The substitution of the conventional trochoid root profile in spur and helical gears by non-conventional root profiles, based on elliptical or Bèzier curves, to reduce the maximum bending stress at the gear root surfaces, has been the subject of an intensive research recently. The application of finite element models in which the load is applied at the highest point of single tooth contact has been mainly used for those studies. However, a complete review of the stress field at the fillet surface and its variation along a cycle of meshing is still missing. The adjacent pair of teeth when it carries part of the load causes compressive stresses on the root area that deserves to be considered for a comprehensive study of the possible benefits of application of non-conventional root profiles. This article focuses on the investigation of the variation of the normal stress in the perpendicular direction to the root line between adjacent teeth. Such variation will be obtained for two cycles of meshing by considering finite element models with five pairs of teeth, so that the effect of load sharing between adjacent pairs of teeth will be considered. This study will provide the variation of the alternating range normal stresses on the root surface in longitudinal direction of the gear teeth for tooth root profiles based on Hermite, elliptical, and Bèzier curves and their comparison in terms of the mentioned stresses with those obtained for conventional root profiles. Several numerical examples corresponding to an existing design of a helical gear drive show the reduction of the maximum principal stress is possible using non-conventional root profiles, although an increment of the alternating component of the normal stress occurs, which may lead to the reduction of the expected fatigue life of the gear drive.

### 1 Introduction

Gear drives have experienced an increasing demand of power density, leading to look for alternative root profiles to reduce bending stresses. A considerable number of methodologies and proposals for application of non-conventional root profiles has appeared in the literature during the last years. Some methods are based on the application of non-predefined curves with a considerable number of control points, whereas other methods are based on predefined curves such as ellipses, Bèzier curves, or Hermite curves, where the number of control variables is limited.

Among the first methods, a first attempt to reduce bending stresses in spur gear drives is presented in [1] where a random adjustment of the root nodes along the radial direction from a fillet center is proposed.

Here, a bi-dimensional finite element model based on one tooth with a given load at the highest point of single tooth contact (HPSTC) is considered. Later, in [2], B-splines are considered to assure continuity in the optimized root profile and reduce the number of control points in respect to the previous work. Here, a bi-dimensional boundary element model based on one tooth with a given load at the HPSTC is considered again, showing reductions up to 19 percent from the bending stresses of the original fillet. In [3], cubic splines with a given number of control points and an optimization process are considered to obtain a structurally optimal root profile. Here, a bi-dimensional finite element model with one tooth and a given load allows the root stresses to be determined. An iterative process is applied to modify the curvature at the control points through some weights that depend on the value of bending stress at such points.

Among the second group of methods, in [4] an elliptical root profile is proposed, showing reductions of root stresses either increasing the form radius or reducing the root radius. Here, a bi-dimensional boundary element model with a given load at the HPSTC is considered. Later, in [5], a Hermite curve is proposed as root profile for non-generated gear-tooth surfaces in spur gears and straight bevel gears, allowing a small number of user-controlled parameters, the initial and final tangent weights of the curve, to reduce bending stresses. In [6], asymmetric root profiles with a larger root edge radius in the driving side are proposed, showing reductions up to 10 percent in bending stresses. In [7], a Bèzier curve with just one control parameter is proposed, showing increments on power capacity from 10 percent to 30 percent. In [8], cubic splines with the control at three parameters (slopes and position of the lowest root point) are proposed showing increments on power capacity from 20 percent to 26 percent.

In all the above-mentioned works, the focus was on the reduction of the maximum bending stress (usually the von Mises stress or the maximum principal stress in the fillet area). Furthermore, most of these works base their studies on a given load condition (load at the HPSTC). In this article, a comprehensive investigation of the bending stresses along the cycle of meshing is provided paying attention not only to the reduction of the maximum bending stress that a non-conventional root profile may provide, but also to the variation of the bending stresses along the cycle of meshing. The results show that, although the maximum bending stress may be reduced, its location in many cases is shifted toward the root line where compressive stresses appear during the cycle of meshing. An investigation of the alternating component of the stress at the root line that is normal to the longitudinal direction of the teeth shows that, in many cases, the non-conventional root profile causes an increment of such a component and, therefore, a decrement in the fatigue life of the gear is expected. Consequently, those gears with a reduced rim thickness would require special attention.

