. Alshennawy, 2014, has adopted a machine vision system coupled with CCD camera as a reverse-engineering tool for developing gear spare parts. Suitable algorithms have been developed for extracting and inspecting the mechanical component. Data have been extracted from the image and used to construct a 3D model and 2D drawing. It has been observed that the detecting of straight lines, holes, and circles are faster and more reliable. However, the accuracy of extracted data from the images are very important for reproducing the component. Charles D. Schultz, 2010, has brought out a methodology for the reliable measurement, evaluation, re-design, and manufacture of replacement parts for gear boxes and industrial machinery.

In the estimation of gear tooth geometry, profile shift/modification factor is an imaginary parameter used to represent the thickness of gear tooth. Negative profile shift leads to smaller tooth thickness and tip diameter and positive profile shift corresponds to the larger tooth thickness and tip diameter. It is to be noted that profile shift, real tooth thickness, and real tip diameter may not match precisely, due to various design and technological considerations. Therefore final drawings of spare parts must contain tooth thickness and tip diameter based on direct measurements and calculations of meshing quality (interference, undercut, overlap, minimum tooth thickness at the tip circle of a gear, etc.).

Of the above methods, the analytical method is more accurate, scientific, and does not involve complex measurements and iterations. Hence, the main focus of this paper is to investigate the used and broken spur gear of a CNC machining center spindle gearbox and estimate the basic gear parameters using standard measuring instruments followed by the use of analytical gear equations as given in DIN 3960 and graphical construction. Typically, the evaluation of gears used in the old gearbox of a CNC machining center, implies challenges to the engineers, since the gears used would be of higher accuracy, and the backlash would be minimum. In general CNC machine tool manufacturers use modified gears.

Gear modifications are carried out by different means and each would influence different parameters. Both the gear and pinion can be provided positive or negative correction. Gears are usually modified to avoid undercutting or to maintain a desired center distance in a gear box. In addition to that, the gear and pinion are now positively corrected to achieve several beneficial effects. These positively corrected gears have better strength at the root and the flank of the tooth. Due to positive correction, the tooth thickness at the root increases, thereby resulting in greater load carrying capacity. In case of a corrected profile, the active profiles formed from the involute curve are generated from the same base circle, but this time a different portion of the curve, which is farther away from the base circle. The main advantage of these gears with an involute profile is that even at an extended center distance, it continues to follow the law of conjugate action and continues to transform uniform angular velocity ratio.

In another type of correction, the mating pair of gears receives equal correction factors, but these two factors are algebraically of opposite signs. Normally, the pinion and gear are provided with positive and negative correction, respectively. In this case, the center distance remains unaltered, it remains the same. Thicker pinion teeth are ensured, and the gear tooth also does not become significantly weak. This correction is used in the case where the reduction ratio is very large.

### Description of Component

There was a requirement to reverse engineer, reproduce, and replace a broken gear component of an old gearbox of a typical horizontal CNC machining center. A schematic of the gear pair is illustrated in Figure 1. The machine was manufactured in 1983 and no spare was available as the machine tool manufacturer closed down the company, thus necessitating the proposed study. The center distance between axis of the gear broken and the axis of the mating gear is 205, ±0.036 mm as shown in Figure 1.

The photographic view of the broken gear is shown in Figure 2.The basic data of the gear pair are measured using standard measuring instruments as seen in Table 1. The common data that can be directly measured are number of teeth, tip diameter, and root diameter for both the gear and pinion. The base tangent length across a fixed number of teeth (number of teeth to be obtained using DIN 3960) was measured using a flange micrometer.

The module of an unknown gear cannot be measured directly. It can be derived from base tangent length. In the case of a corrected gear, the profile modification factor is to be derived. This factor is found out from the measured base tangent length. It is also related to the center distance between the gear pair. An alternative method is also adopted to find the profile modification from the tip diameter of the gear and the pinion.

### Methodology of Gear Geometry Evaluation

The procedure for finding out the profile modification factor and other data is illustrated in Figure 3, with reference to DIN 3960.

