## 3 Design Solution for FBG Inclinometer for Cantilever-Beam-Type Hydraulic Support
### Mechanical Analysis of FBG Inclinometers for Cantilever-Beam-Type Hydraulic Supports  
When the upper beam of the cantilever beam undergoes tilting, the pendulum beneath it remains in a vertical state under the influence of gravity. This leads to the occurrence of local strain at the junction between the upper beam of the cantilever beam and the pendulum. As a result, the two gratings symmetrically arranged on one side of the grating undergo stretching, while the grating on the other side experiences compression. The inclination angle change is then calculated based on the variation in the central wavelength of the tandem grating upon reflection [35; 36]. The simplified model diagram of the cantilever beam's force distribution is presented in Figure 5, which clearly depicts the mechanical behavior and the relationship between the components during the inclination process. Figure 4: Shape diagram of FBG inclinometer for hydraulic support. Figure 5: Mechanical model of cantilever beam structure. When the angle of inclination is \(\theta\), according to the knowledge of mechanics of materials, there is:  
\[mgl_{m}\sin\varphi=F\bullet d \tag{21}\]  
The formula is \(F=k\bullet\Delta\), where:  
\[k=\frac{3 EI}{I^{3}};\Delta=d\bullet\sin(\theta-\varphi) \tag{22}\]  
where \(I\) is the moment of inertia of the cantilever beam; \(E\) is the elastic modulus of the cantilever beam; \(\varphi\) is the pendulum angle of the cantilever beam; \(m\) is the mass of the pendulum. After resolving it, the following is obtained:  
\[kd^{2}(\theta-\varphi)=mgl_{m}\varphi \tag{23}\]  
Style:  
\[\varphi=\frac{kd^{2}\theta}{kd^{2}+mgl_{m}} \tag{24}\]  
The strain from the free end of the beam is as follows:  
\[\varepsilon=\frac{k\Delta x}{EI}\bullet\frac{h}{2} \tag{25}\]  
To wit:  
\[\varepsilon=\frac{khdx}{2 EI}\bullet\frac{mgl_{m}}{kd^{2}+mgl_{m}}\theta \tag{26}\]  
From the FBG sensing principle, there is:  
\[\Delta\lambda=\alpha_{\varepsilon}\varepsilon+\alpha_{T}\Delta T \tag{27}\]  
where \(\alpha_{\varepsilon}\) and \(\alpha_{T}\) are the strain and temperature sensitivity coefficients of the grating, respectively. As the center wavelength of the two gratings is very close, it can be considered that the \(\alpha_{\varepsilon}\) and \(\alpha_{T}\) of the two gratings are the same for the two gratings symmetrically arranged on both sides of the top beam of the cantilever beam; the two gratings are subjected to the same amount of strain change, with opposite signs; so, the following is observed:  
\[\Delta\lambda_{1}-\Delta\lambda_{2}=\alpha_{\varepsilon}\varepsilon+\alpha_{ T}\Delta T-(\alpha_{\varepsilon}\varepsilon_{2}+\alpha_{T}\Delta T)=2\alpha_{ \varepsilon}\varepsilon_{1} \tag{28}\]  
The following is assumed:  
Figure 5:0  
The collation gives the following formula for calculating the inclination of the hydraulic support FBG inclinometer:  
Figure 5:1  
where \(\theta\) is the tilt angle measured by the hydraulic support FBG inclinometer, \({}^{\circ}\);  
\(\lambda_{1}\) is the test wavelength (small wavelength), nm;  
\(\lambda_{10}\) is the wavelength of 1 grating 0 point, nm;  
\(\lambda_{2}\) is the test wavelength (large wavelength), nm;\(\lambda_{20}\) is the 2 grating 0 point wavelength, nm;  
\(K\) is the coefficient of inclination, \(k=\frac{bIdx}{2 EI}\bullet\frac{mgln}{mgI_{m}+kd^{2}}\). The value is a constant, nm/\({}^{\circ}\). Tilt direction: when \(\theta\) is positive, the inclinometer tilts to the right; when \(\theta\) is negative, the inclinometer tilts to the left. When the inclination angle is of less than 5 degrees, owing to the pre-tensioning force applied to the gratings prior to installation, the two gratings symmetrically fixed on both sides of the beam can be regarded as having the same absolute value of strain but with opposite signs. However, when the inclination angle exceeds 5 degrees, for the two gratings symmetrically fixed on both sides of the beam, one side experiences stretching, while the other side undergoes compression.