## 3 Test and Result Analysis of Three-Dimensional Bending Test
#### 3.1.1 Curvature Continuity Algorithm  
Due to the limitation of the number of measuring points and the fixed length of the scraper, only a small amount of discrete data can be obtained from each point in the actual measurement. These data points are often difficult to reconstruct the plane shape of the scraper conveyor. At present, the commonly used interpolation methods are nonlinear interpolation and linear interpolation. The nonlinear curvature interpolation methods mainly include polynomial interpolation and cubic spline interpolation. The following will compare and analyze the interpolation effect of the linear interpolation and the cubic spline interpolation. Linear interpolation method  
Assuming that there are \(n\) fiber gratings in the sensor, the central wavelength variation of each grating is demodulated by the demodulator, the curvature of each grating is calculated, and the curvature array is formed, which is recorded as \(a=[k_{1},k_{2},\ldots,k_{i},\ldots;s_{1},s_{2},\ldots,s_{i},\ldots]\), where \(k\) represents the curvature and \(s\) represents the arc length from each point to the starting point. By inserting m points between two adjacent fiber grating strings, w interpolation points can be obtained:  
\[w=(n-1)\times m+n \tag{12}\]  
Curvature variation \(\Delta k\) between interpolation points:  
\[\Delta k=k_{i+1}-k_{i},i\in[1,n) \tag{13}\]  
The curvature between two known measuring points changes uniformly, and the curvature change of each interval is  
\[\Delta l=\Delta k/(m+1) \tag{14}\]  
The arc length and curvature relationship of each point after interpolation is  
\[k=M\cdot s+N \tag{15}\]  
Therefore, the curvature \(k_{ij}\) and arc length \(s_{ij}\) at the \(i\) th and \(i+1\)th interpolation points and two adjacent grating points should satisfy the following:  
\[ \begin{array}{l}k_{ij}=M_{i}\cdot s_{ij}+N_{i},i\in[1,n),j\in[0,m+1)\\ \left\{\begin{array}{l}M_{i}=\frac{k_{i+1}-k_{i}}{s_{i+1}+s_{i}}\\ N_{i}=\frac{s_{i+1}k_{i}-s_{i}k_{i+1}}{s_{i+1}-s_{i}}\end{array}\right.\end{array}  \tag{16}\]  
where \(i\) and \(j\) are positive integers. Cubic spline interpolation method  
Assume that in a group of discrete data points, the arc length is divided into small segments. Then, in the range of (a, b), the cubic spline function is set to \(k(s),s=a<s_{0}<s_{1}<\cdots<s_{n}=b\), and the corresponding curvature is \((k_{0},k_{1},\ldots,k_{n})\). The interpolation curve of the cubic function is defined by the piecewise function. Using the \(n+1\) discrete points measured by the grating fiber sensor, \(n\) intervals can beconstructed. The following conditions are the basic conditions for solving the undetermined coefficients:  
1. In each segment of the division, we must ensure that \(k(s)=k_{i}(x),i=(0,1,2,\ldots,n)\);
2. To meet \(k(s_{i})=k_{i}\);
3. Spline function of the first derivative and second derivative exist, and continuity ensures that the spline function itself is continuous and smooth. Therefore, the relationship between curvature and arc length is as follows:  
\[k(s)=a_{i}+b_{i}(s-s_{i})+c_{i}(s-s_{i})^{2}+d_{i}(s-s_{i})^{3} \tag{17}\]  
where \(a_{i},b_{i},c_{i},d_{i}\) are four undetermined coefficients, \(s_{i}\) is the arc length of the \(i\)th dividing point from the starting point. Curve interpolation and continuity are defined as follows:  
\[\left\{ \begin{array}{l}k_{i}(s_{i})=k_{i}\\ k_{i+1}(s_{i+1})=k_{i+1}\end{array} \right. (i=0,1,\ldots,n-1) \tag{18}\]  
The continuity of curve differential is defined as follows:  
\[\left\{ \begin{array}{l}k^{\prime}_{i}(s_{i+1})=k^{\prime}_{i+1}(s_{i+1})\\ k^{\prime}_{i}(s_{i+1})=k^{\prime}_{i+1}(s_{i+1})\end{array} \right. (i=0,1, \ldots,n-1) \tag{19}\]  
Combining Equations (18) and (19), combining the boundary conditions, and satisfying the Equation \(k^{\prime\prime}(0)=0,k^{\prime\prime}(s_{\text{sum}})=0\), the values of each coefficient can be obtained, and the functional relationship between the arc length and the curvature at each point on the curve can be obtained.