Effect of Gear Design Parameters on Stress Histories Induced by Different Tooth Bending Fatigue Tests: A Numerical-Statistical Investigation

Abstract

The characterization of new materials for enabling gear design is definitely a fundamental objective in the gear industry and research. Single Tooth Bending Fatigue (STBF) tests can be performed to speed up this process. However, it is well known that STBF tests tend to overestimate material strength compared to tests performed directly on meshing gears (MG) which, in turn, require an excessively long test time. Therefore, it is common practice to use a constant correction factor $f_{korr}$ of 0.9 to translate STBF results for designing actual MG (e.g., via ISO 6336). Recent works involving a combination of Finite Element Models (FEM) and multiaxial (non-proportional) fatigue criteria based on the critical plane concept have highlighted that the assumption of considering $f_{korr}$ as a constant independent of the gear design parameters leads to inaccurate results. However, in previous studies, no correlation between $f_{korr}$ and gear design parameters has emerged. In the present paper, the influence of the normal pressure angle (${\alpha }_n$), the profile shift coefficient ($x^{*}$), and the normal module ($m_n$) on $f_{korr}$ was investigated by analyzing FEM simulations with the Findley fatigue criterion. 27 gear geometries were studied by varying the above 3 parameters in 3 levels (full factorial DOE). These geometries were simulated in both MG and STBF configurations. The results of the 54 FEM simulations were analyzed by applying the Findley fatigue criterion and the corresponding $f_{korr}$ were calculated. The correlation between $f_{korr}$ and ${\alpha }_n$, $x^{*}$ and $m_n$ was investigated using the Analysis of Variance (ANOVA) technique. The results show that the only gear design parameter influencing $f_{korr}$ is $x^{*}$ hence, a regression model for $f_{korr}$ including $x^{*}$ has been developed. This latter has been then adopted for calculating and comparing $f_{korr}$ values from other combination of the parameters found in literature, giving good correspondence.

1. Introduction

In mechanical systems, gears transmit mechanical power between different components in terms of torque and rotational speed [1]. Their operating principle is based on the meshing of teeth, which are subject to different failure modes [2,3,4]. For example, under high contact pressure and/or insufficient lubrication, the sliding/rolling contact between the teeth flanks could lead to wear [5,6], scuffing [7], or (micro-)pitting [8,9]. Furthermore, the contact between the teeth flanks leads the stresses at the tooth root radius to vary continuously, pulsating from zero to a maximum ${\sigma }_F$ [10,11]. The failure mode related to this fatigue loading condition, which could lead to catastrophic consequences due to a crack nucleation and propagation in the fillet region [12,13], is known as Tooth (Root) Bending Fatigue (TBF) [10].

In gear design, the service life of a tooth subject to TBF can determined by calculating the Tooth Bending Strength (TBS) [14,15]. To evaluate the TBS, the standards propose a simplified strength criterion. Indeed, according to ISO 6336-3 [16], by defining the permissible bending stress in the fillet region as ${\sigma }_{FP}$ and the maximum stress due to pure bending in the fillet region as ${\sigma }_F$, the relation ${\sigma }_F\le {\sigma }_{FP}$ has to be satisfied (considering the life factor $Y_{NT}=1$) to ensure an infinite life to the designed gear. On the one hand, different analytical methods to calculate ${\sigma }_F$ can be found, e.g., ISO 6336-3 [16], ANSI/AGMA [17]. On the other hand, ${\sigma }_{FP}$ is proportional to the material strength ${\sigma }_{Flim}$ (method B of ISO 6336-3 [16]). According to ISO 6336-5 [18], for specific materials and/or treatments ${\sigma }_{Flim}$ has to be estimated experimentally.

In the literature, it is possible to distinguish three different types of tests to estimate ${\sigma }_{Flim}$: (1) Meshing Gears (MG) tests, e.g., [15,19], (2) Single Tooth Bending Fatigue (STBF) tests, e.g., [20,21,22], and (3) notched specimens tests, e.g., [23,24,25].

With respect to MG tests, the specimen is itself a gear made with the material and treatment to be characterized. It meshes with (at least) one more gear, see [15,19]. Through MG tests, it is possible to reproduce the actual loading conditions during operation and, thus, the exact stress history in the fillet region [26,27]. Accordingly, it is possible to characterize the fatigue behavior of the tested material (and treatment) very reliably using this method [28]. Indeed, the TBS obtained via MG tests (${\sigma }_{FlimMG}$) is the value of ${\sigma }_{Flim}$ as defined in the standards. However, running tests on MG is relatively expensive, time consuming, and requires specific test rigs, e.g., [15,29]. The above-mentioned limitations are mainly due to the fact that, for each gear tested, a single value of ${\sigma }_{FlimMG}$ can be estimated since the breakage of a tooth makes the specimen unusable for the acquisition of other data [26,27].

Specimen costs and the duration of the experimental campaign (with respect to MG tests) can be drastically reduced exploiting notched specimens. This kind of test can be performed on any universal testing machines. However, since there is no meshing, these tests do not reproduce a stress history that is faithful to the gears operating conditions. In addition, the different geometry (with respect to the gears teeth) further compromises the accuracy of the results that, to be used in the standards, require the implementation of a set of corrective coefficients [30].

