# A Study of Magnetic Mill Productivity
## 7 Development of Interpolation Algorithm  
The accuracy of the approximation of the field of forces and moments acting on the millstone in the rotating magnetic field of the mill inductor will obviously determine the adequacy level of the entire model, which will describe the dynamic behavior of the millstones in the WC. Let us consider the basic approaches involved in the algorithm of interpolation of the field of forces and moments. Using the results of modeling (Section 6), tabulated functions (TF) were obtained, which describe the field of forces \(\overline{F}_{m}\) and torque \(M_{m}\), acting on the millstone, depending on a number of specific arguments, in particular, the position of the millstone in the WC, its orientation in it, etc. After analyzing the obtained results, we have left only 2 such  
Figure 6: Dependencies of the torque \(M_{m}\) on the angle of rotation of the inductor. The values of TF corresponding to the nodal values of the arguments are combined into a column vector  
\[\overrightarrow{F}_{mt}=\left\|F_{m1}\quad\ldots\quad F_{m6}\right\|_{*}. \tag{12}\]  
After solving the vector equation  
\[T\cdot\overrightarrow{c}=\overrightarrow{F}_{mt}, \tag{13}\]  
relative to the vector unknown \(\overrightarrow{c}\), a column of polynomial coefficients is obtained (8). Therefore, the algorithm for obtaining the interpolated value of the TF \(F_{m}[\gamma,r_{m}]\) for arbitrary given values \(\gamma_{z}\) and \(r_{mz}\) consists of sequentially performing the following operations:  
* calculation of the elements of the [PERSON]'s matrix (11);
* formation of a column of TF values (12);
* solutions of the linear vector Equation (13) regarding the unknown \(\overrightarrow{c}\),
* calculating the value of the Taylor's series (9) based on \(\gamma=\gamma_{z}\), \(r_{m}=r_{mz}\);
* calculating the interpolated value of TF as a multiplication \(\overrightarrow{T}[\gamma_{z},r_{mz}]\cdot\overrightarrow{c}\). It should be noted that the interpolated values of the other components of the vector TF \(\overrightarrow{S}_{m}\), namely, the force direction \(\alpha_{m}\) and the torque \(M_{m}\), are similarly found from the corresponding scalar TF \(\alpha_{m}[\gamma,r_{m}]\)\(M_{m}[\gamma,r_{m}]\). This algorithm is suitable for TFs containing an arbitrary number of nodes with an arbitrary mutual location of nodes in the template. Verifying its operability and evaluating the accuracy of the interpolation of the above-mentioned vector function \(\overrightarrow{S}_{m}(\overrightarrow{z})\) was conducted based on the TFs obtained by the methods described in Section 6. The dependence of the interpolated value of the force modulus \(\overline{F}_{m}\) on the angular displacement \(\gamma\) is shown in Figure 8a with a solid-line curve. In this case, the angle of inclination of the longitudinal axis of the millstone was \(\alpha_{m}=90^{\circ}\). The dashed curves shown on the same graph are similar dependencies that correspond to the deviation of the angular displacement within the limits \(\pm 5^{\circ}\). Here, it should be noted that the magnitudes of forces and moments are related to the unit length of the millstone in the direction normal to the calculation plane. This is due to the two-dimensional formulation of the problem of calculating the magnetic field. Therefore, their dimensions are N/m and Nm/m, respectively. Figure 7: Formation of the TF template. The graph in Figure 8b contains the dependence of the modulus of the same force \(\overline{F}_{m}\) on length of the radius vector \(r_{m}\) (solid-line curve) and similar dependencies in case of deviations \(r_{m}\) within \(\pm 5\) mm (dashed curves). The graphs of Figure 8c,d are interpolation dependencies of force direction \(\alpha_{m}\) on the angular displacement \(\gamma\) and the length of the radius vector \(r_{m}\), respectively. Figure 8e,f is the dependencies of the torque \(M_{m}\) on the above-mentioned arguments.