## 2 Materials and Methods
### Proposed System: FBG Sensors Applied to Health Monitoring of Rolling Mill  
The monitoring activity can be carried out starting from signals retrieved by the FBG sensors after a preliminary analysis performed by a unit installed in proximity of the rolling mill. External actions acting on the optic sensors, such as strain, temperature, and vibration, cause a variation in the Bragg wavelength [51; 52]. The \"Bragg gratings\" allow setting up local, intrinsic, and absolute sensors to be operated even with multiplexed signals. The fiber core is treated to exhibit a periodic modulation of the material refractive index. The local sensors look like a sequence of mirrors, namely the Bragg gratings. Their refractive index is intentionally modified. They are distributed along the fiber and are variably spaced to each other, only partially reflecting the signal injected [53]. The diffractive phenomenon allows each grating to reflect a signal at a frequency related to spacing. This geometry allows the detection of any changes in temperature [54], strain, and pressure. The sensor plays the role of a dielectric mirror, able to reflect a specific frequency of light while transmitting all of the other ones. If one analyzes separately several monitoring options, details allow for identifying some key issues of design. Particularly, measuring the displacement of signal peak in the dynamic response spectrum measured by the FBG sensor allows detecting the amplitude of the external agent and tracing its effect. The main parameter is the wavelength reflected by the Bragg grating, \(\lambda_{B}\), which complies with the Bragg condition:  
\[\lambda_{B}=2n_{\mathit{eff}}\Lambda \tag{1}\]  
where \(\Lambda\) is the grating pitch, being the distance between grating targets, and \(n_{\mathit{eff}}\) represents the effective refractive index of the given propagation mode. This is similar to the refractive one, but it is approximated with negligible error for the wave propagating inside transversely limited media as a function of the refractive index of the core and cladding of the optical fiber as in [51]:  
\[n_{\mathit{eff}}=n_{\mathit{cladding}}+\left(1.1428-\frac{0.996}{v}\right)^{2}\Delta n \tag{2}\]  
where \(\Delta n=n_{\mathit{core}}-n_{\mathit{cladding}}\) and \(\
u\) is a function of \(\Delta n\) and of the wavelength \(\lambda\) of light. Several physical actions applied to the sensor, such as strain, temperature, and pressure, change the Bragg wavelength \(\lambda_{B}\), and the measure of peak shift in the spectrum allows for calculating the corresponding change in the physical action. Particularly, if \(X\) is the generic physical signal, its influence upon the Bragg wavelength is:\[\Delta\lambda_{B}=\frac{d\lambda_{B}}{dX}\Delta X=\lambda_{B}\Bigg{(}\frac{1}{n_{ eff}}\frac{dn_{eff}}{dX}+\alpha\Bigg{)}\Delta X \tag{3}\]  
where the first term within brackets represents the normalized sensitivity on the selected physical agent \(X\) (i.e., temperature, pressure, strain... ) of the effective refractive index, while \(\alpha\) is a parameter related to the wavelength shift, expressed as a function of the physical signal \(X\). A typical FBG optical sensor has a physical length of a few millimeters, a reflected bandwidth lower than 1 nm, and it might achieve the 100% reflection of the peak luminous intensity [52]. In strain measurement, the Bragg wavelength changes linearly [53], i.e., the proportionality factor denoted as strain sensibility coefficient, \(K_{e}\), is defined as:  
\[\Delta\lambda_{e}=K_{e}\Delta\varepsilon\approx[0.78\lambda_{B}]\Delta\varepsilon \tag{4}\]  
Since \(\lambda_{B}\) is approximately equal to 1550 nm, a typical value of \(K_{e}\) is 1.2 nm\(\cdot\)m/mm [54].