The geometry of plastic gears used today is usually based on conventional steel gears, which are bound to the restrictions of the machining production of gears. The injection molding process provides more design freedom here. In this work, simulative results in terms of tooth root stress in plastic gears with various tooth root fillet designs are shown. The simulation method is based on finite element analysis and considers the different fiber orientation as well as the complex material behavior of short fiber reinforced plastics.

The analysis includes fully rounded, elliptical, and bionic tooth root fillets. The calculation is carried out with homogeneous material (unreinforced thermoplastics) as well as with short-fiber reinforced plastics. In addition to the tooth root stress in the initial state, results are also presented for the geometry changed by abrasive wear during operation. For the initial condition, the results of the finite-element analysis are compared with analytical calculation methods.

The study shows the potential of stress reduction by specific geometrical adjustments as well as the change of the maximum tooth root stress over the running time of plastic gears. The research results can be an aid in the design of plastic gears to achieve a higher tooth root load carrying capacity.

### 1 Introduction

Plastic gears have been used in low-load drives such as watches, toys, and inexpensive household appliances since the mid-20th century [1]. Plastic gears, however, are also increasingly used in applications such as in the automotive industry [2]. The main advantage over steel gears is the low manufacturing cost of mass production by injection molding [1]. Other advantages include the good damping properties and the associated lower noise emissions [3], the low susceptibility to corrosion and the ability to self-lubricate [4]. Compared to steel gears, the main disadvantage is the lower transmittable power with the same size. The fiber reinforcement of thermoplastics with glass or carbon fibers can increase the transmittable power enormously [6].

To maximize economic efficiency, the design usually tries to achieve the most compact size possible so that energy efficiency and efficiency can be increased due to the lower weight with mobile gears and material consumption can be reduced. At the same time, however, the gearing must be able to withstand the loads during operation and, depending on the application, a corresponding level of safety against failure must be ensured. Tooth breakage is a typical type of damage that can lead to gear failure as a result of the tooth root load capacity being exceeded. The actual stress state in the tooth root is complex. The stress distribution depends on the manufacturing process and the force contact point. It is influenced by the notch effect of the tooth root rounding and, in the case of external gears, usually has its maximum at the cross section resulting from the 30-degree tangent to the tooth root fillet to the tooth centerline.

Compared to steel gears, some special features have to be considered in the computational design of plastic gears. For example, due to the low modulus of elasticity, the contact path increases significantly under load, which is investigated in [7] and [8]. In addition, the material description for thermoplastics with or without fiber reinforcement is much more complex than for metallic materials. In the range relevant to gears, the stress-strain behavior is non-linear, strongly temperature-dependent and influenced by other factors such as loading speed and time, fiber orientation, and relative humidity [9].

In the following, a study is presented that investigates the tooth root stresses of different tooth root fillet design geometries. A trochoidal tooth root geometry is created by the hobbing process commonly used for steel gears. By systematically optimizing the tip rounding of the tool used, the tooth root stress can be reduced [10, 11]. Regardless of the manufacturing process, various other approaches exist for reducing stress by optimizing the tooth root geometry. One possibility is iterative approaches that search for an optimum by successively adapting the geometry and then calculating the stress numerically [12, 13]. Concrete geometric shapes used to construct optimized tooth root curves include cubic splines [14], so- called B-splines [15], circular arcs, and ellipses. Through the latter, gears made of the high-performance plastic PEEK are able to bear up to 15 percent higher loads compared to trochoidal tooth root geometry [16].

The simulation model presented later combines a finite element analysis to determine the gear stress and deformation with a tribological simulation to analyze wear. Abrasive wear alters the tooth geometry, which in turn affects the mechanical stress. This interaction becomes calculable through the time-resolved simulation model.

In this study, tooth root geometries with trochoidal, elliptical, fully rounded, and bionic profiles for steel- plastic gear pairings are investigated on two different gear variants to reduce the maximum bending stress in the tooth root surface. Unreinforced polyamide (PA46) and fiber-reinforced polyamide (PA46GF30) will be used for the plastic gear. Each of these materials is assigned a corresponding material model, which will be an isotropic elasto-plastic model for PA46 and an anisotropic elasto-plastic model for PA46GF30.In addition to the tooth root bending stress in the initial state, results are also presented for the geometry changed by abrasive wear during operation, see section 5.

