## 2 Materials and Methods
### Proposed System: FBG Sensors Applied to Health Monitoring of Rolling Mill
Experimental evidence demonstrated that response remains linear with strain, without hysteresis up to 400 \({}^{\circ}\)C.  
The FBG sensors allow measuring vibration by detecting the frequency of dynamic excitation through a sort of firm of the working machine in operation, and the amplitude of dynamic response as the reflected signal exhibits a shift in frequency depending on the loading condition. Dynamic measuring needs employing a dynamic interrogator, being capable of sampling the signal typically at 1 kHz and even higher frequency, and thus the system cost increases. The longitudinal strain of fibers is exploited for this measurement to assure a suitable sensitivity of the sensor to signal. To make more effective the monitoring system, in the literature is known the use of vibration sensors, based on a small flexible structure equipped with fiber optics, to be connected to the monitored structure, which exploits the forced vibration of a small cantilever (Figure 3) [55]. A contactless measurement is even performed, by focusing the sensor on a thin vibrating element, equipped with a small permanent magnet, whose vibration is excited by the rotating shaft, without direct contact. More in general, a displacement sensor is even based on a couple of fibers, one transmitting and one receiving light, respectively, which focus on a target in motion to detect its relative position with respect to the sensor location. The FBG sensors are even used in monitoring pressure. Measuring pressure lower than 1 bar is difficult. In contrast, when pressure changes quite fast, these sensors allow monitoring even a sudden evolution of operating conditions. Pressure is measured through the Poisson's effect on the fiber's material. If the fiber is assumed to be sufficiently long to avoid any additional local effect of ends, strain is related to pressure applied on the lateral surface as follows:  
\[ \begin{array}{ll}\varepsilon&=\frac{\Delta P}{E}(1-2v)\\ &\Delta\lambda_{B}=K_{P}\Delta P\end{array}  \tag{5}\]  
Figure 3: FBG-based finite vibration sensor used to emphasize the dynamic excitation of monitored structure [55]. where \(\varepsilon\) is the axial strain of fiber, \(\Delta P\) is the gradient of pressure in time, \(E\) is the Young modulus of the fiber's material (typically \(E\approx\) 72 GPa, in silicon composites, or \(E<\) 30 GPa in case of polymeric coating with relevant thickness) and \(v\) is the Poisson's coefficient. The sensitivity on pressure change is given by \(\Delta\lambda_{B}\), and in standard fibers, can be quite low, although, in the literature, some examples of polymeric coatings can be found leading to a coefficient \(K_{P}\) of up to 34 nm/MPa. It is known that by introducing the temperature sensibility coefficient, \(K_{T}\), Equation (3) becomes:  
\[\Delta\lambda_{T}=K_{T}\Delta T \tag{6}\]  
being linear only within a limited range of temperature values (typically below 100 \({}^{\circ}\)C). When a greater temperature range is expected, it is preferred to express the temperature sensitivity coefficient as a function of the thermal expansion coefficient of fiber, \(\alpha_{0}\), and of thermo-optic coefficient, \(\beta_{0}\), as \(K_{T}=\lambda_{B}\) (\(\alpha_{0}+\beta_{0}\)), by resorting to a second-order formula for \(\beta_{0}\), which allows describing its sensitivity on temperature [56]. It is worth noticing that temperature variation of \(n_{\mathit{eff}}\) deeply depends on the characteristics of the fiber, and even a small difference in material and production technique can strongly affect those coefficients, as in [57], where another application of FBG sensors is applied to steelmaking, but limitedly to the copper plate for melting and cooling. Equation (3) clearly states that the signal of a single FBG sensor does not distinguish the contribution induced by strain and by temperature, respectively, to the measured value. In the case of vibration detection, the temperature variation normally occurs at a frequency that is largely lower than the vibration frequency.