## 4 Results and Discussion  
The main outcome of the experimental activities described in Section 3 is represented by the estimation of the sag, as a function of the tensile load imposed to the cable, by means of the proposed method and the wireless sensors for the angular measurements. Estimated values (red curve in Figure 5) are compared with sag values measured through the linear position sensors installed at the midspan, which are taken as reference values (blue curve). The third curve (yellow curve) plotted represents the sag obtained with the classical formulation, which approximates the cable as a parabolic curve (Equation (1)), having considered known the tensile load imposed to the cable. The focus is firstly put on the sag curve computed through the parabolic expression assuming the known tensile load value (yellow curve). This curve is nearly superimposed to the curve of measured reference sag values up to very low tensile loads, where a slight discrepancy can be observed. On the other hand, the proposed method allows to obtain a satisfactory prediction of the sag trend varying the tensile load. However, in some cases, a difference between estimated and measured values is present. This behavior, particularly noticeable at higher tensile loads where the sag variation is lower from step to step, can be explained by analyzing the angle values estimated through the proposed method. Since the sag is directly proportional to the angle apart from a constant (Equation (16)), the curve presents the same trend of sag one as visible in Figure 6. In this case, the angle at the span extremity measured by the sensor (blue curve) and the angle evaluated through the catenary equation in the same position on the basis of the sag measured as reference (red curve) are compared [33]. Figure 5: Comparison of the sag values estimated through the proposed method (red curve) with the reference measured values (blue curve) and the ones from the parabolic model (yellow curve) during Test 1. In this case, error bars reported on the measured angle curve \(\theta_{mis}\) (blue curve) represent the maximum errors made by the sensor in the measurement phase and were computed through Equation (22). \[\theta=\theta_{mis}\pm\frac{\Delta\theta}{2}, \tag{22}\]  
where \(\Delta\theta\) is the difference between the maximum and minimum measurable angles \(\theta_{max}\) and \(\theta_{min}\) according to Equation (23). \[\Delta\theta=\theta_{max}-\theta_{min}, \tag{23}\]  
The maximum and minimum measurable angles \(\theta_{max}\) and \(\theta_{min}\) are a function of the accelerometer resolution \(acc_{sens}\), according to Figure 7, and are computed, respectively, through Equations (24) and (25). The value of \(acc_{sens}\) is reported among the accelerometer main features in Table 1.  
\[\theta_{max}=\arctan\left(\frac{\sin\theta+acc_{sens}}{\cos\theta-acc_{sens} }\right), \tag{24}\]  
\[\theta_{min}=\arctan\left(\frac{\sin\theta-acc_{sens}}{\cos\theta+acc_{sens} }\right), \tag{25}\]  
Figure 6: Comparison between the angle values measured by the sensors and the reference ones computed through catenary equation in correspondence of the sensor position during Test 1. Error bars represent the uncertainty of the sensor measurements. Figure 7: Schematization of angles in the vertical plane for the computation of \(\theta_{max}\) and \(\theta_{min}\), used to obtain the sensor uncertainty. This error may vary depending on the angle's actual value. However, since the angle value is always small in the performed tests, the error can be considered constant along all the range of interest. As can be noticed in Figure 6, the reference curve lays in between the measured one and the error bars, meaning that measurements are consistent with the expected sensor behavior. Nonetheless, an accelerometer with a better amplitude resolution would have allowed to reproduce the sag trend with more accuracy. A similar behavior is visible by analyzing Test 2 results, reported in the sag comparison (Figure 8) and angle comparison (Figure 9).