An excellent compromise between the time required for the experimental campaign, the costs related to the specimens (and the test equipment) and the accuracy/reliability of the results can be found in STBF tests [20]. STBF tests consist of the application of two coaxial forces (fixed directions and sinusoidally variable amplitudes) to two teeth of the sample (a gear made of the material and treatment to characterize). More specifically, in STBF tests, two anvils (having parallel faces and mounted on a universal testing machine) load two specific teeth of the above-mentioned gear, see [31,32,33]. The loading condition leads to the forces to be tangent to the base circle and, at the same time, normal to the tested teeth flanks (Wildhaber property [1]). Since in STBF loading condition, the gear has not to mesh and/or to rotate, the damage/absence of specific teeth (not the tested one) does not compromise the application of the load. Therefore, through STBF tests, it is possible to perform multiple measurement on the same gear. This is a clear economic advantage towards the experimentation on MG. In addition, in STBF tests the lubrication is not required since there is no meshing [34].

However, in STBF tests, the stress history induced in the fillet region presents some differences with respect to those induced in MG and to those modeled in the standard to estimate the TBS. First, in MG the rolling/sliding contact, the possibility of having multiple pairs of meshing teeth leads to the contact force to vary in both direction and amplitude [26,27]. This leads to have, in the fillet region, a multiaxial and non-proportional stress history [35]. Conversely, in STBF tests the applied forces have a constant direction and vary in amplitude (sinusoidally). Therefore, such amplitude trend fails to reproduce the uneven force sharing due to the engagement or disengagement of additional pair of meshing teeth [26,27]. In addition, in STBF tests, the angle between the loaded tooth axis (${\alpha }_{Fen}$) and the applied forces can be considerably different from the one in the Outer Point of Single pair tooth Contact (OPSC) that, in turn, according to the standard, it is exploited to calculate ${\sigma }_F$ on MG. Second, in STBF tests, to avoid undesired movement of the specimen, a compressive load is usually present. A common ratio between the minimum and maximum applied load is R = 0.1, e.g., the STBF tests described in [29,36,37,38,39]. It modifies the average stress in fillet region since, conversely, in MG the above-mentioned ratio in usually equal to zero [19,20,40,41].

Based on the theoretical differences before mentioned and experimental results aimed at estimating the difference between ${\sigma }_{FlimMG}$ and ${\sigma }_{FlimSTBF}$, it is possible to state that a correction coefficient is needed since ${\sigma }_{FlimSTBF}\ne {\sigma }_{FlimMG}$ [42,43]. To consider this effect, Rettig in [42] and Stahl in [43] suggested to exploit a constant correction coefficient ($f_{korr}=0.9$) defined as $f_{korr}={\sigma }_{FlimMG}/{\sigma }_{FlimSTBF}$. However, experimentally, $f_{korr}$ has been estimated without investigating the effect of gears geometry and material [26,42,43]. Numerically, in [26,44] the scholars highlighted that considering this coefficient constant is inaccurate. Indeed, preliminary results show that varying materials, load ratio (R), and some gear design parameters, $f_{korr}$ varies from 0.65 to 1.1. However, in [26,44] the relationship between the gear design parameters and $f_{korr}$ has not been systematically studied.

Therefore, in the present paper, a specific Design of Experiment (DOE) was created to study the effect of specific gear design parameters on $f_{korr}$ using the numerical approach proposed in [26,44]. In Section 2, a background related to the implementation of multiaxial fatigue criteria on gear and to the mathematical model for implementing the Findley criterion on Finite Element Models (FEM) results is presented. In Section 3, the method used to establish the gear geometries to be simulated and the settings of the FEM simulations are shown. The statistical elaboration of the results and the related discussion are presented in Section 4. Conclusions can be found in Section 5.

2. Background

2.1. Studies Related to the Implementation of Multiaxial Fatigue Criteria on Gear

In the literature, several multiaxial-fatigue criteria can be found, e.g., [45,46,47,48,49,50,51,52,53,54,55,56,57,58]. However, papers studying TBF through multiaxial-fatigue criteria are limited. For example, the Sines fatigue criterion [58] combined with Finite Element Analysis (FEA) of a gear has been used in [21]. However, the Sines criterion can be applied exclusively for proportional loading condition. In [27,59] spur gears were studied through the elaboration of FEA results with the Crossland fatigue criterion [46]. In [60] hypoid gears were studied combining FEM simulations and the Liu and Mahadevan fatigue criterion [61]. Both of the above-mentioned criteria are capable of considering non-proportional loads, but it is interesting to notice that the fatigue criterion proposed by Liu and Mahadevan is based on the critical plane concept. The idea behind the fatigue criteria based on the critical plane concept is the following: the crack due to the fatigue loading propagates on a specific plane on which both shear and normal (with respect to the plane) stress components contribute to the failure [47].

Additional studies that have used fatigue criteria based on the critical plane concept to study gears are [26,44,62,63]. In [26], an approach to estimate $f_{korr}$ by applying Findley’s criterion [46] to the results of specific FEA was proposed. In [43], the method proposed in [26] has been extended to include other criteria based on the critical plane concept such as Papadopoulos [53,64], Matake [48], McDiarmid [50], and Susmel et al. [65]. In [62,63] numerical and experimental results have been compared to determine the most appropriate criteria to study TBF in gears. Results have shown that the Findley’s criterion is the most prone criterion to predict the crack behavior in terms of nucleation position and propagation direction [62,63]. Therefore, in the present paper the Findley’s criterion has been exploited.

2.2. General Framework to Calculate $f_{korr}$ by Means of FEM Simulation and Findley’s Criterion

As mentioned in the introduction, the load/stress histories in the MG and STBF conditions are different and therefore the damage caused by the two tests (for each loading cycle) is different. For each test, the damage can be quantified through the implementation of fatigue criteria based on the critical plane, in this specific case the Findley criterion which was proven to be the most accurate for such analyses [62,63]. Indeed, it can be proven that $f_{korr}$, defined as the ratio of ${\sigma }_{FlimMG}$ and ${\sigma }_{FlimSTBF}$, is equivalent to the ratio between the maximum damages (calculated through Findley) in the STBF condition and the MG condition [26].

