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 -
The equation of natural vibrations of the seed at the end of the fiber strand has the form
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
where:
Subsequently, the seed is hit with an impulse
where
Putting (2) and (3) in (1) we obtain
For the value
in (4), factorization is used for a small quantity
Let us further represent the pattern of small deformation of the thread
where ω – the circular frequency of impacts of the beater on the seed, and
the square of the natural frequency of the system vibrations.
For further transformations of (8) we use 2t = ωt.
Then, from (8) we obtain
If we put parameters
we get the final form:
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
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
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
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],
Then the sought circular frequency g of the natural vibrations of the system is as follows:
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
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,
Solutions of the Mathieu equation (12) are oscillatory in nature; their main properties depend on the specific values of parameters
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
The area of stable movements on the Ains-Strett diagram is practically concentrated in the small segment of the
According to the values of parameters
From (11) it can be seen that
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
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,
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.
The theoretical results obtained for the generalization of the technological process in roller ginning can be used in the design and construction of cotton processing machines and in choosing optimal parameters for the construction of beater mechanisms.
The optimal frequency of vibrational striking turned out to be in the range