A MATHEMATICAL MODEL OF THE CUTTING PROCESS OF THE SUGAR BEET LEAFY TOPS WITHOUT A TRACER / МАТЕМАТИЧНА МОДЕЛЬ ПРОЦЕСУ БЕЗКОПІРНОГО ЗРІЗУ

A method is proposed for the assessment of the effect of the soil surface irregularities upon the loss of mass and the residues of the leafy tops during the sugar beet harvesting process. A new mathematical model is constructed and dependences of the loss of mass and the top residues are obtained depending on the distribution parameters of the soil surface irregularities. The established dependences of the loss of the root crop mass and the residues of the leafy tops on the root crops depending on the statistical distribution parameters of the soil surface irregularities and the cutting height of the tops without a tracer allowed justification of an agrotechnically acceptable level of mathematical expectation of the soil surface irregularities and the cutting height of the root crops without a tracer. АНОТАЦІЯ Запропонована методика оцінювання впливу нерівностей поверхні ґрунту на втрати маси і залишки гички на головках коренеплодів в процесі збирання цукрового буряку. Розроблена нова математична модель та отримані нові формули втрат маси і залишків гички в залежності від параметрів розподілу нерівностей поверхні ґрунту. Встановлені залежності втрат маси коренеплодів і залишків гички на головках коренеплодів в залежності від параметрів статистичного розподілу нерівностей поверхні ґрунту і висоти безкопірного зрізу гички дозволили обґрунтувати агротехнічне допустимий рівень математичного очікування нерівностей поверхні ґрунту і висоти безкопірного зрізу коренеплодів цукрового буряку. INTRODUCTION The production of sugar beet and sugar is an important branch of agriculture in the leading states of Europe and America (Marlander B. et al, 2018; Maurus K, et al., 2018, Rajaeirfar M. et al., 2019). Despite the high technical level of development of the modern beet harvesting machinery, there are problems in the process of sugar beet production that need quick solution (Lilleboe D., 2014, Merkes R., 2001). In particular, at the stage of the sugar beet harvesting there are significant losses of the crop (more than 5%), and soiling of the pile of root crops (more than 5%) (Carenko O., Jacun S., 2000; Hoffmann C., 2018). These disadvantages are caused by many factors. However, the main ones are the imperfect technical level of the working tools separating the leafy tops of the sugar beet. It has been found out that increased soiling of a root crop pile with the leaves only by 1% above the agrotechnical requirements reduces the sugar output by 0.1%, and, when storing in clamps with a leafy content of about 4%, the average daily sugar losses amount to 0.02% (Martynenko V.,1996, Zuev N.,1988; Bulgakov V. et al., 2016). Therefore, an important scientific and technical problem is to reduce the loss of mass and residues of the leaves on the root crops. It has been clarified that the soil surface irregularities have a significant impact upon the accuracy of the cutting height of the sugar beet tops without a tracer; however, there are no studies on how they affect the quality indicators of the separation process of the beet tops (Pogorelyi L, Tatjanko N., 2004; Ivančan, S. et al., 2002). 1 Bulgakov V. , Prof. Ph.D. Eng.; Ivanovs S. , Ph.D. Eng.; Pascuzzi S. , Prof. Ph.D.; Boris А. , Ph.D. Eng.; Ihnatiev Ye, Ph.D. Eng. Vol. 59, No. 3 / 2019 INMATEH – 34 A mathematical model is known for predicting the losses of the sugar-bearing mass and the amount of the top residues (Martynenko V., 1996, Bulgakov V. & al., 2015) on the root crops depending on the cutting height without a tracer and the statistical distribution of the protrusion heights of the root crop heads. However, a significant disadvantage of this model is the assumption that the surface of the soil is absolutely flat. Under real conditions, due to imperfect agrotechnics of soil cultivation and many other factors, there are significant soil surface irregularities that have an adverse effect upon the regulation accuracy of the cutting height without a tracer. Therefore, establishment of dependencies between the soil surface irregularities and the loss of mass of the root crops, and the foliage residues on them is an important scientific and practical task. The aim of this work is to develop a mathematical model of cutting the sugar beet tops without a tracer, taking into account the soil surface irregularities. Application of such a model in designing will help reduce the loss of mass of the crop and the residues of the leafy tops in the heap of the root crops. MATERIALS AND METHODS In order to establish a relationship between the soil surface irregularities and the performance quality of the top removing machines, the following assumptions were made: for the base (reference) surface we choose a conditional averaged plane of the soil irregularities (Fig. 1), in relation to which the irregularities of the soil were measured, and their distribution does not contradict the normal law. The heights distribution of the protruding root crop heads relative to the soil surface does not deny the normal law either. The top cutting tools move in parallel to the averaged plane of the soil surface irregularities. Let us consider the cutting process of the sugar beet tops without a tracer, taking into account the soil surface irregularities on which they are located. Since in the well-known model (Bulgakov et al., 2015) an absolutely flat surface of the soil is studied, then under real conditions, due to the soil irregularities (depressions and protrusions), an error occurs in setting the height of cutting. To state this error, we consider the following design flow chart (Fig. 1). Let the location of an arbitrary sugar beet root, located in the soil, has deviation from the soil surface Δh. It can take positive values (protrusions) and negative values (depressions). We will denote the protrusion height of the root crop head relative to the averaged plane as hk, and the protrusion height of the root crop head relative to the soil surface as h. The root crop cutting plane relative to the averaged line of the soil surface irregularities will be at height hz. Fig. 1 – A design flow chart of cutting the sugar beet tops without a tracer, taking into account the soil surface irregularities 1 – the averaged plane of the soil surface irregularities; 2 – the surface of the soil irregularities To determine the losses of the sugar-bearing mass and the amount of the residues of the tops of an arbitrary sugar beet root, we will determine the thickness of the cut layer of the root crop head hzk. To determine the loss of the sugar-bearing mass and residues of the tops of an arbitrary sugar beet root crop, we will determine the thickness of the cut layer of the root head hzk.   zk k z h h h , (1) or     ( ) zk z h h h h . (2) Values Δh and h constitute a distribution system of random values the geometric interpretation of which can be represented using the distribution density (Kallenberg, 2005). Vol. 59, No. 3 / 2019 INMATEH – 35 The distribution density of a system of two random values according to (Kallenberg, 2005) is determined by the expression:       2 ( , ) ( , ) F h h f h h h h , (3) where F (Δh, h) – the probability function of a combination of random values Δh and h. We select an arbitrary region from the range (A, B; Ag, Bg) of size n. We take the area of the protrusion height from A to B, and the area of the soil irregularity distribution from Ag to Bg, and divide them into n intervals. Then the probability of occurrence of a region with borders (hi, hi+1) (Δhj; Δhj+1) according to (Bulgakov et al., 2015) will be equal to: