Investigation of the Process of Separation of Fibers From Seeds During Roll Ginning

Many scientists have been engaged in issues of capturing fibers at the edge of a knife in a ginning machine. [1, 2, 3, 4]. Another paper considers the speed ratio of the flyers that are delivered into the ginning zone and the working drum, as well as the process of capturing fibers. [5]

In a number of works [6, 7], there is an analytical description of the force effects arising between the stationary knife and the gin working drum during the passage of cotton fiber in the contact zone.

To compile a differential equation of the specific pressure forces from the drum and the knife on the raw material between them, we will use the following method [8].

The issue of the safety of cotton seeds during the roller ginning process is one of the subjects of research on the dynamic processes of the primary processing of raw materials.

It is of interest, both theoretically and in applied terms, to analyze the process of seed beating in the so-called resonance mode. This mode allows to reduce the magnitude of the shock pulse on the seed and, consequently, the degree of damage to the seeds. At this moment, the seeds at the end of the strand move only under the action of an elastic restoring force - cx, where c is the elasticity coefficient of the fiber strand, and x – the coordinate of the displacement of the seed along the line of the strand tension.

The equation of natural vibrations of the seed at the end of the fiber strand has the form

1 md2xdt2+CX=0

This kind of movement inevitably occurs in the production process during vibrations of the machine body, which creates conditions for the occurrence of own vibrations of the system.

The coefficient of elasticity c of the fiber strand can be approximately obtained using Hooke’s law in the form [12, 13]:

P=EFlΔl

where: P – external force, l – length of the fiber, E – modulus of elasticity, and F – cross section of the fiber.

2 C=EFL;

Subsequently, the seed is hit with an impulse S for a short period of time t = 0.001 s. In this case, the strand fibers receive an additional small length increment l₁ for the period before the next hit, so that

3 l=l0+l1

where l₀ – initial length of the fiber.

Putting (2) and (3) in (1) we obtain

4 md2xdt2+EFl0+lx=0

For the value

1l0+l;

in (4), factorization is used for a small quantity l₁/l₀ as follows:

5 1l0+l=1l0(1+l1l0)−1=1l0(1−l1l0+….);

Putting (5) in (4), we obtain

6 md2xdt2+EFl0(1−l1l0)x=0

Let us further represent the pattern of small deformation of the thread l₁ in (6) in the form

7 l1=A⋅cos⁡ωt

where ω – the circular frequency of impacts of the beater on the seed, and A is the amplitude of the vibrations. Then from (6) we obtain

8 md2xdt2+γ2(1−Acos⁡ωtl0)x=0

9 γ2=EFl0m=cm;

the square of the natural frequency of the system vibrations.

For further transformations of (8) we use 2t = ωt.

Then, from (8) we obtain

10 md2xdτ2+4γ2ω2(1−Acos⁡2τl0)x=0

If we put parameters a and q in (10)

11 α=4γ2ω2;2q=4Aγ2l0ω2;

we get the final form:

12 md2xdτ2+(1–2qcos⁡2τ)x=0

This is the Mathieu differential equation in its generally accepted form (3).

According to (12), it is possible to analyze the problem considered in the process of roller ginning.

First, we note the following fact: The direct calculation of the value of c according to [12,14,15] is difficult and can lead to errors for the following reasons.

When calculating the dynamic separation of the fibers from the seed upon the impact of the beater, it should be considered that in the real process the fibers in the strand are very heterogeneous and unevenly stretched when they are pulled into the slot of the contact gap of the knife-drum. The separation of the fibers from the seed after the impact of the beater does not occur simultaneously, but sequentially, either by individual fibers or by their groups, as a rule, after 3-4 strokes of the beater [12]. Therefore, the active part of the fiber bundle section in a dynamic process can be as little as 0.001 (or less) of the total fiber bundle section.

Therefore, in the future, a more accurate calculation will be carried out using experimental data for the value of c. In research by R. Korabelnikov [16, 17, 18], the so-called coefficient of compliance d of volatile (seeds with fibers) in the process of roller ginning was determined experimentally. This coefficient is defined as the reciprocal of the stiffness coefficient c; δ = 1/c mm/N.

The average value of the compliance coefficient was d = 0.17 mm/N for raw cotton of the VA-440 variety. This allowed to perform further indirect determination of the stiffness coefficient of volatiles c.