Several numerical examples considering a helical gear drive where the pinion and the gear corresponds to a sun gear and a planet gear, respectively, of a planetary gear transmission, are being considered for this study. Different types of root profiles have been implemented in the planet gear: conventional, elliptical, Hermite curve, and Bèzier curve. The initial design of the gears is provided with micro-deviations on the active tooth surfaces. However, it does not allow increments in the form radius. This reduces the control parameters of the to-be-implemented curves to the root radius and the slope of the tangent at the fillet in its junction with the active profile. Additionally, the Hermite curve allows two additional control parameters through the weights of the initial and final tangents.

The results were obtained by applying tooth contact analysis for two pitch angles of rotation of the pinion and stress analysis using finite element models with five pairs of teeth.

### 2 Implementation of non-conventional root profiles

Gear-tooth surfaces can be generated as envelopes to the family of generating surfaces using the modern theory of gearing [9]. However, nowadays, new procedures to obtain gear tooth surfaces from non-generating processes such as forging, additive manufacturing, or molding, make possible new types of root or active tooth profiles. The geometric modeling of the three types of non-conventional root profiles considered in this investigation — namely Hermite, ellipse, and Bèzier root profiles — is presented later. In such models, it is supposed the active tooth surfaces are already obtained throughout a generating or a non-generating procedure.

#### 2.1 Hermite root profile model

Figure 1 shows a Hermite curve defined between points P_{0} and P_{1}. The active profile has been already defined in coordinate system S_{g}(x_{g}, y_{g}). Point P_{0} is obtained as the point of the active profile whose radius is defined by the user as ρ_{P0}, verifying the equation:

Here, u is the active profile parameter and ρ_{P0} is the actual root form radius of the gear.

Point P_{1} can be determined in system S_{g} as

Here, ρ_{P1} is defined by the user and can be the actual root radius of the gear, f is a portion of the root land in the tooth from the border point R to point P_{1}, and N_{g} is the number of teeth of the gear. The variable f allows the root land to be modified.

Once points P_{0} and P_{1} are determined, the design of the Hermite curve profile between them requires determination of the unit tangents at such points. Tangent **t**_{0g} can be defined as

to keep tangency with the active tooth profile, although it could be defined in a different way as

to keep tangency with the fillet instead of the active profile. Here, λ is the root profile parameter corresponding to the fillet profile and can be determined using an equation similar to (1) with parameter λ instead of u. The tangent obtained through Equation 4 may be different from that obtained through Equation 3 if the gear is provided with undercutting. Furthermore, tangent **t**_{0g} could be defined in a different way using an auxiliary angle γ_{0} (see Figure 1) as

Finally, tangent **t**_{1g} can be defined as

A Hermite curve profile is then obtained as a function of parameter t

Here, w_{0} and w_{1} are the weights of the Hermite curve. The design variables in the proposed Hermite root profile model are ρ_{P0}, ρ_{P1}, f, γ_{0} (in case of application of Equation 5), w_{0} and w_{1}.

#### 2.2 Elliptical root profile model

The variables in this root profile model are ρ_{P0}, ρ_{P1}, f, and γ_{0} (in case of application of Equation 5). Once points P_{0} and P_{1} are defined through Equations 1 and 2, respectively, an ellipse is defined considering: (i) its major axis passes through point P_{1} (see Figure 2), (ii) points P_{0} and P_{1} belong to the ellipse, and (iii) the tangent **t**_{0} (either defined by angle γ_{0} or by a tangent to the active profile or the root profile) is also tangent to the ellipse.

A coordinate system S_{e}(x_{e} , y_{e}) is defined in the center of the ellipse with its axis y^{e} directed along its major axis. Semi-lengths a and b of the major and minor axes of the ellipse, respectively, are unknown and can be determined by solving the following set of equations

Here, (x_{e}^{(P0)}, y_{e}^{(P0)}) are the coordinates of point P_{0} in system S_{e}, and (t_{0ex}, t_{0ey}) are the components of vector **t**_{0} in system S_{e}. These components can be obtained from vectors **r**_{g}^{(P0)} and **t**_{0g} by coordinate transformation from system S_{g} to system S_{e}

Where

and **L**_{eg} being a matrix 3 × 3 that results from removing the last row and the last column of matrix **M**_{eg}. Once a and b are determined, the root profile is derived in coordinate system S_{e} as

and in coordinate system S_{g} as

#### 2.3 Bèzier root profile model

The variables of the Bèzier root profile model are also ρ_{P0}, r_{P1}, f, and γ_{0}. Once points P_{0} and P_{1} are defined through equation (1) and relation (2), respectively, a Bèzier curve can be defined using four points P_{0}, P’_{1}, P_{2}, and P_{3} (see Figure 3) as

with t ∈ [0.0, 0.5] to define the curve from point P_{0} (t = 0) to point P_{1} (t = 0.5).

