ГРАФОАНАЛИТИЧЕСКОЕ ИССЛЕДОВАНИЕ КИНЕМАТИЧЕСКИХ ПАРАМЕТРОВ ПЛАНЕТАРНО-РЫЧАЖНОГО МЕХАНИЗМА ЭЛЛИПТИЧЕСКОГО БАРАБАНА НОВОГО ТИПА, РАЗРАБОТАННОГО ДЛЯ ВЕРТИКАЛЬНО-ШПИНДЕЛЬНОГО ХЛОПКОУБОРОЧНОГО АППАРАТА, С ИСПОЛЬЗОВАНИЕМ ПРОГРАММЫ КОМПАС-3D

УДК 621.01

Abstract

The article presents the development of a graph-analytical method for studying the lever mechanism of an elliptical drum used in vertical-spindle cotton pickers. The aim of the work is to determine the degree of influence of the structure and geometry of the individual mechanism links on its kinematics, applying classical methods of theoretical mechanics, the theory of mechanisms, and machine mechanics. The cam–lever mechanism of the elliptical drum is replaced by an equivalent four-bar linkage, and the instantaneous centre of velocity method is used to obtain the linear and angular velocities of the characteristic points for a given mechanism position. The results show that the geometric parameters, particularly the major axis of the ellipse, significantly affect the spindle velocity within the picking and separation zones. Compared with purely analytical approaches, the proposed method is clear and convenient, enables rapid evaluation of design alternatives, and supports the rational design of advanced cotton harvesting machines.

Аннотация

В статье представлена разработка графоаналитического метода исследования рычажного механизма эллиптического барабана вертикально-шпиндельного хлопкоуборочного аппарата. Целью работы является определение степени влияния структуры и геометрии отдельных звеньев механизма на его кинематику с применением классических методов теоретической механики, теории механизмов и механики машин. Кулачково-рычажный механизм эллиптического барабана заменён эквивалентным четырёхзвенным механизмом, и для определённого положения механизма с помощью метода мгновенного центра скоростей определены линейные и угловые скорости характерных точек. Результаты показывают, что геометрические параметры, в частности большая ось эллипса, существенно влияют на скорость шпинделя в зонах съёма и отрыва хлопка. По сравнению с чисто аналитическими подходами предложенный метод обладает наглядностью и удобством, позволяет оперативно оценивать варианты проектирования и способствует обоснованному выбору рациональных параметров перспективных хлопкоуборочных машин. Метод развивает инженерную интуицию при оценке возможностей механизма по его кинематической схеме и может быть эффективно применён в условиях высокой урожайности.

Keywords: vertical-spindle cotton picker, elliptical drum, planetary–lever mechanism, graph-analytical method, kinematic analysis, spindle velocity, instantaneous centre of velocity.

Ключевые слова: вертикально-шпиндельный хлопкоуборочный аппарат, эллиптический барабан, планетарно-рычажный механизм, графоаналитический метод, кинематический анализ, скорость шпинделя, мгновенный центр скоростей.

Introduction

Serial vertical-spindle cotton harvesting machines are equipped with drums carrying 12 spindles, with a diameter of 292 mm measured along the spindle centerline. One of the main drawbacks of these machines is a noticeable decrease in picking efficiency when the crop yield is high (above 40 centners per hectare), where the completeness of harvesting may decline significantly (in some cases by 5–6%).

This limitation can be primarily attributed to the insufficient number of spindles simultaneously operating within the picking zone in a 12-spindle apparatus, as well as the inability of the spindle kinematic regimes to fully satisfy the requirements of the harvesting process [1, 2].

To overcome this drawback, the idea of structurally developing the conventional spindle drum mechanism has been proposed, aiming to increase the number of spindles simultaneously engaged in the picking zone and to enable control over their kinematic regimes [3, 4]. According to this concept, structural modification of the conventional spindle drum transforms its planetary mechanism into a lever-cam planetary mechanism [5, 6].

In the kinematic analysis of lever and cam mechanisms, analytical methods are most commonly employed. For this purpose, a generalized coordinate function describing the motion law of the driving link is derived based on the principles of analytical mechanics. By taking the first derivative of this function with respect to time, the velocity of the driving link can be obtained, while the second derivative yields its acceleration.

The main advantage of this method lies in the possibility of automating the analysis using computer tools and obtaining results with a sufficient level of accuracy. However, for certain mechanisms, deriving the mathematical expression of the motion law and conducting kinematic analysis based on it can be quite complex. Even a minor error in the derivation of formulas may lead to completely incorrect results or the inability to obtain results at all.

Over the past three decades, computer graphics has advanced to a level that allows the kinematic parameters of virtually any lever mechanism to be determined using graph-analytical methods with arbitrary accuracy. One of the key advantages of the graph-analytical approach is the ability to visually observe how velocities and accelerations vary, which facilitates the development of more efficient engineering solutions [1].

Materials and Methods

Previous studies on the kinematic analysis of elliptical drum mechanisms have yielded certain mathematical expressions describing the motion laws of spindles [2, 3]. These expressions enabled researchers to investigate spindle motion characteristics and kinematic regimes in relation to the technological process of cotton harvesting, as well as to obtain recommended parameters for the rational dimensions of the drum mechanism links. However, the resulting analytical expressions are complex in form and require the use of highly sophisticated computational software for their solution.

In the present work, a graph-analytical method for investigating the kinematic regimes of an elliptical drum using computer graphics is proposed.

Figure 1. Cam–lever mechanism: 1 – connecting rod; 2 – slider; 3 – hinge; 4 – hinge guide

The elliptical drum mechanism is based on a cam–lever mechanism (Figure 1). This mechanism can be interpreted as a crank–slider mechanism in which the slider moves along a curvilinear path (in this case, an ellipse).