### 2 Gear geometry

In this study, investigations are carried out on two different gear sizes, see Table 1. The gear pairs can be described as a standard gear geometry of size 1 & 2 (module 1 and 2 mm) according to the guideline VDI 2736 [17] and will be referred to as “VDI1” and “VDI2” in the following work. For both gear sets, one of the gears is made of steel 16MnCr5 (1.7131/AISI 5110). The plastic gears are made of Polyamide 46 (PA46) and of fiber reinforced Polyamide 46 containing 30 percent of short glass fibers (PA46GF30). Table 1 shows the most important geometry parameters and Table 2 shows the load used in the following study. Here the output torque *T*_{ab} and the drive speed *n*_{ab} is given for the driven plastic gear. Figure 1 shows the contours of the contours of the gearing.

### 3 Tooth root fillet

Within the scope of this study, different tooth root geometries are investigated on the plastic gear. The analysis includes conventional trochoid, fully rounded, elliptical, and bionic tooth root fillets. In the following chapters, the non-conventional tooth root fillets are described.

#### 3.1 Elliptical tooth root

The elliptical tooth root curve is calculated in such a way that, given the root circle diameter *d*_{f,e}, it merges into the tooth flank on both sides in a continuously differentiable manner at the root form diameter *d*_{Ff}.

Equation 1 shows the general equation for the ellipse with the coordinates (*x*_{e}, *y*_{e}) according to [18]

In the coordinate system according to Figure 1, the semi-axis *b*_{e} coincides with the y-axis. The coordinates of the center of the ellipse Me are therefore *x*_{Me} = 0 and *y*_{Me} = *d*_{f,e}/2 + *b*_{e}. The equation for the ellipse thus results in:

Since only the lower elliptical curve forms the tooth root curve, the following applies to the coordinates *y*_{zf,e}(*x*_{zf,e}) of the elliptical tooth root curve:

For a continuously differentiable transition of the tooth flanks into the elliptical curve at the point *P*_{dFf} and on the opposite side, Equations 4 and 5 must be fulfilled [18].

Here, according to [19], the following applies to the coordinates of the transition point *P*_{dFf}:

To determine the still unknown semi-axes *a*_{e} and *b*_{e} of the ellipse, this system of equations is solved numerically. Thus, the elliptical tooth root curve *y*_{zf,e}(*x*_{zf,e}) for *z*_{zf,e} ∈ [–*x*_{dFf}, *x*_{dFf}] can be described unambiguously.

The described elliptical tooth root profile is therefore only dependent on the tooth root diameter *d*_{f,e}. However, the tooth root stress can also be optimized by varying the profile factor *ℎ**_{fP}, which will shift the position of the Point *P*_{d}_{Ff} .

As a starting point for the parameter study in chapter 5.1, the tooth root diameter *d*_{f} of the reference gear was chosen. Depending on the module *m*_{n}, the root diameter was then varied with the factor *e*_{f}, see Equation 8.

### 3.2 Fully rounded tooth root

The calculation of the fully rounded tooth root curve can be derived from the elliptical tooth root curve. However, since a circular arc is only described by a radius *r*_{v} instead of the two semi-axes *a*_{e} and *b*_{e} of the ellipse, the degrees of freedom are limited to the x and y coordinates of the center point *M*_{v}(*x*_{Mv}, *y*_{Mv}) and the radius *r*_{v}. This allows the root circle diameter *d*_{f,v} of the fully rounded tooth root curve cannot be varied for a continuously differentiable transition into the tooth flanks, since it is determined by the other conditions. A general circle equation with the coordinates (*x*_{v}, *y*_{v}) is therefore as follows

Due to the position of the coordinate system according to Figure 3, the following applies in the same way as for the ellipse: *x*_{Mv} = 0 and *y*_{Mv} = *r*_{v} + *d*_{f,v}/2 . This results in *y*_{v}(*x*_{v}) for the circle equation

As with the ellipse, only the lower circular arc forms the tooth root curve. The following therefore applies for the coordinates *y*_{zf,v}(*x*_{zf,v}) of the fully rounded tooth root curve

In contrast to the elliptical tooth root curve, the root diameter *d*_{f,v} cannot be specified and is calculated as follows

The root circle diameter is therefore determined by the position of the point *P*_{d}_{Ff} and the still unknown radius *r*_{v} of the circular arc. This can be calculated from the requirement of the continuously differentiable transition of the involute into the root curve with Equation 13.

Compared to the elliptical tooth root, no numerical equation system has to be solved in this case, instead the tooth root geometry can be determined with purely analytical equations.

As already mentioned at the beginning, there are no degrees of freedom for the parameterization of the fully rounded tooth root. However, the tooth root stress can still be optimized by varying the profile factor *ℎ*^{*}_{fP}, which will shift the position of the Point *P*_{d}_{Ff} .

