## 2 Materials and Methods  
### Background: Loosening Torque  
Figure 1 shows the free body diagrams of a screw during tightening and untightening operations showing the different torques involved. \(T_{T}\) and \(T_{L}\) are, respectively, the applied external tightening and untightening (loosening) torques. \(T_{h}\) and \(T_{t}\) are the resistance torques developed under the screw head and in the thread contact surface due to friction forces, respectively, both acting in the opposite direction to the screw rotation. Finally, \(T_{p}\) is the pitch torque generated by the axial load acting on the helical surface of the threads, always acting in the screw loosening direction. Thus, the external tightening and loosening torques \(T_{T}\) and \(T_{L}\) can be expressed in terms of the following twisting moment equilibrium equations:  
Based on the Motosh methodology and including the 3D effect of both the helix and thread profile angles the approximate torque tension relationship is to give [34; 35]:  
\[T_{T}=F_{p}\cdot\mu_{h}\cdot r_{h}+F_{p}\cdot\mu_{t}\cdot\frac{r_{t}}{\gamma \cdot cos\alpha}+F_{p}\cdot\frac{tan\beta}{\gamma}\cdot r_{t} \tag{3}\]  
\[T_{L}=F_{p}\cdot\mu_{h}\cdot r_{h}+F_{p}\cdot\mu_{t}\cdot\frac{r_{t}}{\gamma \cdot cos\alpha}-F_{p}\cdot\frac{tan\beta}{\gamma}\cdot r_{t} \tag{4}\]  
where the first term is \(T_{h}\), the second one \(T_{t}\) and the third one \(T_{p}\). The more accurate expressions for (3), (4) are given in [32]. In Equations (3) and (4), \(F_{p}\) is the screw preload, and \(\mu_{h}\) and \(\mu_{t}\) are respectively the friction coefficients of the screw head and thread contacts. Besides, as illustrated in Figure 2, \(\alpha\) is the half-angle of the thread profile, \(\beta\) is the helix angle, and \(\tau_{h}\) and \(\tau_{t}\) are respectively the screw head and thread effective contact radii obtained by considering the preload to be uniformly distributed under the bolt head and on the thread surface:  
\[r_{h}=\frac{2}{3}\cdot\frac{r_{tmax}{}^{3}-r_{tmin}{}^{3}}{r_{tmax}{}^{2}-r_{ tmin}{}^{2}}\approx\frac{r_{tmax}+r_{tmin}}{2} \tag{5}\]  
\[r_{t}=\frac{2}{3}\cdot\frac{r_{tmax}{}^{3}-r_{tmin}{}^{3}}{r_{tmax}{}^{2}-r_{ tmin}{}^{2}}\approx\frac{r_{tmax}+r_{tmin}}{2} \tag{6}\]  
where \(r_{tmax}\) and \(r_{tmin}\) are the maximum and minimum radius of the screw head contact surface, and \(r_{tmax}\) and \(r_{tmin}\) are the maximum and minimum radius of the screw thread contact surface (see Figure 2). Finally, the parameter \(\gamma\) is:  
\[\gamma=1-\mu_{t}\cdot\frac{sin\beta}{cos\alpha} \tag{7}\]  
Thus, Equation (3) allows putting into relation the applied tightening torque and the preload achieved in the screw, and similarly with the loosening torque (Equation (4)). Nevertheless, Equations (3) and (4) do not consider the effect of an external transverse force on the tightening and loosening  
Figure 1: Twisting moment equilibrium during: (**a**) tightening (**b**) untightening.  
torques. As mentioned in the Introduction section, this force contributes to overcome the resisting friction torques. Consequently, the loosening torque needed to untighten the screw when an external transverse force is acting on it will be smaller than the value predicted in (4).  
In this sense, [PERSON] et al. [31; 32] developed an analytical model to study self-loosening in screwed joints under transverse cyclic loading, which experimentally proved to give accurate results for the case of two sliding plates, similar to Junker test machine [28; 29]. The effect of the screw head flexibility was further implemented in the model by [PERSON] et al. [33]. This model allows understanding the mechanical foundations of the phenomenon and quantifying the effect of the design, manufacturing and operational parameters involved, thus enabling to take the correct decisions in terms of screw head and thread geometry, friction coefficient, tightening torque, and other relevant parameters. As a drawback, its applicability is not straightforward, as the analytical model must be solved via numerical integration.