Therefore, a general framework to estimate $f_{korr}$ for a specific combination of gear geometry and material involves two steps.

• The first step relies on setting up FEM simulations of MG and STBF tests on the same gear. The loading conditions (force applied to the anvil in STBF simulation and torque applied to the MG simulation) have to lead to the same ${\sigma }_F$ according to [16]. In addition, to model a critical condition, ${\sigma }_F$ should be similar to a presumed ${\sigma }_{FP}$. In this way, the results of the FEA related to the STBF and the ones related to MG are equivalent from the standard’s perspective [16].
• The second step involves the calculation of the maximum damage reached in the fillet region according to the Findley criterion. With this respect, the stress history tensor ${\displaystyle \stackrel{=}{\mathit{\sigma }}}\left(t\right)$ (Equation (1)) has to be extracted for each position along the fillet region for both the FEM analysis performed. In this way, the points of the fillet region can be analyzed through the Findley criterion. Indeed, elaborating the various $\stackrel{=}{\mathit{\sigma }}\left(t\right)$ it is possible to detect the most critical point and calculate the maximum damage parameter for each type of test.

  $$\stackrel{=}{\mathit{\sigma }}\left(\mathit{t}\right)=\left[\begin{array}{ccc}{\mathit{\sigma }}_{\mathit{x}\mathit{x}}\left(\mathit{t}\right)& {\mathit{\tau }}_{\mathit{x}\mathit{y}}\left(\mathit{t}\right)& {\mathit{\tau }}_{\mathit{x}\mathit{z}}\left(\mathit{t}\right)\\ {\mathit{\tau }}_{\mathit{y}\mathit{x}}\left(\mathit{t}\right)& {\mathit{\sigma }}_{\mathit{y}\mathit{y}}\left(\mathit{t}\right)& {\mathit{\tau }}_{\mathit{y}\mathit{z}}\left(\mathit{t}\right)\\ {\mathit{\tau }}_{\mathit{z}\mathit{x}}\left(\mathit{t}\right)& {\mathit{\tau }}_{\mathit{z}\mathit{y}}\left(\mathit{t}\right)& {\mathit{\sigma }}_{\mathit{z}\mathit{z}}\left(\mathit{t}\right)\end{array}\right]$$

  (1)

In the following of this section, the concepts behind the fatigue criteria based on the critical plane are presented together with the procedure to elaborate $\stackrel{=}{\mathit{\sigma }}\left(t\right)$ according to the Findely’s one to acquire the maximum damages parameters and, therefore, to calculate $f_{korr}$. The mathematical modelling behind this procedure is explained in more detail in [62].

In Figure 1, a plane defined by its normal vector n (having spherical coordinates ${\varphi }_n,{\theta }_n$) is shown. This plane passes through the point to be evaluated in terms of damage due to fatigue. To this regard, $\stackrel{=}{\mathit{\sigma }}\left(t\right)$ has to be extracted in the above-mentioned point. The total stress acting on this plane can be represented by a vector having time-dependent modulus and direction (${\mathit{P}}_{\mathit{n}}$) that, in turn, can be defined through Equation (2).

$${\mathit{P}}_{\mathit{n}}\left({\varphi }_n,{\theta }_n,t\right)=\stackrel{=}{\mathit{\sigma }}\left(t\right)\mathit{n}\left({\varphi }_n,{\theta }_n\right)$$

(2)

${\mathit{P}}_n$ can be decomposed into ${\mathit{\sigma }}_{\mathit{n}}$ and ${\mathit{\tau }}_{\mathit{n}}$. The former has fixed direction (normal to the plane) and time-dependent modulus. The latter is tangential to the plane and have a time-dependent modulus and direction.

In Figure 2, it is possible to note that, when periodic stresses are in play, ${\mathit{P}}_{\mathit{n}}$ describes a three-dimensional closed curve whose projection on the plane is the positions of ${\mathit{\tau }}_{\mathit{n}}$ assumed in a load cycle (${\Gamma }_n$ in Figure 2). In addition, along a cycle, it is possible to individuate ${\sigma }_{n,min}$ and ${\sigma }_{n,max}$ (Figure 2).

However, for implementing fatigue criteria, it is necessary to identify the value of the alternate tangential stress ${\tau }_{n,a}$ and its average value ${\tau }_{n,m}$. These values represent the entire load cycle in terms of shear acting on the studied plane. With this respect, the most common method to summarize ${\Gamma }_n$ into two scalars (${\tau }_{n,a}$ and ${\tau }_{n,m}$) is the Minimum Circumscribed Circle (MCC) [64]. In this approach, represented in Figure 3, ${\tau }_{n,a}$ is calculated as the radius of the smallest circle that can entirely contains the curve ${\Gamma }_n$, while ${\tau }_{n,m}$ can be defined as the distance between the center of the circle and the origin of ${\mathit{\tau }}_{\mathit{n}}$.

