Methodology for calculating the angular vibrations of blades on a separator machine rotor and developing its mathematical model

Introduction. In the recommended technology for separating cotton seeds from hulls using a separator machine, the main working body of the separator is the drum and the shaft mechanism passing through it. Striking blades are installed at a 45-degree angle along circular rows on the outer surface of the rotating shaft using elastic supports. In this arrangement, the striking blades rotate together with the shaft in a helical pattern along the length of the shaft, forming a spiral line. When the striking blades interact with the cotton seed hulls, they not only separate them into fractions but also transport specific fractions along the length of the drum [1, 2]. During operation, the striking blades generate angular oscillations due to the elastic supports. Furthermore, by selecting the required amplitudes and frequencies of these oscillations, it becomes possible to effectively separate the cotton seed hulls into fractions. Therefore, a theoretical study of the small angular oscillations of the striking blades on the shaft of the separator machine is considered important [3]. Methodology for constructing the calculation scheme and mathematical model of the angular oscillations of the blades on the shaft of the separator machine. During the operation of the separator, the striking bladed shaft rotates with an angular velocity of 7.33 s⁻¹. In this process, the striking blades, due to a certain center of mass and changes in inertia forces, cause a change in the moment of inertia of the drum. Striking blades can mainly be manufactured in two different versions [4]: − striking blades with elastic supports that operate only once per rotation; “Развитие науки и технологий” Научно–технический журнал №5/2025 4 Техника, технология ва жиҳозлар − striking blades with elastic supports that are symmetrically mounted for continuous rotation. In such a process, the load is evenly distributed in the second variant. For this variant of efficient operation of the shaft's striking blades, the calculation scheme is shown in Figure 1. To determine the influence of geometric parameters, as well as the elastic-dissipative properties of the elastic support of the striking blades, the angular velocity of the drum and shaft is not taken into account. In this case, according to the calculation scheme (Figure 1), the system is considered as a beam oscillating around the O axis, interacting with the cotton seed hulls. To obtain the mathematical model of the angular oscillations of the striking blades on the shaft of the separator machine, Lagrange's equation of the second kind was used. The angular displacement φb of the striking blades relative to their hinge axis was chosen as the generalized coordinate. In this case, the kinetic and potential energy of the system, as well as the dissipative Rayleigh function [5], are taken in the following form: Figure 1. Calculation scheme of the striking blades on the shaft of the separator machine.    = = ; 2 1 2 2 1 b Аl c P  = ; 2 1 2 2 1 b Аl в F  = 2 2 2 b b b b l m J T      ; 3 2 1 2 (1) where, φb – is the moment of inertia of the striking blades with respect to the hinge O; l, l1 – are the length of the striking blades and the radius of the elastic support's location relative to the axis of rotation; cA, vA – are the stiffness and dissipation coefficient of the elastic support; mb – is the mass of the striking blades. In this case, we obtain the following recommended differential equation [6, 7] for the small angular oscillations of the striking blades on the separator shaft for separating cotton seed hulls into fractions: 2 l t F F l в l c l m b А b А b b     + = + +   . ) sin ( 3 0 1 2 1 2 1 (2) Let's reduce the obtained differential equation to the standard form, and it will look like this: l t F F p n b b b ) sin ( 2 1 0 1 1 2 0     + = + +   (3) 2 0 1 0 l m F F 2 1 l m l в n b = . 3 2 1 2 0 l m l c P А = ; 3 2 1 1 1 l m F F А = ; 3 ; 3 2 2 b = where, b b According to the solution of this differential equation, for effective and stable operation of the cotton seed hull separation process and the separator machine, we obtain the following [8]: (4) where, B – is the amplitude of the striking blades' oscillations: ); sin(    + = t B b + + = . 2   1 0 1 1   2 2 0    − = P n arctg l t F F B + − )) sin( ( ; 4 ) ( (5) 2 2 2 2 2 0 n P Analysis of the influence of parameters on the vibration range of the striking blades on the separator shaft. To solve the problem of determining the small angular oscillation range of the separator shaft's striking blades for separating cotton seed hulls into fractions, it was carried out with the following numerical values of the optimal parameters [9, 10]: ωb=7,33 s-1; F1=(0,5÷1,5) N; cA=(1,5÷4,5)‧103 N/m; vA=(2,6÷3,2) Ns/m; mb=(0,04÷0,24) kg; l=0,240 m; l1=0,190 m; rb=0,05 m; rs=0,6 m. “Фан ва технологиялар тараққиёти” Илмий–техникавий журнал №5/2025 5 Техника, технология и оборудование The graphical dependence of the change in the angular oscillation range of the striking blades on the change in technological load is shown in Figure 2. The analysis of the obtained graphs shows that with an increase in the constant component of the technological load on the striking blades of the separated cotton seed fractions from 0.4 to 1.5 N at ωb=20 s⁻¹, the small angular oscillation range of the separator's striking blades increases nonlinearly from 0.53º to 2.39º. With an increase in the angular velocity up to 35 s⁻¹, the oscillation range ∆φb increases from 1.26º to 7.42º. To ensure the values of ∆φb within the limits of (2.5°÷5.0°), the recommended values of the technological load are F1 ≤ (1.2÷1.4) N: 1-ωb=20 s⁻¹; 2ωb=25 s⁻¹; 3- ωb=30 s⁻¹; 4- ωb=35 s⁻¹. Figure 3 shows the graphs of the dependence of the change in the range of angular oscillations of the impactor blades on the change in the stiffness coefficient of the elastic support. Analysis of the constructed dependence graphs of the parameters in Figure 3 shows that with an increase in the values of the linear stiffness coefficient of the impactor blades support from 0.8·103 N/m to 4.5·103 N/m, the range of small angular oscillations of the impactor blades at vA=2.6 Ns/m decreases from 7.6° to 2.55° according to the nonlinear dependence, and at vA=3.2 Ns/m, the values of ∆φb decrease from 3.86° to 1.09°. This is explained by an increase in the stiffness of the elastic support and a decrease in its deformation, which leads to a decrease in the values of ∆φb. The recommended values are: cA=(2,0÷2,5)·103 N/m; vA=(2,5÷2,8) Ns/m. Figure 2. Graphs of the dependence of the change in the angular oscillation range of the striking blades on the change in technological load Figure 3. Graphs of the dependence of the change in the angular oscillation range of the striking blades on the change in the stiffness coefficient of their elastic support. 1 – bA = 3,2 Ns/m; 2 – bA = 3,0 Ns/m; 3 – bA = 2,8 Ns/m; 4 – bA = 2,6 Ns/m; Figure 4. Graphs of the dependence of the change in the angular oscillation range of the striking blades on the change in the mass of the separator's striking blades. 1–cA = 1,4103 N/m; 2–cA = 1,8103 N/m; 3–cA = 2,2103 N/m; 4–cA = 2,6103 N/m; “Развитие науки и технологий” Научно–технический журнал №5/2025 6 Техника, технология ва жиҳозлар “Фан ва технологиялар тараққиёти” Илмий–техникавий журнал №5/2025 7