## 3 Materials and Method  
### 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 addendum 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 \(\pi\). 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 \(20^{\circ}\) while, in the USA, the standard \(\alpha_{n}\) is \(25^{\circ}\)[1]. However, applications in which \(\alpha_{n}\) is between \(14^{\circ}\) and \(20^{\circ}\) 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 radius 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.