In order to derive the locations of points P’_{1}, P_{2}, and P_{3}, a coordinate system S_{b} is set with its origin O_{b} located at point P_{1} and its axis y_{b} directed along the radial direction. A coordinate transformation from system S_{g} to system S_{b} allows points P_{0} and P_{1}, and tangent **t**_{0}, to be defined in S_{b}.

Here,

and **L**_{bg} is a matrix 3 × 3 that results from removing the last row and the last column of matrix **M**_{bg}. Determination of point P’_{1} in system S_{b} requires the solution of a system of two equations

Equation 22 is set from Equation 16 considering coordinate transformation **r**_{b} = **M**_{bg}**r**_{g} and coordinate r_{by} with t = 0.5. Since points P_{2} and P_{3} are symmetrically located to axis y_{b} from points P_{0} and P_{1}‘, respectively (see Figure 3), the following relations are required in Equation 22

### 3 Implementation of micro-deviations on the active tooth surfaces

Micro-deviations of the active tooth surfaces adjust the location of the bearing contact and assure a function of transmission errors with a limited peak-to-peak value. Micro-deviations may be provided either along the profile direction or the lead direction of the active tooth surface. Once the tooth surface is defined in its coordinate system S_{g} by position vector **r**_{g}(u,v), a user-defined function can be applied to modify such a vector

Here, (u,v) are the surface parameters, **n**_{g} is a unit normal defined at a surface point given by coordinates (u,v), and f(u,v) is a user-defined function that may implement a tip relief, a root relief, a profile crowning, a back relief, a front relief, a lead crowning, and so on. Several user-defined functions can be added to the same tooth surface with linear, circular, or parabolic progressions. Standard ISO 21771:2007 illustrates a wide variety of possible micro-deviations of the active tooth surfaces.

### 4 Tooth contact and stress analysis

Tooth contact analysis (TCA) is performed following the algorithm that was proposed in [10] and later applied in [11,12]. It assumes the active tooth surfaces are rigid and is based on the minimization of the distance between three active tooth surfaces of the gear and three active tooth surfaces of the pinion, considering 21 contact positions distributed along two pitch angles of rotation of the pinion. The distance to be minimized is obtained from a kinematic point of view, considering the rotation of the gear to approach the pinion tooth surfaces. Once the contact is achieved, a marking compound thickness of 0.0065 mm allows the contour of the bearing contact for that position to be determined. The contour of the bearing contact is obtained as the loci of pinion and gear tooth surface points whose distance is equal to 0.0065 mm. The function of transmission errors is also obtained as:

where φ_{1} and φ_{2} are the angles of rotation of the pinion and gear for each contact position, angles φ_{10} and φ_{20} are the angles of rotation of pinion and gear, respectively, for a reference position, and N_{1} and N_{2} are the tooth numbers of pinion and gear, respectively.

Stress analysis is performed using the finite element method. The finite element models are built following the methodology illustrated in [9] and do not assume any location of the load on the active tooth surfaces. Figure 4 shows an example of a finite element model where a torque is applied to the pinion reference node whereas the gear reference node is held at rest. Both lateral sides and the bottom part of pinion and gear rims constitute rigid surfaces that are rigidly connected to the respective reference nodes. The model considers 21 positions of contact of the pinion and gear distributed along two pitch angles of rotation of the pinion.

The normal stress at the root line in the border between the central tooth and its adjacent one has been determined. This has required a coordinate transformation from the gear reference system to a local-node system, for each node, in order to obtain the stress along the local-node x axis. This x axis is perpendicular to the root line (see Figure 5).

