# An IoT-Based System for Monitoring the Health of Guyed Towers in Overhead Power Lines
## 4 Analytical Model of the Tower  
The current framework includes a complete set of sensors to make the solution all-purpose. Consequently, the designer can follow the current hardware guidelines in this research work but limit the used sensors for its algorithmic approach. Furthermore, to demonstrate modern approaches, we consider a simplified model based on vectorial analysis that helps us better understand the tower's behavior and ML techniques. While it is impossible to capture the flow details and other perturbations around the actual structure within this method, the overall displacement and forces are well reproduced, reflecting practical information captured by the sensors. Furthermore, synthetic data are helpful for the ML approach since they are practically possible to create unlimited data for ML training. Figure 14 illustrates a simplified analytical model of the mockup tower. The V-shaped guyed tower simplifies to a T-shaped model where cables attach at the top side edges. There are four monitored cables. Each cable has a tension sensor to read the forces and an accelerometer to read the gravity vector information. The forces, denoted in the plot by the spring symbol over the cable, follow [PERSON]'s law, where the force needed to extend or compress a spring by some distance scales linearly to that distance. Thus, the tension force \(F_{cb}\) over a cable is as follows:  
\[F_{cb}=-kx \tag{1}\]  
where \(k\) is a constant factor characteristic of the cable, and \(x\) is the total deformation of the cable concerning the original length of the cable. On the other hand, the quantity of interest from the accelerometer is the gravity vector \(\mathbf{g}\). The vector \(\mathbf{g}\) is an essential reference because it is always pointing to the same vertical direction when the device is static. In this way, it is possible to calculate the angle \(\theta_{cb3}\) of the direction of the Cable 3 and the vector \(\mathbf{g}\). In addition, notice from Figure 14 the explicit correlation between tension, given by Equation (1) and the angle \(\theta_{cbx}\) of each cable. Figure 13: User interface samples from the central server: (**a**) View of leaf nodes data from sensor cables, accelerometers, and weather stations, (**b**) View of history data plot of the weather station, (**c**) View of history data plot of cable 1, and (**d**) View of history data plot of the central accelerometer. The accelerometer data must receive careful analysis when dealing with global and local axes reference. The supplied data of the accelerometer device are considered a local axis reference. Thus, the local axis reference must be converted to a global axis. Notice in Figure 15 that the y-axis local reference of the devices is always parallel to the cables and rotated concerning the global axis represented in the left corner of the plot. If the accelerometer data are imported in the global reference, the local axes coincide with the global axes, and the \(\mathbf{g}\) vector points to an arbitrary direction as is shown in Figure 15a. Thus, since we know that \(\mathbf{g}\) vector points to the vertical direction, it is possible to calculate a rotation matrix and apply an operation to achieve the correct reference. Figure 15b illustrates the correct reference when we apply rotation on the local coordinates of the accelerometer. The most important information from the accelerometer is to obtain the angle between the \(y\)-axis and the \(\mathbf{g}\). Furthermore, the most critical requirement is to ensure that at least one axis is parallel to the cable. Since, in this case, the y-axis is parallel, then the \(x\)- and \(z\)-axes relative to the global reference are irrelevant. The angle \(\theta\) between the \(y\)-axis and the \(\mathbf{g}\) is given as follows:  
\[\theta=\arccos\left(\frac{\mathbf{v}\cdot\mathbf{u}}{||\mathbf{v}||\cdot|| \mathbf{u}||}\right) \tag{2}\]  
Figure 14: Analytical representation of the mockup tower. Figure 15: Axes references for (**a**) local and (**b**) global.