In [13] it is also shown that in the process of ginning, the contact of the seed and the blade of the inertial beater is not destroyed, i.e. the motion of the blade and seed is accomplished jointly. Therefore, in the calculations, it is necessary to assume that according to [19,20], m = m₁ + m₂, where m₁ is the mass of the blade of the inertial beater at the interaction point, and m₂ is the mass of the seed with fiber (smaller compared to m₁, Fig. 1).

Fig. 1.

Then the sought circular frequency g of the natural vibrations of the system is as follows:

13 γ=1δ⋅m

When the compliance coefficient d = 0.17 mm/N and the given mass of the blades are m = 1g, the circular frequency of the natural vibrations of the systems will be g = 750 rad/s.

In modern designs of roller ginning machines of the DV-1M type, the frequency of impacts of the beater on the seed is Z = 24-30 strokes/sec.

Then the circular frequency ω of these impacts will be 2π·Z = 6.28 (24-30) = 150-190 rad/s. The calculations assume that ω = 200 rad/s. The length of fiber l stretched between the seed and the contact gap slit is l = 3-15mm. The calculations assume that l = 10mm.

The amplitude A in (7) is determined from A=V/ω;

The speed of impact of the blades of the beater with the seed is 2.5 m/s. Considering that the impact is inelastic, we will assume that the velocity acquired by the seed is V = 0.5 m/s. Then A = 0.002 m.

Given all these data for the following system parameters: γ = 750 rad/s, ω = 200 rad/s, l = 0.01 m, and A = 0.002 m, we obtain that in (11) and (12) the values of the parameters are as follows a = 56-100; q = 5.6-10.

Solutions of the Mathieu equation (12) are oscillatory in nature; their main properties depend on the specific values of parameters a and q. In some cases of this combination, a and q correspond to oscillations limited in amplitude, and in other cases, oscillations with increasing amplitude.

If the amplitudes remain limited, then the system is stable; otherwise, parametric resonance takes place and the system is unstable.

The boundaries between the regions of stable and unstable solutions are presented in the form of an Ains-Strett diagram (Fig. 2), constructed in the plane of parameters a and q. If the image point with coordinates a and q is within the shaded fields of the diagram, then the system is stable. In the unstable system, the image points are located in white fields.

Fig. 2.

The area of stable movements on the Ains-Strett diagram is practically concentrated in the small segment of the a axis, where q < 1. With the exception of points a = 1²; 2²; 3²; 4².., where q = 0.

According to the values of parameters a = 56-100 and q = 5.6-10 obtained, due to large values of q > 1, the dynamic state of the system practically falls into the region of parametric resonance, where the motion is unstable. This means that the oscillations of the seed at the end of the strand increase significantly after the impacts of the beater, which leads to separation of the seed from the fiber.

From (11) it can be seen that a and q parameters can be controlled by changing the g and ω frequencies, as well as the speed of the impact on the seed V and the length l of the pinched fiber. In this case, both a stable and unstable state of the system can be achieved.

Separation of the seed in the resonant mode allows to use a relatively small impact force of the beater, which reduces the possibility of damage to cotton seeds. The combination of specific values of the a and q parameters from (11) are significant for the occurrence of parametric resonance, rather than the magnitude of the beater force.

It is known that for the static separation of the seed from the fiber, a force of P = 30 N is required [14]. The average force is p = 12 N/cm for a rigid beater [13] when ginning raw cotton of the BA-440 variety. For comparison, the critical force leading to the damage to the seed is p = 45 N/cm.

The method presented for the separation of fibers from seeds by an element beater was used in further research [15]. The experiments were carried out to find the optimal value of the parameters for the so-called method of the lower striking of seeds from the edge of a fixed knife.

Using the previously given values for g, l, F and the new value for ω = 113-126 1/s, the parameters of the oscillating system in this case will be a = 140-176 and q = 14-17.6. These a and q parameters (quite a large value for q) on the Ains-Strett diagram (Fig. 1) correspond to a state of instability that facilitates the separation of seeds with a small force.

Based on the above values for the parameters, the operating parameters of the beater in a modern roller ginning machine of the DV-1M type allow to conduct the separation in a resonant mode quite favorable for the preservation of seeds. Apparently, the selection of optimal parameters of the mechanism for the separation of seeds mainly occurs empirically.