## 3 Results and Discussion
The limit values of \(\varphi_{l}\), to which the angle \(\varphi\) tends during the damped oscillatory process, are different from zero (Figure 18). At the force source shaft angle of \(\varphi\to\varphi_{l}\), the engine shaft rotational speed is \(\dot{\varphi}\to\dot{\varphi}_{l}\). Figure 5:5 Dependence \(P=P(\dot{\varphi})\) for different values of \(\rho\). As seen in Figure 20, the greater the number of holes \(n_{2}\), the less the maximum power of the upward lifting force of the piston required. This is obvious from the dependence of the power of the stirring-beating force on the number of holes \(n_{2}\) of the piston (27.2). The maximum deflection angles of \(\varphi\) in Figure 18 define the right limits of the dependence \(P=P(\varphi)\) at various values of \(n_{2}\), and the limit values of \(\varphi_{l}\) the left end limits. The value \(n_{2}\) affects the rotation angle \(\varphi\) in the relation of \(P\)-\(\sin^{3}\varphi\), since \(\varphi=\varphi(\dot{\varphi})\) and \(\dot{\varphi}=\dot{\varphi}(n_{2})\) (Figure 19), and therefore the periodicity of the function is \(\sin^{3}\varphi\). The larger the \(n_{2}\), the larger is the period of the function \(\sin^{3}\varphi\) (Figure 20). Figure 5:6 Dependence \(P=P(\varphi)\) at \(m=25\) kg and various values of \(n_{2}\). The radius of the holes on the piston \(R_{2}\) has little effect on the dependences \(\varphi=\varphi(t)\) and \(\dot{\varphi}=\dot{\varphi}(t)\) (Figures 22 and 23). Therefore, the \(R_{2}\) values are taken to obtain more visual graphs. These graphical dependences have almost the same period of oscillation, but the initial and subsequent amplitudes are larger. The angle of rotation \(\varphi\) over time tends to the limits, which are determined by the relation \(\varphi\rightarrow\varphi_{l}=\arcsin(k_{2}/\omega_{0}^{2})\), where \(\omega_{0}^{2}=\omega_{0}^{2}(R_{2})\). Obviously, \(\varphi\rightarrow\varphi_{l}\) at \(\dot{\varphi}\rightarrow\dot{\varphi}_{l}=0\) (Figure 23). Figure 5:7 Dependence \(P=P(\dot{\varphi})\) at \(m=25\) kg and various values of \(n_{2}\). Summarizing all dependencies \(\varphi=\varphi(t=\tau/\omega_{0})\), an assessment can be made regarding the applicability of the Runge-Kutta method quartic for the numerical solution of the equation of motion Equation (25) or (26). When solving the equation of motion Equation (25), the mesh step was selected in such a way as to ensure the required calculation accuracy. If numerical solutions are available on two meshes \(y(\tau,h)\) and \(y(\tau,rh)\), where \(r>1\), then the error of the solution on the mesh with the smaller step is given by  
Figure 5:8  
where \(p\) is the order of the Runge-Kutta scheme, \(r\) is the ratio of the steps, and \(h\) is the increment step of the variable \(\tau\). The dependence of the solution error of Equation (25) on the dimensionless time at \(r=2\), \(h=2\times 10^{-3}\), \(p=4\) is shown in Figure 5:9  
The dependence of the solution increment \(\Delta_{i}=\varphi\ _{i+1}-\varphi_{i}\) (\(i=1\), \(2\), \(3\), \(n\)) of Equation (25) on dimensionless time is shown in Figure 6:0 The graph in Figure 27 shows that the increment between values of the solution of Equation (25) tends to zero as time increases. Therefore, the solution of Equation (25) is convergent. Figure 6:1 Dependence \(P=P(\dot{\varphi})\) at \(m=30\) kg and various values of \(R_{2}\). As is known, the results of solving the equations of motion Equations (25) and (26) are presented in Figures 2, 6, 10, 14, 18 and 22, as graphical dependencies \(\varphi=\varphi(t)\), for various system parameters. The solution curves of Equation (25) shown in these figures vary slowly and do not exhibit a rapid \"transient phase.\" When developing the mathematical model of the mixing and whipping system for low-viscosity suspensions (for example, kumys viscosity averages \(\eta=1.75\cdot 10^{-3}\) Pa-s), it was assumed that the whipping effect is negligible.