#### 3.3 Bionic tooth root

The calculation of the bionic tooth root rounding is based on [19]. Placing the line of symmetry through the tooth root on the y-axis, the involute for this tooth root geometry in the transition diameter *d*_{r} goes tangentially into a tangent function of the form

and

respectively. These two functions are then joined in the middle of the tooth space by a circular fillet with radius *r*_{b} with a continuously differentiable transition in the points with distance *x*ū to the y-axis.

The angle *γ*_{b} and the radius *r*_{b} can theoretically be chosen freely. However, for an improvement over conventional gearing, *γ*_{b} should in any case be chosen smaller than 65°. In [19], it is stated it has proved particularly preferable to determine the angle according to the relationship in Equation 16, whereby good results can be achieved in a tolerance range of ±20% around this value [19]:

Herein is:

*α*_{t} Transverse pressure angle at the transition diameter *d*_{r} in rad.

In [19] it is stated that for the radius *r*_{b} of the circular path, values in the range of 0.1-0.6 times and preferably in the range of 0.3-0.4 times the normal modulus *m*_{n} have proven to be particularly suitable. A further improvement can be achieved by referring to the tooth space width *S*_{L} as a chord at the transition diameter *d*_{r}. Here, too, values in the range of 0.1-0.6 times and preferably in the range of 0.3-0.4 times the tooth space width *S*_{L} are recommended. For this reason, the factor *b*_{f} is introduced:

### 4 Simulation Method

#### 4.1 Finite Element Analysis (FEA)

The study of load, deformation, and stress distributions for discrete gear mesh positions is performed using the FEA Software Abaqus. The gear geometry is generated using a parametric Matlab script, which includes the geometry of the tooth root fillet and profile corrections. After defining the input parameters (gear geometry, material data, meshing parameters, etc.), the FE model is automatically generated, the FEA is performed, and the relevant data is automatically evaluated so it can be used for further calculations. For reasons of calculation time, simplifications have to be made for the FE model as seen in Figure 5. One of these simplifications is the number of modeled teeth. To further save computation time, the model includes only half the tooth width, and a symmetry boundary condition in tooth width direction was specified. Furthermore, the steel gear was additionally simplified to a rigid shell model, since there is a stiffness difference of up to a factor of 50 between the steel and plastic gears. Figure 5 also shows the mesh of the gear model for the initial geometry. Linear elements with reduced integration points are used.

The gear segments are coupled at their edges with their center of rotation in all degrees of freedom. The center of rotation of the driving steel gear and the driven plastic gear has been restricted in all degrees of freedom except rotation about its own axis. The torque is specified as the output torque on the plastic gear. Furthermore, a rotation angle is specified at the reference point of the steel gear, see Figure 5.

Figure 6 shows the distribution of the equivalent stress (von-Mises-stress S. Mises) and the contact pressure (CPRESS) for an exemplary meshing position, whereby the steel gear is blanked out.

A well-known method of displaying results for gears is to show the respective value over the theoretical contact path (A to E). With increasing load, greater deformation of the tooth pairs involved in meshing occurs, which results in an increase in the effective contact ratio of the gearing and the contact path. The results are displayed via the meshing distance A* to E*, which includes the contact path increase under load. For a simpler presentation of the results, the results for spur gears are reduced to a 2D representation within the scope of this work. For this purpose, for example, the maximum values in the tooth width direction for the respective meshing position are taken to represent the resulting contact pressure. The contact position is determined on the basis of the maximum pressure. The evaluation of the tooth root stress is carried out on the entire tooth root fillet. Here, the maximum tooth root stress is also evaluated and assigned to each contact position.

The focus of this study is on the bending stress. To determine the bending stress, a local coordinate system is placed for each element at the tooth root surface. Here, the 1-direction follows the tooth root surface of the element. The stress in the 1-direction is evaluated and named as tooth root stress. The advantage of this procedure compared to the evaluation of the principal stress is that compressive stresses, e.g., resulting from contact within the tooth root area, are not evaluated. In the context of this study, it can be assumed that due to high wear, there is also contact of the gears within the root fillet.

Figure 7 shows an exemplary result for the maximum tooth root stress (bending stress) over a meshing cycle. It can be seen that the contact path begins significantly earlier than the theoretical start of the contact at point A and ends significantly later than the theoretical end of point E, which shows the contact path increases significantly under load.

#### 4.2 Material model

An isotropic elasto-plastic material model is used to calculate isotropic material like PA46. This requires the input of the elastic modulus *E*, the Poisson’s ratio *v*, and the strain hardening curve (see Figure 10). Since this is a rather well-known model, it will not be described any further here.

