# A Study of Magnetic Mill Productivity
## 8 Mathematical Model of the Dynamics of Millstone Movement in the MM Working Chamber  
All the preliminary mathematical processing conducted is aimed solely at preparing input data for a mathematical model that calculates indicators of the movement of millstones and WS particles located in the rotating magnetic field of the inductor. Any movements of these elements were considered by us to be planar movements, that is, they were described by functions of only two spatial coordinates. The differential equations of motion of the center of mass of a two-dimensional body and the equation of moments for relative motion around an axis passing through the center of mass perpendicular to the plane of motion are as follows  
\[\frac{\mathrm{d}^{2}x_{m}}{\mathrm{d}t^{2}}=\frac{\sum\limits_{i=1}^{n}F_{ix}}{ m};\ \frac{\mathrm{d}^{2}y_{m}}{\mathrm{d}t^{2}}=\frac{\sum\limits_{i=1}^{n}F_{iy}}{m}; \ \frac{\mathrm{d}\omega_{m}}{\mathrm{d}t}=\frac{1}{J}\sum\limits_{i=1}^{n}M_{z}( F_{i})=\frac{M_{z}}{J};\ \ \omega_{m}=\frac{\mathrm{d}\phi}{\mathrm{d}t}, \tag{14}\]  
where \(x_{m}\), \(y_{m}\) and \(\phi\)--coordinates of the center of mass and the angular position of the millstone; \(m\) is the mass of the millstone; \(J\), \(M_{z}(F_{i})\), \(M_{z}\) is the moment of inertia, the moment of force \(F_{i}\) and the main moment of all external forces relative to the axis passing through the center of mass; \(\omega_{m}\) is the angular speed. The mathematical formulation of this problem takes into account the forcing forces, the magnitude and direction of which were calculated based on the interpolation procedure of the vector TF \(\overset{\rightarrow}{S}_{m}(\overset{\rightarrow}{z})\), namely, the modulus of the force, its direction and torque. On the basis of the obtained values, which were updated at each time integration step, force projections \(\overline{F}\) and \(\overline{F_{o}}\) were determined (Figure 2). The values of the projections \(\overline{F_{o}}\) were calculated according to the formulas  
\[F_{ox}=\pm\frac{2M_{m}}{l_{m}}\sin(\alpha_{m}+\gamma_{m});F_{oy}=\pm\frac{2M_ {m}}{l_{m}}\cos(\alpha_{m}+\gamma_{m}). \tag{15}\]  
The points of their application are marked in this figure with numbers 1 and 3. System (14) consists of four equations and contains four unknown functions: \(x_{m}(t)\), \(y_{m}(t)\), \(\phi(t)\), and \(\omega_{m}(t)\). It constitutes the content of the mathematical formulation of the problem of calculating the trajectories of the planar motion of a rigid body of arbitrary shape which is affected by action of external forces and moments. An algorithm for solving such a problem for a system of interacting bodies using the finite element method is considered in [31]. Its software implementation is performed in the Ansys LS-DYNA software complex [32] in the APDL (Ansys Parametric Design Language) programming language. To process contact interactions within the system of bodies, such as touching, sliding, etc., the method of kinematic restriction is used, the essence of which is to block the degree of freedom of nodes responsible for normal movements in relation to the contact line. To preserve the efficiency of algorithm of the method of explicit numerical integration over time of systems of type (14), the masses of the bodies are concentrated (scaled) to such a limit that only the main degrees of freedom of each node [33] remain connected. In order to verify the performance of the created model and to assess its level of adequacy, the trajectories and other indicators of the movement of a set of five millstones and eight WS particles, located in a WC with a diameter of 120 mm, were calculated. Ferromagnetic millstones are subjected to the force of the rotating magnetic field created by the inductor, which was mentioned and described in the mathematical experiment for calculating the field of magnetic forces in Section 6. The dimensions of the millstones are \(d_{m}=1.5\) mm and \(l_{m}=13.5\) mm. The diameter of the particles for grinding is \(\approx\)9 mm.