### 5 Numerical examples

#### 5.1 Gear geometry generation

Table 1 shows the basic data of a helical gear drive where the pinion (the sun gear) is a left-hand helical gear and the gear (the planet gear) is a right-hand helical gear. The corresponding cutters are provided with bottom relief in order to obtain undercut active tooth surfaces. The data of the root form radii are required to obtain the profile parameters u of the active tooth profiles during the virtual gear generation.

Figure 6 shows both rack-cutter normal sections applied in pinion and gear tooth generation. Figure 7 shows both pinion and gear models after virtual gear generation in a different scale.

Micro-deviations for pinion and gear active tooth surfaces are applied through Equation 25. Such deviations modify the active tooth surfaces as it is illustrated in Figure 8. Here, the involute tooth surface and the modified tooth surface for the left side of a pinion tooth and a gear tooth are compared, considering only profile deviations. The same profile deviations are obtained in the right side of the teeth. Figure 9 shows the lead deviations for both tooth sides in the pinion, whereas Figure 10 shows such deviations for the gear.

#### 5.2 Tooth contact and stress analyses of the reference geometry

The results of tooth contact analysis (TCA) of the transmission pinion-gear with the modified tooth surfaces described in previous section are illustrated in Figure 11. A contact path in diagonal direction is observed on both the pinion and gear active tooth surfaces. The unloaded function of transmission errors shows a peak-to-peak value of 8 arc seconds.

The variation of the maximum contact pressure at each tooth pair in the model comprising five pairs of contacting teeth is shown in Figure 12. A total of 21 contact positions distributed along two pitch angles of rotation of the pinion are considered. A torque, T=219,040.7 Nm, is applied to the pinion reference node. A maximum contact pressure of 1,306.2 MPa is observed.

Figure 13 shows the variation of the maximum principal stress in the fillet area of the central teeth of the pinion and gear models. A maximum value of 249.3 MPa is reached for contact position 10 in the gear model. This value represents the reference value to be decreased throughout the use of non-conventional root profiles in the gear model. The gear is the initial focus of this research because it acts as a planet gear in a planetary gear drive.

#### 5.3 Analysis of the Hermite curve root profile

A total of 300 geometries are considered combining the values listed in Table 2. Here, w_{0} and w_{1} are the initial and final tangent weights of the Hermite curve, c_{r} is a coefficient that multiplies the module to be added to the root radius to obtain ρ_{P1}, and γ_{0} is the initial tangent angle (see Figure 1). Radius ρ_{P0} cannot be modified (increased) due to restrictions in the meshing of the gear with the pinion, and variable f is about 0.186% of the pitch angle times the root radius and is not modified.

From the results of 300 analysis, it is observed that the reduction of the root radius hardly has benefits. The main contributors to reduce the bending stress are the tangent weights, although an increment in the initial tangent angle can have also some benefits. Figure 14 shows a comparison of the variation of the maximum principal stress at the fillet area of the central tooth of the gear model for the reference geometry and for the best four geometries (geometry numbers 94, 142, 203, and 251). Geometry 203 (w_{0} = 0.6; w_{1} = 0.4; c_{r} = 0.0; γ_{0} = 0.2498) is the best one with a reduction of the maximum principal stress of 14.6% (212.45 MPa). It is observed that the maximum value of the maximum principal stress is reached at step 10 of a total of 21 steps or contact positions distributed along two pitch angles of rotation of the pinion.

Figure 15 shows the maximum principal stress field at the fillet of the central tooth of the gear model for the reference geometry and for the geometry 203, at step 10. It is observed that geometry 203 provides a uniform distribution of the stress and that makes possible a lower maximum value. However, it can be observed as well that the location of the maximum value is shifted toward the root line. Node 30 on the root line has been outlined in Figure 15a because the variation of the normal stress (normal to the longitudinal direction of the teeth) is illustrated below at this node.

Figure 16 shows the variation of the normal stress at node 30 along the 21 contact positions for the reference geometry. This variation proves the normal stress alternates between positive values (traction) and negative values (compression), and, therefore, an alternating component and a mean component exist. The alternating component is calculated here as (149.45 + 129.57)/2 = 139.51 MPa whereas the mean component is calculated as (149.45 − 129.57)/2 = 9.94 MPa. Since the maximum value of the maximum principal stress is closer to the root line in geometry 203, this may cause an increment of the alternating component at the root line where traction and compression stresses coexist.