According to the mechanism, when the crank OA performs rotational motion, point B—which represents the spindle center and is connected to the crank via the connecting rod AB—moves along an elliptical trajectory. To ensure the motion of the spindle along the ellipse, it would be sufficient to place this point within a groove shaped as an ellipse.

However, in such a configuration, the rollers of the spindle’s friction drive may extend beyond their support, and under the influence of inertia forces arising in transition zones, these rollers may fail. Therefore, it is more appropriate to relocate the roller to point C, which is rigidly attached to the connecting rod AB. The trajectory of this point differs from an ellipse and represents a quasi-elliptical curve.

Results and Discussion

In the graph-analytical study of mechanism kinematics, it is often more practical to replace a cam mechanism with an equivalent lever mechanism [4]. For this reason, the cam–lever mechanism of the elliptical drum is replaced with a purely lever-based mechanism. As such, a four-bar linkage that reproduces the motion law of point B can be considered (Figure 2). The main feature of this mechanism is that, for each position of the driving link OA, the driven link B assumes a different length  (ρ). To determine this length, the instantaneous center of velocity method from theoretical mechanics is applied.

Since point B moves along an ellipse, its velocity vector is tangent to the ellipse at that point. Therefore, a normal to the ellipse is drawn through point B. Meanwhile, point A moves along a circular path, and its velocity vector is tangent to the circle with radius OA. By extending the line OA until it intersects the normal drawn to the ellipse at point B, the instantaneous center is determined.The segment ВО₁ ​obtained in this way represents the instantaneous length of the driven link B.

Figure 2. Construction of the velocity diagram

To construct the velocity diagram, the velocity of point А is first determined:

                                                 (1)

where - ОА is the angular velocity of point A, defined as:

  ;                                                    (2)

Here, n- is the rotational speed of the drum shaft, n=105 rpm, and ОА- is the length of the driving link (crank),ОА=0,105 m.

From equation (2) =10,99 s^-1, and from equation (1):

=0,105·10,99= 1,15 m/s.

Velocity scale

For the velocity diagram, a velocity scale is selected:

                                   (3)

Assuming=40 mm, we obtain:

= 1,15/40=0,02875 .

Construction of velocity vectors

The velocity vector of point В relative to point А ,   is perpendicular to the connecting rod  Therefore, a perpendicular line to ВА is drawn from the end of vector pa. The absolute velocity vector of point В is perpendicular to segment BO₁. Thus, a perpendicular line to BO₁ is drawn from point p. The intersection of these perpendicular lines is denoted as point b. The vector  represents the absolute velocity of point В, while,  represents the relative velocity of point В with respect to point A.

Determination of point C velocity

To determine the velocity of point С the angles formed between lines connecting point С with points  А and В and the line АВ are used (Figure 2). From point а, a line is drawn at angle, and from point b, a line is drawn at angle β. Their intersection defines point с. Connecting point c with point р, the vector  represents the velocity of point С.

Determination of actual velocities

The actual values of the velocities ,  and  are obtained by multiplying the lengths of vectors ,  and  by the velocity scale  [3].

For example, for the considered position of the mechanism:=52 mm,  =31 mm and =37 mm. Thus:

 ==1,495 m/s;

 ==0,89  m/s;

 ==1,06 m/s;

Using the same method, velocity diagrams can be constructed for all positions of the crank and rocker, and the velocity of any point on the connecting rod can be determined.

The ratio of the linear velocity of a point to its distance from the instantaneous center of velocity gives the angular velocity of that point.

Graphical analysis

Based on the calculated velocity values, graphs titled “Dependence of spindle velocity on the major axis of the ellipse” are constructed (Figure 3).

Figure 3. Dependence of spindle velocity on the major axis in an elliptical spindle system

The obtained graphs show that, in an elliptical drum, the geometric parameters of the mechanism links significantly influence the kinematic regimes of the spindle.

In particular, as the major axis of the ellipse increases, the variation in spindle velocity within the picking and separation zones becomes more pronounced. This effect is especially noticeable during the transition of the spindle from the picking zone to the separation zone.

Taking into account the geometric parameters of the drum mechanism links, their optimization and design considerations will form the main focus of future research.

Conclusion

In this study, a graph-analytical method for investigating the kinematic behavior of an elliptical drum planetary–lever mechanism has been developed and demonstrated. The proposed approach enables the determination of linear and angular velocities of characteristic points of the mechanism with sufficient accuracy while maintaining a clear and intuitive analytical framework.

The results show that the geometric parameters of the mechanism, particularly the major axis of the ellipse, have a significant influence on the kinematic regimes of the spindle. An increase in the major axis leads to more pronounced variations in spindle velocity within both the picking and separation zones. This effect is especially critical in the transition region between these zones, where changes in kinematic conditions directly impact the efficiency and quality of the cotton harvesting process.

Compared to purely analytical methods, the graph-analytical approach offers improved visualization of kinematic relationships and simplifies the process of analyzing complex mechanisms, where deriving explicit mathematical expressions may be difficult or impractical. In addition, the method provides a convenient tool for engineering analysis, allowing rapid evaluation of design alternatives and facilitating the selection of rational geometric parameters for the mechanism.

Overall, the developed method expands the capabilities of kinematic analysis for elliptical drum mechanisms and can be effectively applied in the design and optimization of advanced cotton harvesting machines. Future research will focus on refining the geometric parameters of the mechanism and extending the approach to dynamic and experimental validation studies.

References:

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