For the calculation of the anisotropic material PA46GF30, an existing anisotropic or more precisely an orthotropic elasto-plastic material model from Abaqus is used, taking temperature and strain-rate dependence of the material stiffness into account. In the scope of this study, only investigations at room temperature are carried out, which is why the temperature dependence is not discussed in the further course of the work. For a better description, the model is divided into an elastic material model component and a plastic material model component. The elastic material model component requires the nine engineering constants *E*_{11}, *E*_{22}, *E*_{33}, *v*_{12} , *v*_{13}, *v*_{23}, *G*_{12}, *G*_{13}, *G*_{23}. Here, *E* is the Young’s modulus, *v* the Poisson’s ratio and *G* the shear modulus. Furthermore, a transversal isotropic material behavior can be assumed. In this case, the required engineering constants can be reduced to five with *E*_{22} = *E*_{33}, *v*_{12} = *v*_{13}, *G*_{12} = *G*_{13} and *G*_{23} = *E*_{22}/2(1+*v*_{23}).

The plasticity model is used with Hill yield surfaces [25] and the associated plastic flow, which allow for anisotropic yield. Hill’s potential function is a simple extension of the Mises function, which can be expressed as follows

with

and

where

*σ*^{0} is reference yield stress.

*τ*^{0} is reference shear stress *τ*^{0} = *σ*^{0}√3.

*σ*_{–}_{ij} is measured yield stress.

The difficulty in the practical application of the Hill model is the determination of the yield ratios *R*_{ij}. These were determined in [20] for PA46GF30 and are shown in Chapter 4.3.

In order to use the material model, the fiber orientations must be known. In [20], the fiber orientations for fiber reinforced plastic gears were investigated with the aid of CT images on a nanotome (General Electrics). It was shown that the orientation follows the tooth surface, due to the flow direction in the injection molding process, and makes a transition inside the gear, see Figure 8. The color disk shows the fiber orientation.

This characteristic was used to replicate the fiber orientation in the model. For this purpose, the tangent vectors of the tooth geometry were calculated. Then the meshing grid is built up in the FE model. For each element in the FE model, the coordinates at the centroid are determined. Afterwards, the fiber orientation is determined within the tangent vectors of the gear geometry by means of interpolation. The result is shown in Figure 8. A very good agreement between CT measurement and model can be observed.

Once the fiber orientation has been calculated for each element, a local coordinate system is defined for each element where the 1-direction corresponds to the fiber orientation and the 2- and 3-directions are orthogonal to the fiber orientation, see Figure 9.

#### 4.3 Material data

#### 4.3.1 PA46

The data for the calculation of the isotropic plastic PA46 were taken from data sheet [24]. The modulus of elasticity was given as *E* = 3,300 MPa and the Poisson’s ratio as *v* = 0.4. The strain-hardening curve was determined from the stress-strain data of the tensile test and is shown in Figure 10 for 23°C.

#### 4.3.2 PA46GF30

The simulation for fiber reinforces plastic gears is carried out using an anisotropic elasto-plastic material model, taking temperature and strain-rate dependence of the material stiffness into account. Within the scope of this study, only investigations at room temperature are carried out, which is why the temperature dependence is not discussed in the further course of the work.

Fiber-reinforced tensile specimens usually exhibit a 3-layer structure consisting of strongly aligned edge zones and a comparatively randomly oriented middle layer. Thus, all stress-strain curves and characteristic values determined in the mechanical characterization are also the result of this 3-layer structure. For the component simulation, however, only the properties of the highly oriented outer layers are needed as input, so the 3-layer structure of the tensile specimen falsifies the transferability of the characteristic values. To overcome this problem and to provide the simulation with adapted characteristic values as input, a micromechanical simulation series was carried out in [20]. There, extensive tensile tests were carried out on PA46GF30 in the conditioned state at 50 percent relative humidity depending on temperature and strain rate, see Figure 11. Subsequently, the engineering constants were determined with the help of micromechanical simulations. These results are used in this study for the elastic part of the material model and are shown in Table 3 for room temperature.

For the plastic material model component, the Hill flow model is used as described in chapter 4.2. For this purpose, the hardening curves for different orientations and strain rates were evaluated, see Figure 12. The hardening curves for different strain rates are given to the material model in tabular form. The Hill yield ratios *R*_{ij} have also been determined in [20] with *R*_{11} = *R*_{12} = *R*_{13} = *R*_{23} = 1 and *R*_{22} = *R*_{33} = 0.66.

A coefficient of friction of *μ* = 0.2 is specified in [21] for steel/plastic contact. This friction coefficient is used for the FEA and also for the wear calculation.