Figure 17 shows the alternating components of the normal stress along the root line for the geometries being compared. The stress is evaluated at a total of 56 nodes located on the root line. The alternating components reach their higher values around node 30 and are clearly higher in the geometries with Hermite curve (94, 142, 203, 251) than in the reference geometry. Although the mean normal stresses (not illustrated in Figure 17) are positive, their values are low and can be neglected.

Figure 17 shows that geometry 203 provides an increment in the alternating component of about 39.8 percent of that of the reference geometry. Assuming a rim factor Y_{B} = 1 [13] for the reference geometry due to its adjusted rim thickness, this means that geometry 203 would provide a rim factor Y_{B} = 1.398, which is totally undesirable for the considered gear. The adjusted rim thickness in the considered gear is about 57 mm to allocate the rollers without using an outer ring. This is a typical engineering solution to reduce space and weight in large planetary gear drives where the inner surface of the planet rims is used as a raceway for the bearing rollers. This circumstance makes the rim thickness 1.2 times the tooth height and therefore Y_{B} = 1 in the reference geometry.

From the 300 geometries, there are some other geometries where the reduction of the maximum value of the maximum principal stress is lower than the illustrated ones in Figure 14. For example, geometry 118 (w_{0} = 0.4; w_{1} = 0.7; c_{r} = 0.0; γ_{0} = 0.122) provides a reduction of 8.5 percent. However, the increment of the alternating component of the normal stress is 14.9 percent of the reference value and would provide a rim factor Y_{B} = 1.149 that, although it is lower than the one that geometry 203 provides, it may be as well undesirable.

#### 5.4 Analyses of the ellipse and the Bèzier curve root profiles

A total of 30 geometries for each curve type are considered combining the values listed in Table 3. The initial value of γ_{0} (0.017 rad) is the angle of the initial tangent when it is in tangency with the fillet of the reference geometry. The final value of γ_{0} (0.369 rad) is the angle of the initial tangent when it is in tangency with the active tooth profile of the reference geometry. This difference is due to the existence of undercutting in the reference geometry.

In both types of curve, geometry 6 (c_{r} = 0.0; γ_{0} = 0.369) shows the best results. No advantages from reducing the root radius are observed. Figure 18 shows the variation of the maximum value of the maximum principal stress at the fillet of the central tooth of the gear model for the reference geometry and geometries 6 for each type of curve. A reduction of 9.7% (225 MPa) for the ellipse and a reduction of 8.9% (227.2 MPa) for the Bèzier curve are observed. Figure 19 compares the stress field in these two geometries. It is observed that the maximum value of the maximum principal stress is closer to the root line than in the case of the reference geometry (see Figure 15a).

Figure 20 shows the alternating components of the normal stress along the root line for the three above mentioned geometries. An increment of 23.1% (171.7 MPa) is observed for the geometry with the ellipse curve and an increment of 24.2% (173.2 MPa) is observed for the geometry with the Bèzier curve. This would mean a rim factor Y_{B} = 1.231 and Y_{B} = 1.242, respectively, for ellipse and Bèzier curve, assuming Y_{B} = 1 for the reference geometry.

### 6 Conclusions

The performed research allows the following conclusions to be drawn:

• The use of Hermite, ellipse, and Bézier curves as root profiles allows the maximum principal stress to be reduced in the fillet area with reductions up to 14.6 percent, 9.7 percent and 8.9 percent, respectively, for the investigated gear. Other gears may provide different reductions depending on geometric variables such as the module, number of teeth, possibility to increase the form radius, and so on.

• The geometries that provide the highest reduction of the maximum value of the maximum principal stress when applying the above-mentioned curves provide as well a shift of the location of the maximum value towards the root line.

• The investigation of the variation of the normal stress to the root line along the whole cycle of meshing is critical to detect the effect of non-conventional root profiles on the alternating component of the normal stress.

• The geometries with non-conventional root profiles that provide the largest reduction of the maximum principal stress also provide considerable increments of the alternating component of the normal stress to the root line. Such increments advise against the use of non-conventional root profiles in the investigated gear due to its adjusted rim thickness. However, each gear would require a specific investigation.

### 7 Acknowledgements

The authors express their deep gratitude to the Spanish Ministry of Economy, Industry and Competitiveness (MINECO), the Spanish State Research Agency (AEI) and the European Fund for Regional Development (FEDER) for the financial support of research project DPI2017-84677-P.

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