Hence, by varying the parameters (${\varphi }_n$, ${\theta }_n$) it is possible to define multiple planes passing through the studied point and, for each of these planes, the relevant stress values, i.e.; ${\tau }_{n,a}$, ${\tau }_{n,m}$, ${\sigma }_{n,max}$, ${\sigma }_{n,min}$ can be calculated through the aforementioned methods. At this point, establishing specific criteria it is possible to individuate (among the defined planes) the critical one. In the present paper, the spherical coordinates and the related stress’ values are labelled with the subscript c (critical) as in Equations (3) and (4).

$${\sigma }_{c,max}={\sigma }_{n,max}\left({\varphi }_C,{\theta }_C\right)$$

(3)

$${\tau }_{c,a}={\tau }_{n,a}\left({\varphi }_C,{\theta }_C\right)$$

(4)

The principle to establish the critical plane (${\varphi }_c$, ${\theta }_c$) varies according with different criteria. In most of the cases, the critical plane corresponds with the plane having the maximum value of ${\tau }_{c,a}$, e.g., [48,50,53,65]. However, according to the Findley criterion (Equation (5)), the critical planes can be found through Equation (6) by varying $\varphi$ and $\theta$ in the range $\left[0,\pi \right]$. In Equations (5) and (6), $r_{\tau /\sigma }$ is the ratio between the material fatigue limit at symmetrical alternating torsional loading (${\tau }_f$) and the limit at symmetrical alternating bending loading (${\sigma }_f$) (Equation (7)).

The Findley criterion states that the studied point withstands the loading history $\stackrel{=}{\mathit{\sigma }}\left(t\right)$ for an infinite lifetime, if the left-hand side of Equation (5) is less than a constant related to a material property obtained by fatigue testing with pure bending and pure torsion. This left-hand of Equation (5) can be considered the Damage Parameter (DP) of the studied point that, in turn, its maximum value determines the critical plane (Equation (6)). Indeed, the Findley criterion defines the critical planes as the plane in which the maximum DP is present. This definition of the critical plane differentiates it from the other criteria. Indeed, the Findley criterion exploits a linear combination of ${\tau }_{n,a}$ and ${\sigma }_{n,max}$ for the individuation of the critical plane while, as mentioned above, the other criteria consider only ${\tau }_{n,a}$.

$${\tau }_{c,a}+\frac{2r_{\tau /\sigma }-1}{2\sqrt{r_{\tau /\sigma }-r_{\tau /\sigma }^2}}{\sigma }_{c,max}\le \frac{{\tau }_f}{2\sqrt{r_{\tau /\sigma }-r_{\tau /\sigma }^2}}$$

(5)

$$\left({\varphi }_C,{\theta }_C\right)\to \underset{\varphi ,\theta }{\max }\left\{{\tau }_{n,a}\left(\varphi ,\theta \right)+\frac{2r_{\tau /\sigma }-1}{2\sqrt{r_{\tau /\sigma }-r_{\tau /\sigma }^2}}{\sigma }_{n,max}\left(\varphi ,\theta \right)\right\}$$

(6)

$$r_{\tau /\sigma }=\raisebox{1ex}{${\tau }_f$}\!\left/ \!\raisebox{-1ex}{${\sigma }_f$}\right.$$

(7)

Through the evaluation of the fatigue trough the Findley criterion on each position along the tooth root radius (where data on $\stackrel{=}{\sigma }\left(t\right)$ can be obtained via FEA) it is possible to evaluate the related DP and, therefore, the point of the tooth root radius in which the DP assumes the maximum value. This process can be repeated for both simulated conditions (STBF and MG) and it is therefore possible to extract the maximum DP value for each of them. At this point, considering that when the DP reaches the critical value the failure due to fatigue theoretically occur, it is possible to state that the ratio of the maximum DP reached in STBF and MG conditions is representative of $f_{korr}$. Therefore, $f_{korr}$ can be calculated with Equation (8).

$$f_{korr}=\frac{\underset{STBF}{\max }\left\{DP\right\}}{\underset{MG}{\max }\left\{DP\right\}}=\frac{\underset{STBF}{\max }\left\{{\tau }_{c,a}+\frac{2r_{\tau /\sigma }-1}{2\sqrt{r_{\tau /\sigma }-r_{\tau /\sigma }^2}}{\sigma }_{c,max}\right\}}{\underset{MG}{\max }\left\{{\tau }_{c,a}+\frac{2r_{\tau /\sigma }-1}{2\sqrt{r_{\tau /\sigma }-r_{\tau /\sigma }^2}}{\sigma }_{c,max}\right\}}$$

(8)

Implementing the framework presented in this section it is possible to calculate $f_{korr}$ starting from (1) the $\stackrel{=}{\mathit{\sigma }}\left(t\right)$ extracted by the FEM simulation of MG and STBF tests (2) the values of ${\sigma }_f$ and ${\tau }_f$ that can be obtained via standard fatigue tests for new materials and/or from literature, e.g., [66] for most common materials.

In addition, according to the Findley criterion, it is possible to define the Safety Factor ($S_F$) as in Equation (9). $S_F$ is the ratio between the right term and the left term of Equation (5) where the DP assumes its maximum value (and therefore the $S_F$ is the minimum among all the positions of the tooth root radius). More specifically, when the $S_F$ value is close to the unit, it means that the load applied to the tooth leads towards the limit value of fatigue life.

$$S_F=min\left\{\frac{\left({\tau }_f/2\sqrt{r_{\tau /\sigma }-r_{\tau /\sigma }^2}\right)}{\left({\tau }_{c,a}+\frac{2r_{\tau /\sigma }-1}{2\sqrt{r_{\tau /\sigma }-r_{\tau /\sigma }^2}}{\sigma }_{c,max}\right)}\right\}$$

(9)

3. Materials and Method

3.1. Definition of the Main Design Parameter and Design of Experiment

In spur-gear design, the gear geometry can be defined (at least macroscopically) through a specific number of parameters, i.e., the normal module ($m_n$), the normal pressure angle (${\alpha }_n$), the number of teeth ($z$), the face width ($b$), the profile shift coefficient ($x^{*}$), the addendum coefficient ($h_{aP}^{*}$), the dedendum coefficient ($h_{fP}^{*}$), and the root radius factor (${\rho }_{fP}^{*}$).

