Various numerical methods have been suggested for validation of tooth root bending strength. In these approaches, it has always been a challenge to consider local strength — as determined by hardness and residual stresses — in numerical calculations. Therefore, the analytical approaches as described in standards such as ISO 6336-3 prevail in gear rating. This standard, however, does not consider typical or even less typical deviations or variants in the actual tooth root geometry.

Geometrical deviations in the form of grinding notches occur due to lack of or insufficient protuberance. In this case, the individual result may have little, moderate, or even a significant negative impact on the bending strength. Tooth root geometry may also be different from the shape as described in ISO 6336-3 as generated by form grinding or 5-axes milling.

If carried out with caution, these variants obviously do not cause a significant loss in load carrying capacity but are not fully covered by the analytic calculation methods. The presented approach for external cylindrical gears combines the well-established ISO 6336-3 method with a numerical add-on to identify and consider the additional influence of the actual tooth root geometry. The additional stress concentration in the area of the 30-degree tangent is described as a reduction of safety against tooth root breakage.

### 1 Introduction

Tooth root or bending strength is one of the main criteria to validate a gear design for operation in a specific application. The profile geometry of a typical tooth root of an external cylindrical gear as shown in Figure 1 is not a radius but a curve generated by the radius on the tip of the gear hob during the hobbing process. In most cases, this geometry is not further changed in further manufacturing steps such as gear grinding, where only the gear flanks are machined.

Basic approaches of bending strength calculation are, e.g., described in [1] and [2]. The focus in this article is on the method described in ISO 6336-3 [2]. The main principle of ISO 6336-3 follows [3] and [4] based on scientific studies such as [5, 6, 7].

According to ISO 6336-3 [2], the safety factor *S** _{F}* against tooth breakage is defined as the ratio of tooth root stress limit

*σ*

*and actual tooth root stress*

_{FG}*σ*

*:*

_{F}For external cylindrical gears, the maximum tooth root stress under operation is usually expected at the point where a 30°-tangent touches the root fillet, see [6] and Figure 1. ISO 6336-3 [2] describes an experimentally developed analytical approach to calculate *σ** _{F}* . The tooth root stress limit

*σ*

*is typically determined by pulsator testing for different materials, quality grades, and heat treatments. This calculation method has been used for decades and found reliable for many different gear applications with typical tooth root geometry as generated by gear hobbing.*

_{FG}As the first step to determine the actual tooth root bending stress *σ** _{F}*, the nominal tooth root stress

*σ*

_{F}_{0}is calculated. It is the maximum of the tooth root stress at the 30°-tangent when the tooth flank is loaded by the static nominal torque.

Where

*F** _{t}* is the nominal transverse tangential load at reference cylinder per mesh [N].

*b* is the face width [mm].

*m** _{n}* is the normal module [mm].

*Y** _{F}* is the tooth form factor [-].

*Y** _{S}* is the stress correction factor [-].

*Y*_{β} is the helix angle factor [-].

*Y** _{B}* is the rim thickness factor [-].

*Y** _{DT }* is the deep tooth factor [-].

For details on the parameters, refer to ISO 6336-3 [2].

To account for different operating conditions, various load factors are used to finally evaluate the actual tooth root stress *σ** _{F}* .

Where

*σ*_{F}_{0} is the nominal tooth root stress [N/mm^{2}].

*K** _{A}* is the application factor [-].

*K*_{γ} is the mesh load factor [-].

*K** _{v}* is the dynamic factor [-].

*K*_{F}_{β} is the face load factor [-].

*K*_{F}_{α} is the transverse load factor [-].

For details on the load factors, refer to ISO 6336-1 [8].

The calculation methods according to ISO 6336 are evaluated for a tooth root shape generated by gear hobbing with standard profiles. The scope of ISO 6336-3 [2] says: “The given formulae are valid for spur and helical gears with tooth profiles in accordance with the basic rack standardized in ISO 53.” If the tooth root geometry shows slight deviations from this shape, strictly speaking, the methods of ISO 6336 are not applicable. Moreover, the quantitative deviation of the bending strength, be it higher or lower than for the standard profile, cannot be determined.

For root profiles, which are similar to the profiles as described in ISO 6336, it is proposed in this article to use the approach of the standard and only account for the stress difference by a newly defined correction factor, the profile factor *Y** _{P}*.

### 2 Description of computational approach

The profile factor *Y** _{P}* is suggested to account for tooth root geometries different from the generated shape as described in ISO 6336-3 [2]. A modified nominal tooth root stress

*σ*

*is used for further calculation following the formulae of ISO 6336.*

_{F0,P}Where

*σ** _{F0}* is the nominal tooth root stress [N/mm

^{2}].

*Y** _{P}* is the new stress correction factor to account for tooth root geometry [-].