### Module

The module is an important parameter in defining the size of a gear tooth. It cannot be measured directly from the gear but can be calculated using equations of base tangent length. As per DIN 3960, the expression for finding the base tangent length is mentioned in equation (1) and (2). Using these two equations, the expressions for the module can be obtained as equation (3) and (4). Further, equations (3) and (4) are combined, and a single equation (7) is derived to find out the standard module. Pressure angle is also an unknown parameter. The standard pressure angles and the measured base tangent lengths are substituted in equation (7), and the nearest standard module is derived.

where,

W_{k}_{1 }– Actual base tangent length of gear over k_{1} no. of teeth

W_{k}_{2 }– Actual base tangent length of pinion over k_{2} no. of teeth

Then

From Table 2, it is observed that for pressure angle of 20 degrees, the difference between calculated and standard modules is minimum. Hence, it is verified that the pressure angle and the module of both gear and pinion are 20 degrees and 3.5 mm respectively.

### Addendum Modification Coefficient

After finding the values of the pressure angle and module, pitch circle diameters are calculated using standard formula. The center distance is equal to half of the sum of reference diameters. From the calculations, it is found that the center distance is 199.5 mm. If it is less or more than the measured value, there is a positive or negative correction on either or both the gear and pinion. In the case of positive correction, gears are pulled apart by an amount ‘X_{1}+ X_{2}’ times module. In this condition, a higher amount of backlash usually results. Therefore, to minimize the backlash, the gear pairs are brought closer to an intermediate value (center distance modification coefficients, ‘y’ to get a new center distance.)

For the modified gear pair, the center distance is equal to the sum of pitch circle radii of gear and pinion and the sum of the center distance modification coefficient times the module. The center distance coefficient ‘y,’ working pressure angle (αw), and the sum of modification coefficients, X_{1}+ X_{2} (theoretical value) are calculated using equations (8) to (10) as per DIN 3960.

To derive the modification coefficient, different methodologies are available. But, since the study involves remanufacturing of a used one, it is proposed to follow the methods that use the measurement of an existing component. In this study, measured base tangent length (BTL) and tip diameter have been used for finding the coefficients, and pin over diameter was used for verification.

### Addendum Modification Coefficient Using the BTL Method

Using the equations (1) and (2), modification coefficients (X_{1}, X_{2}) are obtained from the measured BTL value. The measured BTL values and calculated values of modification coefficients (X_{1}, X_{2}) are given in Table 3.*

### Addendum Modification Coefficient Using the Tip Diameter Method

Using equations (11) and (12), modification coefficients (X_{1}, X_{2}) are obtained from the measured tip diameter value. The measured tip diameter values and calculated values of modification coefficients (X_{1}, X_{2}) are given in Table 4.

### Validation of Addendum Modification Coefficient

The modification coefficients obtained from the two different methodologies are verified through another method called diameter of the gear and pinion over the recommended roller pin diameter. The measured value and the calculated value are compared. The calculated values are obtained by using equations (13) to (22) and tabulated in Table 5.

### Results and Discussion

From Table 5, it can be seen that the sum of the differences D_{xG} and D_{xP} are minimum in the case of D_{x}_{1}_{G} and D_{x}_{1}_{P}. Thus the profile modification coefficients obtained through the BTL method are closer to the measured value.

The gear and pinion with the above obtained modification coefficients are brought closer by, i.e. 0.1829 mm, to achieve minimum backlash of 0.03 mm, which is equal to DIN Class 5 accuracy standard value that is required for a high accuracy CNC machine tool gearbox and maintaining the center distance of gear pair.

### Conclusion

From the approach of evaluating unknown geometry of high accuracy gear the following conclusions can be drawn:

- A relation between module and pressure angle was established, which is applicable for both the gear and pinion. The different standard value of pressure angles is assumed and the corresponding module was obtained from the relation. The obtained value of the module and the standard value of the module were compared. The actual value of the module was ascertained where the difference is found to be minimum.
- The method of finding the addendum modification coefficient using the BTL and tip diameter methods and verification using pin over diameter is found to be useful for gear parameter evaluation of an unknown gear.
- The BTL method is observed to be better for finding the addendum modification coefficient for the gear pair based on the minima obtained between the calculated and measured values of pin over diameter.
- The amount by which the gear pair is to be brought closer to obtain the desired backlash as per DIN class accuracy can be derived using the above methodology.