$m_n$ is the ratio between the circular pitch (pitch diameter $d_p$ divided by the number of teeth $z$ and π. This parameter is the most important because it defines which gear can mate. The applicability of the standard ISO-6336 [14] is limited to $m_n>2$ mm. For smaller normal modules, specific arrangements have to be made [22,32,33]. On the other hand, STBF tests have been conducted also on gears having $m_n=8$ mm, e.g., [31].

${\alpha }_n$ is the normal pressure angle, the one between the normal to the line going through the two gear centers and the tangent to the two base circles. This parameter defines the inclination of the line of action and is responsible for the different share between bending and compression stresses. Moreover, it affects the bending and the torsion of the shaft on which the gear is mounted/manufactured. According to the ISO, the standard ${\alpha }_n$ is 20° while, in the USA, the standard ${\alpha }_n$ is 25° [1]. However, applications in which ${\alpha }_n$ is between 14° and 20° are common [1].

$x^{*}$ is a dimensionless coefficient that, if multiplied by $m_n$, it provides the offset value (in mm) to be applied to the hobbing tool during the gear cutting. A null value of shift coefficient means that the gear has standard proportions. The shift of the cutting line with respect to the reference one permit the creation of the so-called corrected gears [1]. On the one hand, a positive shift coefficient increases the tooth thickness at the root and therefore the TBS. On the other hand, it reduces the width of the tooth tip (on the outside circle) [1]. In addition, $x^{*}$ affects the operating pressure angle, the efficiency, and/or the center distance [67]. It is known that negative $x^{*}$ have a negative impact on the TBS [1,67]. Indeed, negative $x^{*}$ reduce the section of the tooth in the most critical area, actually increasing the maximum stresses (with respect to the same applied force). In addition, negative $x^{*}$ lead toward undercutting issues [1]. Therefore, in the present study only positive $x^{*}$, i.e., those that increase the TBS have been considered. Typical positive $x^{*}$ are in the range 0–0.5 [1].

$h_{aP}^{*}$ and $h_{fP}^{*}$ are two dimensionless coefficients that, when multiplied by $m_n$, give the value (in mm) of the addendum and dedendum, respectively. The addendum is the difference between the outer radius and the pitch radius while the dedendum is the difference between the pitch radius and the root radius. On the other hand, ${\rho }_{fP}^{*}$ is a dimensionless coefficient that, if multiplied by $m_n$, provides the tip rounding radious of the hobbing tool ${\rho }_{fP}$ (in mm). It is interesting to notice that this area (tooth root radius) is not actively involved in the mating of pairs of teeth and, therefore, it can be machined with different geometries, e.g., elliptical, hypocycloidal [68] and with different strategies, e.g., milling, hobbing [69].

Based on the above-mentioned information along with the discussion presented in the introduction, it is possible to assume that the main differences between the stress/loading conditions in MG and STBF tests are due to the different angles and positions at which the forces are applied. These depend mainly on $m_n$, ${\alpha }_n$ and $x^{*}$. Therefore, in the present study the effect of these three design parameters on $f_{korr}$ has been investigated. With this respect, 27 different gear geometries were created through the combination of the 3 parameters and 3 levels. These levels are:

• $m_n=2;5;8$ mm in compliance with the standard range [14];
• ${\alpha }_n=15;20;25$° to include a relatively small angle (15°), the European standard (20°) [14], and the U.S. standard (25°) [1].
• $x^{*}=0;0.2;0.4$ to study the effects of standard gears ($x^{*}=0$) and positive shift (which improves tooth strength [1]) in a machining range common in many applications [1].

In the definition of the 27 geometries (listed in

Table 1. Geometrical characteristics of the simulated gears.

Gear Geometry		Design Parameters
	${\mathit{m}}_{\mathit{n}}$	${\mathit{\alpha }}_{\mathit{n}}$	${\mathit{x}}^{*}$	$\mathit{z}$	$\mathit{b}$	${\mathit{h}}_{\mathit{f}\mathit{P}}^{*}$	${\mathit{h}}_{\mathit{a}\mathit{P}}^{*}$	${\mathit{\rho }}_{\mathit{f}\mathit{P}}^{*}$
	mm	°	-	-	mm	-	-	-
1	2	15	0	24	10	1.25	1	0.380
2	2	15	0.2	24	10	1.25	1	0.380
3	2	15	0.4	24	10	1.25	1	0.380
4	2	20	0	24	10	1.25	1	0.380
5	2	20	0.2	24	10	1.25	1	0.380
6	2	20	0.4	24	10	1.25	1	0.380
7	2	25	0	24	10	1.25	1	0.317
8	2	25	0.2	24	10	1.25	1	0.317
9	2	25	0.4	24	10	1.25	1	0.317
10	5	15	0	24	10	1.30	1	0.380
11	5	15	0.2	24	10	1.25	1	0.380
12	5	15	0.4	24	10	1.25	1	0.380
13	5	20	0	24	10	1.25	1	0.380
14	5	20	0.2	24	10	1.25	1	0.380
15	5	20	0.4	24	10	1.25	1	0.380
16	5	25	0	24	10	1.25	1	0.317
17	5	25	0.2	24	10	1.25	1	0.317
18	5	25	0.4	24	10	1.25	1	0.317
19	8	15	0	24	10	1.30	1	0.380
20	8	15	0.2	24	10	1.25	1	0.380
21	8	15	0.4	24	10	1.25	1	0.380
22	8	20	0	24	10	1.25	1	0.380
23	8	20	0.2	24	10	1.25	1	0.380
24	8	20	0.4	24	10	1.25	1	0.380
25	8	25	0	24	10	1.25	1	0.317
26	8	25	0.2	24	10	1.25	1	0.317
27	8	25	0.4	24	10	1.25	1	0.317