#### 4.4 Wear simulation

A local correction of the flank by the corresponding wear heights *δ*_{V} is carried out in order to consider the changing flank geometry due to wear in the simulative examination of the gear drive. The corrected flank geometry is then re-meshed in the FE model and used for the next runtime iteration. This results in a changed contact situation between the gear pair. To ensure wide-ranging validity, the model for simulating local wear should be physically based and is therefore based on the energetic wear calculation approach according to Fleischer [22] and was presented for the wear calculation of plastic gears in [26]. This approach provides for a linear relationship between the solid friction work *W*_{R} and the wear volume *V*_{V}:

Here, *e*_{R}* is the wear energy density, which has to be determined on the basis of experimental investigations. It describes the material-specific resistance to abrasive wear and is higher the more energy must be mechanically introduced into the material to remove individual material particles. For the experimental determination, the applied solid friction work is related to the total wear volume according to Equation 26. This results in an average wear energy density, which is equally used in the wear calculation for all points of the calculation mesh of the gear flank. To calculate the wear volume according to Equation 26 requires knowledge of the friction work *W*_{R} in the contact between the gears as well as the wear energy density *e*_{R}*. The friction work *W*_{R} is calculated locally at each point of the computational grid. With the local friction force Δ*F*_{RF} and the local friction distance Δ*s*_{R}, the local wear volume Δ*V*_{V} can be written as follows:

The friction force Δ*F*_{RF} can be calculated with the locally acting line loads *w*_{b}, the width of a single contact line element Δ*b*_{H} and the friction coefficient *μ*_{Gr}

After substituting Equation 28 into Equation 27, the following happens

The friction coefficient depends on the material of the plastic gear as well as the steel gear, the contact temperature as well as other influences. The friction distance Δ*s*_{R} is calculated according to [23] via the sliding speed at the point of contact *v*_{g} and the contact time Δ*t*. However, the contact time Δ*t* is determined from the FEA.

If the local wear volume is normalized with the contact area at the calculation point Δ*A* = 2*a*_{H} ∙ Δ*b*_{H} and the approximation of the wear volume with Δ*V*_{V}/Δ*A* ≅ *δ*_{V} is taken as the sum of cuboids with base area Δ*A* and height *δ*_{V}, Equations 29 and 30 can be used to give the wear height with Equation 31.

The wear simulation was validated in [20] with experimental results and good agreement was obtained at high loads between simulation and experiment. Steel-plastic pairings with PA46GF30 and a similar gearing to VDI2 were also investigated here. The friction energy density was chosen to be *e*_{R}* = 1.45 ∙ 10^{13} Nm/m^{3} for high loads.

### 5 Results

As described in Section 3, the geometry of the tooth root fillet curve depends in most cases on at least one parameter. The aim is to find a good combination of these parameters, to reduce the maximum bending stress at the gear root surfaces. For this purpose, a parameter study was carried out on two different gear sets using the nominal geometry. The best parameter combinations are compared to each other. A wear simulation was carried out using the best combination of parameters to evaluate the tooth root stresses over the running time for fiber reinforced plastic (PA46GF30), see Section 5.2.

#### 5.1 Tooth root stress with nominal gear geometry

The optimization of the tooth root stress depends on two parameters in the case of the conventional trochoid root profile (*ℎ**_{fP} and *ρ**_{fP}), the bionic root profile (*γ*_{b} and γ_{b}), the elliptical tooth root profile (*e*_{f} and *ℎ**_{fP}) and on one parameter in the case of the fully rounded tooth root profile (*ℎ**_{fP}). The parameters were varied until contact was made at the base of the tooth or when the stress increased significantly.

#### 5.1.1 PA46

The results of the parameter study for the maximum bending stress *σ** _{F}* with the material PA46 are shown in Figure 13. As expected, the results for the trochoid show a decreasing stress with increasing

*ρ**

_{fP}both for the results from the FEA and for the calculation according to the VDI 2736. The parameter

*ℎ**

_{fP}shows only a slight influence on the tooth root stress for size VDI1 and a tendency toward decreasing stresses with decreasing

*ℎ**

_{fP}for size VDI1 and a tendency towards decreasing stresses with increasing

*ℎ**

_{fP}for size VDI2.

Comparing the FEA results of the trochoid with the results of the analytical calculation according to VDI 2736, it can be seen that for VDI1 there is good agreement, whereas the VDI 2736 overestimates the stresses by about 20 percent for VDI2.