### References

- Kumar, A.; Jain, P.K.; Pathak, P.M. (2014). Machine element reconstruction using integrated reverse engineering and rapid prototyping approach. Proceedings of the 26th All India Manufacturing Technology, Design and Research Conference, IIT Guwahati, Assam, India. December 12-14, 2014 123, 1-5.
- Grimsley, P. (2003). Software solutions for unknown gear. Gear Solutions. June 2003, 16-23.
- Innocenti, C. (2007). Simple techniques for measuring the base helix angle of involute gears. In Proceedings of the 12th IFToMM World Congress, Besançon, France, June 18-21, 2007, 406-412.
- Belarifi, F. and Bayraktar, E. Benamar, A. (2008). The reverse engineering to optimise the dimensional conical spur gear by CAD. Journal of Achievements in Materials and Manufacturing Engineering. 31 (2), 429-433.
- Schultz, C. D. (2010). Reverse Engineering. In proceedings of the AGMA Fall Technical Meeting. AGMA Technical Paper 10FTM09. 9pp. Milwaukee, Wisconsin.
- G. Gonzalez Rey, A. G. Toll and C. I.E.R. Gonzalez, A Procedure to determine the unknown geometry of external cylindrical gears, Gear Solutions, Feb 2016.
- Jadhav, S. and Sandooja, A. (2012), “Analytical Approach to Gear Reverse Engineering (Spur and Helical),” SAE Technical Paper 2012-01-0812, 2012.
- A. Alshennawy, (2014), A Reverse Engineering Technique for Reproducing Spare Parts using Computer Vision System, International Journal of Scientific and Engineering Research, Volume 5, Issue 10, October-2014.
- Charles D. Schultz (2010), Reverse Engineering, AGMA Technical Paper 10FTM09.

### Notations

*Z** _{1}* – Number teeth on gear

*Z** _{2}* – Number teeth on pinion

*m* – Module of gear and pinion

*X** _{1}* – Modification on gear

*X** _{2}* – Modification on pinion

*α* – Pressure angle

*α** _{w}* – Working pressure angle

*d** _{1}* – Pitch circle diameter of gear

*d** _{2}* – Pitch circle diameter of pinion

*d** _{b1}* – Base circle diameter of gear

*d** _{b2}* – Base circle diameter of pinion

*d*_{α}* _{1}* – Tip circle diameter of gear

*d*_{α}* _{2}* – Tip circle diameter of pinion

*d** _{r1}* – Root circle diameter of gear

*d** _{r2}* – Root circle diameter of pinion

*W** _{k1}* – Actual base tangent length of gear

*W** _{k2}* – Actual base tangent length of pinion

*a* – Standard center distance between the gear and pinion

*a** _{w}* – Modified center distance between the gear and pinion

*y* – Center distance modification coefficient

*d** _{g}* – Measuring pin diameter

*S** _{1}* – Gear tooth thickness for the corrected gear

*S** _{2}* – Gear tooth thickness for the corrected pinion

*d** _{p}* – Distance between the pin for the even number of teeth

*d’** _{1}* –Distance between the pin for the odd number of teeth

*M** _{a}* – Diameter over pin

*α** _{1}* – Pressure angle of the involute profile at the pin center for gear

*α** _{2}* – Pressure angle of the involute profile at the pin center for pinion

D* _{xG}* – Difference between the measured and calculated value of pin over diameter on gear

D* _{xP}* – Difference between the measured and calculated value of pin over diameter on pinion

**ABOUT THE AUTHORS** Prasmit Kumar Nayak, A. Velayudham, and C. Chandrasekaran are with the Combat Vehicles Research and Development Establishment in Avadi, Chennai, India. Nayak can be reached at prasmit1@gmail.com, Velayudham at vel_sivam@yahoo.com, and Chandrasekaran at ccs931955@gmail.com.