) the other parameters (i.e., $z$, $b$, $h_{fP}^{*}$, $h_{aP}^{*}$, ${\rho }_{fP}^{*}$) have been kept constant within the limits of the feasibility of the profiles. More specifically, $z=24$, $b=10$, $h_{aP}^{*}=1$ for all the gear geometry. $h_{fP}^{*}=1.25$ for all geometries except gear geometry number 10 and 19 (Table 1) where, due to profile manufacturability, $h_{fP}^{*}$ has been selected equal to 1.30. Moreover, for the same reasons mentioned above, ${\rho }_{fP}^{*}$ has been selected 0.380 except for geometries with ${\alpha }_n=25$ (7–9; 16–18; 25–27 in Table 1) where ${\rho }_{fP}^{*}=0.317$. The values of $h_{aP}^{*}=1$, $h_{fP}^{*}=1.25$, and ${\rho }_{fP}^{*}=0.380$ are standard values according to [14]. It is worth mentioning that in the studied geometries the profile is constant along the whole width $b,$ i.e., there is no profile modification [70].

In Figure 4, it is possible to see the 27 tooth profiles created according to the properties listed in Table 1. Each tooth profile has been associated with the number of the geometry to which it corresponds. In order to simplify visibility, all the profiles have been made homogeneous through a scale factor inversely proportional to the $m_n$. In the figure, it is possible to notice how the profiles vary according the three design parameters studied.

3.2. Finite Element Analysis

The 27 gear geometries presented in the previous section have been simulated via FEM in MG and STBF condition with the aim of obtaining the stress histories ($\stackrel{=}{\mathit{\sigma }}\left(t\right)$) in the tooth root radius. In this way, as it is explained in more detail in the next section, it has been possible to study stress histories by applying Findley’s fatigue criterion.

Each gear geometry has been modelled through the open-source software Salome-Meca/Code_Aster. In the present study, 2D simulations have been performed to speed-up the calculation time of the 54 simulations (27 modeling the MG conditions and 27 modeling the STBF ones). In Figure 5, it is possible to see an explanatory model of STBF condition and, in Figure 6, an explanatory model of MG condition is represented. Two-dimensional simulations are also supported by the fact that in previous studies (in which fatigue criteria based on critical planes were applied to gears, e.g., [62,63]) it has always been observed that the critical plane is perpendicular to the frontal view of the gear. Moreover, in this study, ideal geometries have been exploited and defects are not taken into account.

In both the configurations, the loaded teeth have been modeled through quadrangular elements. The tooth root radii have been discretized by 31 nodes as well as the teeth flanks. The remaining part of the gears have been modeled through triangular elements having larger dimensions. This allowed reducing the number of total mesh elements taking into account the overall stiffness of the simulated gears.

In MG simulations (Figure 6), the gears have been simulated in their whole geometry. On the teeth involved in the gearing, the mesh density has been increased as mentioned above. In the STBF loading condition (Figure 5), it was possible to exploit the symmetries. Indeed, in STBF, the gear is stationary and there is a symmetry in the plane passing through the center of the gear and parallel to the anvil’s contact faces (yellow line in Figure 5).

Since the teeth are loaded cyclically due to the meshing with other teeth (MG) or with anvils (STBF), non-linear simulations have been set up discretizing the loading cycle in 40 time steps. The modeled material is linear isotropic having a Young modulus $E=205\mathrm{GPa}$ and a Poisson’s ratio $\nu =0.3$. In all the simulations, the state of stress never exceeded the yielding. Therefore, the non-linearity of the simulation is related to the contact that, as visible from Figure 5 and Figure 6, involves the points (highlighted in green) of the two components according to the simulation. In other words, the nodes on which contact can occur are the tooth flanks in MG and between the tooth flank and the anvil in STBF. In the contact, a friction coefficient of 0.05 have been set. However, the effect of friction has a negligible effect on the tooth root bending fatigue.

With respect to the MG simulations, the gears have been positioned fixing their axes of rotation at a theoretical center distance. The teeth were positioned half an angular step out of phase (since both wheels have the same number of teeth) in order to ensure meshing. The motion has been assigned to the driving gear (the top gear in Figure 6) and the resistant torque to the driven one (the bottom gear in Figure 6) accordingly. In MG simulations, multiple teeth can engage simultaneously as in the real tests and, therefore, the stiffness of the entire meshing arc was taken into account in MG simulations [26].

With respect to the STBF simulations, the symmetry was set as fixed constrain and a pulsating force varying sinusoidally with $R=0.1$ was applied to the anvil as in the real tests. The maximum force in STBF and the torque applied in MG simulations have been set in order to lead to the same ${\sigma }_F$ according to the standard [16]. Moreover, force or torque should lead to ${\sigma }_F≌{\sigma }_{Flim}$ for simulating the conditions where the crack nucleates and propagate leading to the fracture at approximately 10⁶ cycles [63]. In the present study, the material considered is the 39NiCrMo3. This typical gear steel presents ${\sigma }_{Flim}=281\mathrm{MPa}$ [18], a bending fatigue limit ${\sigma }_f=367\mathrm{MPa}$ [66,71], a torsional fatigue limit ${\tau }_f=265\mathrm{MPa}$ [66,71], and an ultimate tensile strength ${\sigma }_R=856\mathrm{MPa}$ [66,71].