The bionic tooth root shows a reduction in bending stress with a small *b*_{f}. The angle *γ*_{b}, however, should be selected large for this gearing in combination with the material pairing steel/PA46. There are also no significant differences in the stress behavior between the VDI1 and VDI2 gearing. Regarding the bionic tooth root profile, an optimum was determined with PA46 for the geometry VDI1 in the range at *b*_{f} = 0.1 and *γ*_{b} = 35° and for the geometry VDI2 at *b*_{f} = 0.1 and *γ*_{b} = 40°.

Regarding the elliptic tooth root, it seems to be important to choose the correct value of *ℎ**_{fP} for each *e*_{f}. Both parameters should be chosen preferably small, as can be seen in Figure 13.

The fully rounded tooth root can only be influenced by changing the parameter *ℎ**_{fP}, whereby a small value is advantageous for the bending stress reduction.

Figure 14 shows a comparison of the maximum tooth root stresses for each best case of the respective tooth root geometries. The elliptical tooth root leads to the lowest bending stress in the tooth root of both the VDI1 and VDI2 gear variants. The bionic tooth root shows a low tooth root stress in the VDI2 variant, whereas, in the VDI1 gearing, the bionic tooth root shows the highest stress. Another noticeable feature is that all optimized profiles show only minor differences in the max. bending stress. In VDI1, the trochoid shows a 2.66% higher stress, the bionic profile shows a 5.73% increase and the fully rounded profile a 2.62% increase compared to the elliptical profile. In VDI2, the trochoid shows a higher stress of 6.43%, the bionic profile by 0.56% and the fully rounded by 10.33% compared to the elliptic tooth root.

#### 5.1.2 PA46GF30

The results of the parameter study for the maximum bending stress *σ** _{F}* are shown in Figure 15. A very high stress reduction can be achieved by the optimization for all profiles.

As expected, the results for the trochoid show a decreasing stress with increasing *ρ**_{fP} both for the results from the FEA and for the calculation according to the VDI 2736. The parameter *ℎ**fP shows only a slight influence on the tooth root stress for size VDI1.

Comparing the FEA results of the trochoid with the results of the analytical calculation according to VDI 2736, a different behavior is seen here, as seen for PA46. The stresses in the VDI1 variant are underestimated by up to 25 percent by VDI2736. The results of variant VDI2, on the other hand, show acceptable agreement.

The bionic tooth root shows a reduction in bending stress with a small *b*_{f} for VDI2, whereas with the gearing VDI1 this tendency can only be seen at smaller angles *γ*_{b}. In both cases an optimum in the range of *γ*_{b} = 30° could be achieved.

Regarding the elliptical tooth root, an optimum has been achieved for VDI1 and VDI2 at the smallest values of *ℎ**_{fP} and the factor *e*_{f} = –0.2 and *e*_{f} = –0.4.

The fully rounded tooth root can only be influenced by changing the parameter *ℎ**_{fP}, whereby a small value is advantageous for the bending stress reduction. At the gear geometry VDI2, the value *ℎ**_{fP} could not be further reduced because contact occurred in the tooth root at *ℎ**_{fP} ≤ 1.2.

Figure 16 shows a comparison of the maximum tooth root stresses for the best case of the respective tooth root geometries. It can be seen that the elliptical or fully rounded tooth root shows the lowest bending stress in the tooth root of the VDI1 gearing. However, regarding the gear VDI2, the bionic tooth root leads to the lowest bending stress. Another noticeable feature is all optimized profiles show only minor differences in the maximum bending stress. In VDI1, the trochoid shows a 2.12% higher stress and the bionic profile a 6.96% increase compared to the fully rounded tooth root. In VDI2, the trochoid shows a higher stress of 7.3%, the ellipse by 4.84% and the fully rounded by 2.63% compared to the bionic tooth root.

#### 5.2 Tooth root stress over runtime

Plastic gears are subject to very high wear, especially in dry operation. Figure 17 shows an exemplary simulation result for the gear geometry VDI2 with a bionic tooth root fillet. The load case is specified in Table 2. At the beginning of the wear process, wear occurs mainly at the areas toward the root and tip of the profile. This is due to the elongation of the line of action under load. After a certain amount of wear has been removed, the changed contact situation also causes wear on the pitch diameter *d*. It can also be seen that the contact pattern of the tooth flank increases with increasing wear toward the direction of the tooth root and a wear notch is formed in the tooth root fillet, which promotes failure at this location. Therefore, a stress-optimized initial tooth root geometry does not necessarily mean it is still the best option when wear has altered the gear geometry.

