## 2 Materials and Methods
### Proposed System: FBG Sensors Applied to Health Monitoring of Rolling Mill
In static or quasi-static analysis, this becomes a problem since the two sources of strain are coupled. Therefore, it is required to resort to a measuring system consisting of at least two sensors:  
\[\left( \begin{array}{c}\Delta\lambda_{1}\\ \Delta\lambda_{2}\end{array} \right)=\left[ \begin{array}{cc}K_{1t}&K_{1T}\\ K_{2t}&K_{2T}\end{array} \right]\left( \begin{array}{c}\varepsilon\\ T\end{array} \right)=\left[K\right]\left( \begin{array}{c}\varepsilon\\ T\end{array} \right) \tag{7}\]  
where \(K_{i\epsilon}\) and \(K_{iT}\) are the strain and temperature sensitivity coefficients of the \(i\)-th sensor, respectively. Excluding the special case in which every sensor is sensible only to one of the two above-mentioned perturbations, two configurations are foreseen. When the measurement of one sensor, at least, is affected only by one of the two physical perturbations, i.e., one element of matrix \(K\) is null, that sensor becomes the reference to uncouple the two signals composing the system response. When the sensitivities of the two sensors are just moderately different, and the determinant of matrix \(K\) is defined, i.e., not null, a numerical solution to the above Equation (7) is found. It is worth noticing that using FBG sensors, whose sensitivities have both non-null values, implies resorting to the second case here described above. Producing FBG sensors with different sensitivity is possible, for instance, by exploiting fibers with different diameters spliced together. A technical difficulty arises in splicing together fibers whose diameter is different. Moreover, even inscribing the FBG in the fibers' core is difficult. This type of configuration cannot be applied in embedded fiber because both sensors would result in being bound to the structure and undergoing the same strain, and thus matrix \(K\) would be singular. Those are the main limitations of the proposed approach, but a technical solution is available. Resorting to two FBG sensors, made with different Bragg wavelength \(\lambda_{B}\) allows handling those problems. According to Equation (3), FBG sensors exhibiting different central wavelengths also have different strain and temperature sensitivity coefficients. Therefore, inscribing both sensors on the same fiber and using an interrogator capable of detecting a multitude of wavelengths allow solving Equation (7), provided that all parameters of sensors are known. To avoid the case of badly conditioned matrix \(K\), a difference of at least 10% in Bragg wavelength between the two sensors is required [58]. Some suitable values of \(\lambda_{B}\) have already been tested in some commercial FBG sensors and result in 850 and 1300 nm, or 1300 and 1550 nm, respectively. Therefore, the configuration here described above looks suitable for a coupled measurement without a real problem of the distinction between monitored physical entities. Moreover, the overall size of sensors corresponds to the dimensions of fibers. This allows for embedding sensors in the mechanical system and sometimes even in the material of components. They look minimally invasive, and they allow avoiding a significant structural weakening effect. Finally, the wavelength division multiplexing (WDM) technique allows exploiting several sensors [59], composing a network applied to the monitored system, and thus the cost and complexity of the monitoring system are significantly reduced.