For each simulation, the $\stackrel{=}{\mathit{\sigma }}\left(t\right)$ of all the positions within the tooth root region of the loaded tooth have been extracted (red area in Figure 5 and Figure 6). These points are the ones in which the crack has higher probability to nucleate. Since the performed simulation are 2D, ${\tau }_{xz}\left(t\right)$; ${\tau }_{zx}\left(t\right)$; ${\tau }_{yz}\left(t\right)$; ${\tau }_{zy}\left(t\right)$ are equal to zero due to the plane stress condition.

3.3. Calculation of $f_{korr}$ through the Findley’s Fatigue Criterion

The $\stackrel{=}{\mathit{\sigma }}\left(t\right)$ has been extracted for each grid node (31 nodes) belonging the tooth root radius in each simulated condition (STBF and MG). These represent an entire load cycle discretized in 40 time steps. The approach presented in Section 2.2 has been implemented to calculate $f_{korr}$. More specifically:

• For each node the critical plane has been identified using the cylindrical coordinates because the case is 2-dimensional. This means that the angle defining the representation plane of the gears has been kept constant and equal to 90°;
• On each critical plane (of each node) the DP and, therefore, the $S_F$ was calculated;
• On each tooth root radius, the maximum DP and, therefore, the minimum $S_F$ was individuated.

Based on the above-mentioned values evaluated for the MG and STBF conditions of the same gear geometry, it has been possible to calculate $f_{korr}$.

Results of the $S_F$ (related to the MG and STBF) and the $f_{korr}$ of each gear geometry are presented in

Table 2. Applied Torque (in MG), applied Force (in STBF), calculated safety factors and $f_{korr}$.

Gear Geometry	Torque in MG (Nm)	Force in STBF (N)	${\mathit{S}}_{{\mathit{F}}_{\mathit{M}\mathit{G}}}$	${\mathit{S}}_{{\mathit{F}}_{\mathit{S}\mathit{T}\mathit{B}\mathit{F}}}$	${\mathit{f}}_{\mathit{k}\mathit{o}\mathit{r}\mathit{r}}$
1	75.35	3720	1.11	0.97	0.88
2	75.85	3429	1.07	1.04	0.97
3	75.9	4800	1.11	1.03	0.93
4	82.6	4737	1.09	0.95	0.87
5	81.95	5071	1.12	0.97	0.86
6	81.35	4027	1.14	1.10	0.97
7	85.1	4605	1.16	1.04	0.89
8	84	4877	1.15	1.11	0.96
9	82.7	5196	1.24	1.23	0.99
10	480.2	9377	1.10	0.97	0.88
11	496	8829	1.04	1.02	0.99
12	494.5	9964	1.09	1.06	0.97
13	534.5	11,206	1.08	0.97	0.90
14	529.5	12,401	1.10	1.00	0.90
15	524.5	10,256	1.13	1.10	0.97
16	546.5	11,818	1.15	1.02	0.89
17	538.5	12,377	1.16	1.09	0.95
18	530	13,365	1.20	1.11	0.93
19	1207	14,822	1.10	0.98	0.89
20	1246	13,957	1.05	1.04	0.99
21	1242	15,729	1.10	1.08	0.98
22	1342	17,684	1.08	0.99	0.91
23	1328	19,535	1.12	1.02	0.91
24	1315	16,159	1.13	1.12	0.99
25	1371	18,457	1.16	1.04	0.90
26	1350	19,187	1.16	1.10	0.95
27	1328	20,927	1.24	1.15	0.93

. It is interesting to notice how, in the majority of cases, safety factors assume values approximately the unit. This is a symptom of the correct setting of the forces/torques (using the ISO 6336 approach) combined with the good ability of the Findley criterion to characterize the DP.

4. Results and Discussion

In order to obtain a robust model of the corrective factor, the acquired data must follow a normal distribution [72] thus, a normality test of $f_{korr}$ values has been performed (Figure 7). The plot of Figure 7 shows the percentage of the normal probability of the data residuals. The central straight line depicts the cumulative probability, while the two curves represent the 95% confidence interval boundaries. For respecting the normality assumption, data residuals should stay inside the confidence interval curves and closed to the cumulative probability line, showing a symmetric behavior respect to it. Therefore, from Figure 7, it can be noticed that the normality assumption distribution of the data is validated, thus the results of the succeeding analysis can be considered robust and reliable.

The results of the ANOVA analysis, performed on the data based on the DOE plan are reported in

Table 3. Analysis of variance for $f_{korr}$.

Source	DoF	SS	Adj SS	Adj MS	F-Value	p-Value
$m_n$	2	0.000576	0.000576	0.000288	0.30	0.749
${\alpha }_n$	2	0.002253	0.002253	0.001127	1.18	0.357
$x^{*}$	2	0.029913	0.029913	0.014957	15.60	0.002
$m_n$ × ${\alpha }_n$	4	0.008682	0.008682	0.002171	2.26	0.151
$m_n$ × $x^{*}$	4	0.002614	0.002614	0.000653	0.68	0.624
${\alpha }_n$ × $x^{*}$	4	0.010961	0.010961	0.002740	2.86	0.096
Error	8	0.007670	0.007670	0.000959
Total	26	0.062668

.