To determine the profile factor *Y** _{P}*, the following approach is suggested: By numerical calculation, the tangential tooth root stress is firstly determined for the standard gear profile at the 30°-tangent. Secondly, the maximum tangential tooth root stress for the deviating tooth root geometry is evaluated. The ratio of maximum local tangential stress of the variant to the reference stress at 30°-tangent defines the profile factor

*Y*

*. The tangential stress (parallel to the surface) is considered as a relevant parameter, as this is the maximum principal stress in the tooth root area.*

_{P}Following this approach, the profile factor is defined as follows:

Where

*max*(*σ** _{t}*) is the maximum tangential stress of the actual root profile [N/mm

^{2}].

*σ*_{t,0}^{30°} is the tangential stress at the 30°-tangent of the reference design [N/mm^{2}].

### 3 Description of simulation model and boundary conditions

Numerical computation takes place in a two-dimensional space: The plane strain theory [9, 10] describes the static behavior of a single gear tooth in the transverse section as shown in Figure 1. Small deformations and linear material behavior is assumed throughout all computations. Since the comparison between two linear models defines the factor *Y** _{P}*, no absolute (realistic) value for the actual load is required. A unit point load is applied to the model on one flank as shown in Figure 2 at a location that is typical for the outer point of single pair tooth contact.

The numerical implementation is done by means of the boundary-element-method (BEM) [10, 11, 12]. As a result, only a one-dimensional (closed) description of the final contour is needed to set up the simulation model. The underlying geometry is illustrated in Figure 2. In total, three gear teeth are considered. The load application point is on the right flank of the center tooth. A 120° segment of an arbitrary wheel body is considered as indicated in Figure 2. The corresponding interface boundaries to the remaining (not modeled) wheel body are constrained, i.e., no deformation in any direction is possible. The same applies to the internal radius of the wheel body; see blue line in Figure 2.

The evaluation of the results takes place at the center tooth root on the unloaded side, where the stress level can be assumed to be maximum.

### 4 Application and results

A standard gear tooth with a profile as described in ISO 6336 is taken as a reference; the rack profile with the main dimensions is shown in Figure 3; the main gear data are summarized in Table 1.

Four further design variants are studied in this article. The first three variants (V1, V2, and V3) are practical examples for tooth root fillets as generated by alternative manufacturing; the last variant (V4) represents an unfavorable manufacturing strategy (especially smaller radius of curvature at 30°-tangent) and serves as a further example. Figure 4 shows the evaluation area of variant one (V1) with details of the deviation to the reference profile. Figure 5 shows the corresponding details for all four variants.

For the reference and all variants, the tangential stress values along the tooth root fillet are calculated numerically using the BEM. The results are summarized in Figure 6. The height of the blue area over the root contour represents the level of tangential stress. To evaluate the profile factor *Y** _{P}*, the stress level at the 30°-tangent is taken from the reference profile whereas for all variants, the highest stress level along the profile is used. Corresponding to common theory, in all calculations, the location of the maximum stress value is very close to the 30°-tangent and thus, from a practical point of view, an evaluation at the 30°-tangent might also be sufficient. However, the referring profile factor

*Y*

*is calculated for each variant using Equation 5. The results are shown in Table 2. For additional information on the reference profile, the profile factor*

_{P,i}*Y*

*is calculated comparing the maximum stress and the stress at the 30°-tangent.*

_{P,ref}The results show the following:

• For realistic applications, the influence of a slight deviation from the reference profile is within 5 percent of the tangential tooth root stress, V1 … V3.

• With this moderate influence, it is considered reasonable to apply the calculation method of ISO 6336-3 [2] and take account for additional stress concentration using the introduced profile factor *Y** _{P}*.

• The maximum tangential stress for the reference profile is slightly higher than at the 30°-tangent, but the difference (0.5%) is negligible.

• For one variant (V1), the maximum tangential stress is even lower than the value at the 30°-tangent of the reference. To be on the safe side, it is recommended to use Equation 6.

If the tangential stress exceeds the reference by more than 5 percent (e.g. V4), the use of a profile factor *Y** _{P}* is not recommended without further experience and studies; i.e.,

*Y*

*> 1,05 should not be used but rather the profile chosen to be closer to the reference.*

_{P}### 5 Conclusion

Bending strength calculation for standard gear profiles according to ISO 6336-3 [2] has been approved to be reliable for many decades. For slightly deviating profile shapes, the standard does not describe a method. This article provides evidence that with minor numerical effort, the tangential stresses for any profile shape can be correlated with the standard reference and hereby a factor defined to take into account a higher stress level in the tooth root area. This factor is considered for a modified calculation of the actual bending stress safety factor *S** _{F}*.

An example with several variants for the shape of the tooth root fillet was presented, referring adjusting factors for the tooth root stress determined and hereby validated that the impact on bending strength is moderate and can therefore be accounted for by the numerical approach presented.

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