With the best parameter combinations of each tooth root profile, a wear simulation was therefore carried out on the VDI2 gear geometry with the load case from Table 2 in order to evaluate the tooth root stress over the running time. The results are shown in Figure 18. It can be seen that, for all geometries, a high increase in stress can be observed with increasing running time due to the reduced cross-section and the wear notch of the teeth due to wear. In addition, the force application point shifts, and the force application angle changes when the geometry changes due to wear. In contrast to the other tooth root geometries, the fully rounded tooth root shows the highest bending stresses over the running time. The other tooth root profiles, however, are hardly distinguishable from each other. This is due to the fact that the highest stress shifts to the wear notch, see Figure 17. The trochoidal, elliptical, and bionic tooth root geometries coincide in the area where the wear notch is formed and form an almost identical wear notch, which is why hardly any difference can be observed during the running time. However, the wear notch of the fully rounded tooth root starts almost in the middle of the rounding. This ensures the remaining tooth root geometry below the wear notch can only counteract a small amount of bending loads.

### 6 Summary

In this study, tooth root fillet geometries with trochoidal, elliptical, fully rounded, and bionic profiles for steel-plastic pairings were investigated on two different gear sizes. Unreinforced polyamide (PA46) and short fiber-reinforced polyamide (PA46GF30) were used as the plastic material. An elasto-plastic material model was used for PA46, and an anisotropic elasto-plastic material model accounting for inhomogeneity and fiber orientation was used for PA46GF30. In addition to the tooth root bending stress in the initial state, results were also presented for the geometry changed by abrasive wear during operation.

Regarding the initial condition of the gears, a parameter study was performed for each profile to identify the optimum parameters of every fillet geometry for low bending stress. It was shown that a significant stress reduction was achieved for all profiles through optimization, although no clear optimum tooth root profile could be determined. A clearly different behavior in the optimum parameters between fiber-reinforced material and non-fiber-reinforced material could also be shown. Fiber-reinforced material must therefore have a different geometry than non-fiber-reinforced material. Depending on the material and the gearing variant, the highest stress reduction was achieved with the help of the fully rounded and bionic tooth root.

To consider wear, the finite element analysis was coupled with a transient wear simulation. The investigations were carried out on a gear variant made of short fiber reinforced polymer (PA46GF30). It was shown that, for this variant, the fully rounded tooth root exhibits a significantly higher stress over the running time. In contrast, hardly any difference was found between the bionic, elliptical, and trochoidal tooth root during the running time. This is due to the fact that the highest stress shifts to the wear notch. The trochoidal, elliptical, and bionic tooth root geometries coincide in the area where the wear notch is formed and form an almost identical wear notch, which is why hardly any difference can be observed during the running time.

Analytical methods for gear design reach their limits when inhomogeneous materials are considered. For this reason, it is recommended that numerical methods be used to optimize the tooth root geometry of gears made of fiber-reinforced plastic, where complex material behavior can be represented. It should also be noted these are purely simulative studies, and the results should be supported with experimental verifications in the future.

