# A Study on the Dynamic Behavior of a Sieve in an Industrial Sifter
## 3 Materials and Methods
### Determination of Equations of Motion  
The equations of motion of the sieve body were obtained using the principle of virtual work [38]. Figure 4: Possible layouts of the axes of rotary vibrators in the double-frequency screen. The position of equilibrium of the screen body in the center of mass \(S_{0}\) was related to a right-handed system of x, y, and z coordinates (Figure 5). Relative to such a system, the screen body has 6 degrees of freedom in space. However, due to the lack of stimulating forces acting in the z-direction, the considered system was limited to the XY plane transverse to the operating plane of the screen (perpendicular to Figure 5). The damping in the considered system was omitted. The angles of rotation of the solid around the transverse axis z are denoted by \(\varphi_{z}\). The subject of the considerations is the movement of the center of mass of the screen body with the mass M and the moment of inertia for the transverse axis (z) equal to \(J_{z}\). The screen is suspended by a set of four identical springs with vertical stiffness \(k_{y}\) and longitudinal stiffness \(k_{x}\). The derivation of the equations was conducted in two stages. Initially, the system did not consider the forces forcing motion and it was first considered as a system for free vibrations. The system of free motion equations of the sifter body based on [PERSON]'s second law of dynamics in the matrix notation has the Equation (2). \[\underline{B}\cdot\underline{\dot{s}}+\underline{K}\cdot\underline{s}=0, \tag{2}\]  
where:  
\(\underline{B}=\) inertia matrix of the system under consideration [kg],  
\(\underline{K}=\) matrix of elasticity of the system [N/m],  
\(\underline{s}=\) vector of displacements of the screen body \(\underline{s}=\{x,y,\varphi_{z}\}\) [m]. Figure 5: Variables used during determining the equations of motion. Equation (2) is a homogeneous form of the full equation of motion of the considered system, which is similar to that presented in [39]. To determine Equation (2), the screen body was deflected from the equilibrium position, giving it successively displacements following the positive signs of the individual six coordinates. The positions of the body after a deflection from the equilibrium position in three consecutive directions are marked in Figure 5 with different colored lines. Each deflection of the body from the equilibrium position is accompanied by the appearance of reaction forces in the elastic elements of the system. These forces and their moments determine the elements of the elastic matrix of the system in Equation (2). The left sides of equations of motion (according to Figure 2) lead to Equations (3)-(5):  
\[\sum P_{ix}=M\cdot\ddot{x}+4\cdot P(x)+P_{1}(\varphi_{2})=M\cdot\ddot{x}+K_{x} \cdot x+K_{x}\cdot A_{0}\cdot\varphi_{2}, \tag{3}\]  
\[\sum P_{iy}=M\cdot\ddot{y}+4\cdot P(y)=M\cdot\ddot{y}+K_{y}\cdot y, \tag{4}\]  
\[\sum M_{iz}=J_{z}\cdot\ddot{\varphi_{2}}+4\cdot P(\varphi_{2})\cdot B_{1}+P_{ 1}(\varphi_{2})\cdot A_{0}+4\cdot P(x)\cdot A_{0}=J_{z}\cdot\ddot{\varphi_{2}} +K_{1}\cdot B_{1}^{2}\cdot\varphi_{2}+K_{x}\cdot A_{0}^{2}\cdot\varphi_{2}+K _{x}\cdot A_{0}\cdot x, \tag{5}\]  
In the second stage of deriving the motion equations for the screen, the operation of electric motors generating centrifugal forces causing screen vibrations was considered. Let \(\underline{P}\) denote the vector of forces and moments of stimulating forces acting on the dynamical system under consideration. Then, the total equation of forced motion of the screen body has the Equation (6)  
\[\underline{\underline{B}}\cdot\dot{\underline{\bar{x}}}+\underline{\underline{ K}}\cdot\dot{\underline{s}}=\underline{P}, \tag{6}\]  
In the second stage of deriving the equations of motion, the \(\underline{P}\) vector components were determined. For this purpose, the principle of superposition was applied, adding, in turn, the stimulating forces from individual motors forcing the screen to move.