In the source column, it is possible to view that the single effects of the geometrical parameters and their interactions have been evaluated. DoF, SS, Adj SS, and Adj MS columns represent, respectively, degrees of freedom, sum of squares, adjusted sum of squares, and adjusted mean squares of the relative source. The F-value column is calculated as the ratio between the adjusted mean squares of the relative source and of error, and represent the test statistics for verifying or rejecting the hypothesis that the source has influence on the response ($f_{korr}$). In practice, if the F-value is lower than the related percentile of F-distribution, it means that the source influence on the response is negligible; if it is higher, on the contrary, the source significantly affects the response. A most useful evaluation of the significance of the source parameter is the analysis of the p-value. This latter is calculated as a function of F-value, the related F-distribution percentile, DoF of the source, and DoF of the error. The p-value represents the probability of obtaining test results at least as extreme as the results observed, under the assumption that the null hypothesis $H_0$ is correct. A very small p-value means that such an extreme observed outcome would be very unlikely under the null hypothesis. A small p-value implies therefore that the null hypothesis is false or something unlikely has occurred. In a formal significance test, the null hypothesis is rejected if the p-value is less than a pre-defined threshold value, depending on the selected confidence interval, which is referred to as the significance level. It is common practice to set the confidence interval to 95% [72], as optioned in the present analysis, thus the significance level for the p-value is equal to 1 − 95% = 0.05. With this evidence (p-values < significance level), the alternative hypothesis $H_1$ should be considered correct.

The results of Table 3 underline that the only gear design parameter affecting $f_{korr}$ is the profile shift coefficient $x^{*}$ (p-value = 0.002), while the influence of the other parameters and their interactions is negligible (p-values > 0.05). This is also noticeable in Figure 8 where the main effect plots are reported and displays how $f_{korr}$ behaves as a function of each parameter. Every point in the plots of Figure 9 represents the average of the values of $f_{korr}$ related to the particular value of the process parameter, while the bands indicate the dispersion of the $f_{korr}$ measurements. The high slope of the graph related to $x^{*}$ highlights the great influence of it on the corrective factor, while to low inclination of the graphs, together with the dispersion bands, for $m_n$ and ${\alpha }_n$ indicates again their negligible contributions. Moreover, Figure 8 suggests that an increase of $x^{*}$ leads to a rise of $f_{korr}$.

The lack of significance of $m_n$ and ${\alpha }_n$ parameters respect to $f_{korr}$ may be noticed also observing the surface plots of Figure 9, that depict the evolution of $f_{korr}$ in the variation range of $x^{*}$ and $m_n$ (Figure 9a), and of $x^{*}$ and ${\alpha }_n$ (Figure 9b). In both plots, in fact, it is clearly visible how the main inclination of the surface is related to the axis representing the $x^{*}$ variation, while for both $m_n$ and ${\alpha }_n$ axes the surface slope is trivial.

Following the aforementioned considerations, a statistical regression considering only $x^{*}$ parameter has been performed, leading to the formulation of a linear mathematical model of $f_{korr}$ reported in Equation (10). Indeed, starting from the above-mentioned evidence it was possible to define a relation between the profile shift coefficient $x^{*}$ and the correction factor $f_{korr}$.

$$f_{korr}=0.2012·x^{*}+0.8945$$

(10)

The goodness of this relation ($R^2=0.878$) was proved with the application of the abovementioned procedure (RG vs. STBF via FEM) to other 6 configuration (Figure 10) coming from literature [21,22,29,30,31,32,73,74,75]. These 6 configurations are characterized by having values for the $x^{*}$, the ${\alpha }_n$, and the $m_n$ that are not the ones used for the derivation of the regression equation and significantly outside the range used for the DOE ($x^{*}=-0.2÷0.45$, ${\alpha }_n=20$, $m_n=0.45÷8$) (Figure 11). These 6 geometries have been simulated (with material properties of 39NiCrMo3) with the aim of obtaining $f_{korr}$ in [26].

It is interesting to notice how all the additional data used for validation (blue circles in Figure 10), fall on the regression line (red) and show a maximum error of 3% in the prediction of $f_{korr}$. This supports the good accuracy of Equation (10) regarding the properties of the material used.

5. Conclusions

In previous studies, the authors have shown that through the application of FEM combined with advanced fatigue criteria based on the critical plane concept is possible to have a better estimation of $f_{korr}$ for any specific gear design (instead of considering $f_{korr}$ always equal to 0.9).

In the present work, the effect of the gear design parameters on the translation of STBF results into MGs’ strength was investigated. Specifically:

• Three design parameters were included in the analysis: the profile shift coefficient $x^{*}$, the normal pressure angle ${\alpha }_n$, and the normal module $m_n$.
• For each of those parameters, three different levels were considered, and a full factorial DOE (27 designs) was studied.
• The 27 gear geometries were numerically simulated via FEM both in the RG and in the STBF conditions. The calculated stress states for the 2 configurations (for each gear design) were analyzed exploiting a fatigue criterion based on the critical plane concept, i.e., the Findley criterion.
• For each gear design, the applied load was selected according to the ISO 6336 standard. The comparison between the results of the FEM for the RG and the STBF configurations has led the determination of the correction factor $f_{korr}$.
• The systematic approach used in the present work and the amount of data collected, allowed to better understand the possible relations between the gear design parameters and $f_{korr}$.
• An ANOVA has shown that among the three considered parameters ($x^{*}$, ${\alpha }_n$ and $m_n$), only the profile shift coefficient have a statistically significant effect on $f_{korr}$.
• It has been translated in terms of linear regression in Equation (10). To validate the goodness and reliability of this relation, 6 additional gear designs which parameters did not fall in the ranges used for the derivation of the relation between $f_{korr}$ and $x^{*}$ were investigated with the same procedure.
• The results are aligned with the predictions of the linear model with a maximum error of 4%.
• The relation could be possibly used to improve the reliability of the material parameters obtained via STBF tests.
• Future studies will be aimed at further investigating the effect of material strength parameters (e.g., ${\sigma }_f$ and ${\tau }_f$, critical to Findley’s implementation) on the relationship between $f_{korr}$ and $x^{*}$ obtained in this study.