### Bibliography

- McGuinn J, 2016, “Plastic Gearing Continues Converting the Unconverted,” Gear Technology, 36-39.
- Tavcˇar J, Grkman G, Duhovnik J, 2018, “Accelerated lifetime testing of reinforced polymer gears,” Journal of Advanced Mechanical Design, Systems, and Manufacturing 12, 1-13. https://doi.org/10.1299/jamdsm.2018jamdsm0006.
- Parab R, Salve V, Ghadi P, 2018, “Gearbox Noise Reduction using Engineering Plastic Gears,” International Journal of Engineering Research and Technology 7(10), 23-26.
- Singh AK, Singh SPK, 2018, “Polymer spur gear behaviors under different loading conditions: A review,” Proceedings of the Institution of Mechanical Engineers, Part J: Journal of Engineering Tribology 232(2), 210-228, https://doi.org/10.1177/1350650117711595.
- Bravo A, Koffi D, Toubal L, Erchiqui, F, 2015, “Life and damage mode modelling applied to plastic gears,” Engineering Failure Analysis 58, 113-133.
- Crippa G, Davoli P, 1995, “Comparative Fatigue Resistance of Fibre Reinforced Nylon 6 Gears.” J. Mech. Des. 117(1):193-198. https://doi.org/10.1115/1.2826106.
- Hasl C, Oster P, Tobie T. et al., 2016, “Method for calculating the contact ratio under load of plastic spur gears” (In German: Verfahren zur Berechnung der Überdeckung unter Last von Kunststoffstirnrädern.) Forschung im Ingenieurwesen 80: 111–120. https://doi.org/10.1007/s10010-016-0207-8.
- Van Melick H, 2007, “Tooth-Bending Effects in Plastic Spur Gears,” Gear Technology, 24. 58-66.
- Baur E, Osswald T, Rudolph N, 2019, Plastics Handbook. Hanser, München.
- Dong, P., Zuo, S., Du, S., Tenberge, P. Wang, S., Xu, X., Wang, X., 2020, “Optimum design of the tooth root profile for improving bending capacity,” Mechanism and Machine Theory, Volume 151. https://doi.org/10.1016/j.mechmachtheory.2020.103910.
- He,R., Tenberge, P., Xu, X., Li, H., Uelpenich, R., Dong, P., Wang, S., 2021, “Study on the optimum standard parameters of hob optimization for reducing gear tooth root stress,” Mechanism and Machine Theory, Volume 156, https://doi.org/10.1016/j.mechmachtheory.2020.104128.
- Kapelevich, A. L., 2013, “Direct Gear Design.” CRC Press.
- Kapelevich, A., 2020, “Optimal Polymer Gear Design: Metal-to-Plastic Conversion,” Gear Technology, 37(3), May: 40-45.
- Zou, T., Shaker, M., Angeles, J., Morozov, A., 2014, “Optimization of Tooth Root Profile of Spur Gears for Maximum Load-Carrying Capacity,” Proceedings ASME 2014 international design engineering technical conferences and computers and information in engineering conference IDETC/CIE 2014, August 17–20, Buffalo, Doi: 10.1115/DETC2014-34568.
- Xiao, H., Zu, J., Zaton, W., 2005, “Fillet Shape Optimization for Gear Teeth,” ASME 2005 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Doi: 10.1115/DETC2005-84657.
- Landi, L., Stecconi, A., Morettini, G., Cianetti, F., 2021, “Analytical procedure for the optimization of plastic gear tooth root,” Mechanism and Machine Theory, Volume 166, https://doi.org/10.1016/j.mechmachtheory.2021.104496.
- Richtlinie VDI 2736 Blatt 4, Thermoplastic gears – Determination of load capacity characteristics on gears (in German: Thermoplastische Zahnräder – Ermittlung von Tragfähigkeitskennwerten an Zahnrädern.) Berlin: Beuth-Verlag, 2016.
- Thomas, J.; Seibicke, F.; Heil, H.-G.; Zimmer, M.; Huber, P., 2012, Otto, M. et al.: Flank generator. (In German: Flankengenerator.) Abschlussbericht zum FVA-Forschungsvorhaben 604 I (Heft 1017). Frankfurt/Main: Forschungsvereinigung Antriebstechnik e.V.
- Voith Patent GmbH, 2009, “Gearing of a gear wheel” (In German: Verzahnung eines Zahnrads). Patent DE 10 2008 045 318 B3. Roth, Z.; Etzold, M.
- Kassem, W., Gebhard, A., Schmidt, S, Oheler, M., Hausmann, J., 2022, Design of injection molded plastic gears. (In German: Auslegung spritzgegossener Kunststoffzahnräder.) Abschlussbericht zum FVA-Forschungsvorhaben 856 I (Heft 1491). Frankfurt/Main: Forschungsvereinigung Antriebstechnik e.V.
- Richtlinie VDI 2736 Blatt 2, 2014, Thermoplastic gears – Load capacity calculation (In German: Thermoplastische Zahnräder – Tragfähigkeitsberechnung.) Berlin: Beuth-Verlag.
- Fleischer, G. and Gröger H, 1980, Wear and reliability (In German: Verschleiß und Zuverlässigkeit), VEB Verlag Technik, Berlin.
- Linke, H., 2010, Spur gears: calculation, materials, manufacturing. (In German: Stirnradverzahnung: Berechnung, Werkstoffe, Fertigung.) München: Carl Hanser Verlag.
- CAMPUS® Automobil OEM Datasheet: Stanyl
^{®}TW341 – PA46. Available at: https://www.campusplastics.com/mate-rial/pdf/152378/StanylTW200F6?sLg=en. - ABAQUS INC., 2016, ABAQUS Online Documentation.
- Kassem, W. Oehler, M., Sauer, B., 2022, “Investigation of the time-dependent behavior of plastic gears with local tribological simulation,” 4th International Conference on High Performance Plastic Gears 2022, VDI-Bericht Nr. 2389.

^{Printed with permission of the copyright holder, the American Gear Manufacturers Association, 1001 N. Fairfax Street, Suite 500, Alexandria, Virginia 22314. Statements presented in this paper are those of the authors and may not represent the position or opinion of the American Gear Manufacturers Association. (AGMA) This paper was presented October 2022 at the AGMA Fall Technical Meeting. 22FTM13}