Mathematical Modeling of the Weight of Grain Straw during Conveying in Vertical Conveying Reaper

s Analyzing the total weight of cut straws inside the conveyor is important in a vertical conveyor reapers machines to optimize the supply energy into the lugged belt conveyor system. This weight was mathematically modeled by considering the working mechanism of vertical conveyor reapers machines, corelating feeding rate of cut straws from cutter to conveyor and conveying rates of lugged belt conveyor. Then the mathematical model for total weight was defined by determining numbers of cut straws in front of each cutters or knives. The defined equation shows that the feed rate of cut straws into the conveyor is in arithmetic sequence fashion with equal initial value and common difference. The total weight of cut straws equation is an arithmetic series with positive common difference. The equation also shows that the numbers of cut straws in the conveyor increased in arithmetic progression when the numbers of cutters or knives increased. In general, the weight of cut straws in conveyor depends on mass of straw, gravitational acceleration, numbers of cutters or knives, forward speed of reaper, spacing of lugs, angular speed of conveyor pulley, diameter of conveyor pulley and diameter of straw.


Introduction
Vertical conveyor reapers are machines which are used to harvest grain crops such as wheat, barley, rice, soybean, and other cereal and oilseed crops. There are four types of Vertical conveyor reaper machines. These are selfpropelled vertical conveyor reaper, tractor mounted vertical conveyor reaper, self-propelled riding type vertical conveyor reaper and self-propelled reaper binder (Department of Agriculture & Corporation of Mechanization and Technology, n.d.).
self-propelled vertical conveyor reaper machines cut the crops and conveys it vertically to one end and windrows the crops on the ground uniformly using lugged canvas conveyor belts. Tractor mounted vertical conveyor reaper cut the crop and held it in a vertical position and delivered to one side of the machine by lugged belt conveyors and fall on the ground in the form of a windrow perpendicular to the direction of movement of machine. Self-propelled riding type vertical conveyor reaper is a self-propelled unit in which the operator rides on the machine.
Self-propelled reaper binder cut and convey vertically to the binding mechanism where it is tied by the twine and released to the ground in the form of bundles (Department of Agriculture & Corporation of Mechanization and Technology, n.d.).
The weight of a cut grain straw is the sum of the weight of its shoot and spike. The weight of the shoot and spike of a grain depends on types of grain, genetic variety within a grain and environmental constraints. The weight of the shoot has a direct relation with its stem length. Mekonnon (2014) experimental proofed that the weight of barley spikes has direct relation with its spike length and numbers of grains per spike for different genotype barley. Cui et al. (2011) used to evaluate possible genetic relationships between plant height (PH) and plant height components (PHC) using conditional quantitative trait locus (QTL) mapping method. This study shows that plant height depends on genetic basis in wheat.

Shah et al. (2011)
conducted a field experiment to assess the effect of tillage, mulch and time of N application on yield of wheat. This experiment shows that higher plant height was observed in surface seeded, mulched and half N at sprouting and half at CRI plot. And also grain yield (3.62 t ha -1 ) and harvest index (39.35%) were significantly higher in surface seeded plot than conventionally tilled plot due to the higher number of effective tillers m -2 , longer spike length and higher spike weight, grain weight spike-1 and test weight. Similarly, mulched crop had also shown significantly higher grain and straw yield than non-mulched crop due to the longer spike length and higher grain weight spike -1 and test weight. This research proof that the weight of wheat depends on environmental traits. In general, all the Vertical conveyor reaper machines are driven by using different sources of input power. The input power is divided into cutter and conveyor sub systems. The conveyor power depends on the weight of crops conveyed at a time. One of the basic problems in vertical conveyor reaper machines is determining weight of straws to optimize amount of energy needed during conveying straws. The weight of grain straws depends on types of plant, genetic varieties and environmental constraint. The purpose of this study is to develop a mathematical equation which can be applicable to estimate the weight of grain straws during conveying for any types of crops harvested by vertical conveying reaper windrow.

Methods
 Feed Rate Estimation: defining mathematically the feed rate of cut straws into the conveyor using forward speed of machine and diameter of grain straw by studying the feeding mechanism into the cutter, cutting and controlling mechanism of the cutter of the Vertical conveyor reaper machines.
 Conveying Rate Estimation: defining mathematically the conveying rate of cut straw within the conveyor using speed of conveying belt and pitch of lug by studying the flat belt and lug geometry sequence, and star wheel geometry and sequence of the Vertical conveyor reaper machines.
 Finding Numbers of Straws between the First Two Consecutive Lugs: defining mathematically the cut straw conveyed between consecutive lugs. The distribution of numbers of straws examined and analyzed by using feed and conveying rates to get the numbers of straw between two consecutive lugs.
 Finding total numbers of Straws Conveyed at a Time: the total numbers of straws conveyed at a time determined by studying the sequence of the straws in front of each cutter or blade.
 Determining total Weight of Straws: defining mathematically the weight of the whole straw conveyed at a time by determining the initial value and common difference, and considering mass of grain and gravitational acceleration.   Figure 3 shows the schematic diagram of lugged belt conveyor and star wheel combination. In this diagram, the rotation of the camshaft is clockwise. The camshaft drives the conveyor pulley 1 in the clockwise direction. The conveyor pulley drives the lug and conveyor belt assembly in the clockwise direction. The lugged belt conveyor conveys the cut straws to the right together with the star wheel.

3.3.Mathematical Modeling of Weight of Grain
The following assumptions should be bear in mind before developing the mathematical modeling of weight of grain.
Assumption 1: the distribution of crop plants is uniform throughout the field with optimum plant density of crops.
Assumption 2: the star wheels perfectly control the crops feed into the cutters i.e., the star wheel forces the feed crop into the cutters in rows and these rows of straws are close to each other about their diameter.

Figure 4 2D Representation of Conveyor System
The magnitude of conveying force F c applied by the lugs depends on the weight of straw, frictional force between the straw and cutter bar surface, forward speed of the harvester, speed of the lug, pitch of lug and diameter of straw. The kinematics relations of straw and lug shall be first defined in order to get the conveying force F c applied by the lugs. The kinematic analysis can be carried by considering the above four assumptions. Hence, the time t s required to feed one straw into the knife is the ratio of diameter of straw (d s ) and forward speed of the harvester V m .
The numbers of straws (n 1 ) feed into the first lug from the first cutter depends on the feeding and conveying rates. It has direct relation with feeding rate whereas inverse relation with conveying rate. This implies that the numbers of straws become higher when the feeding rate is high on contrary its numbers becomes few when the conveying rate too low. Therefore, the numbers of straws (n 1 ) feed into the first lug from the first cutter becomes the ratio of feeding and conveying rates.

-------------------------------------------------------------------------------------------------5
Equation 5 shows the numbers of straws conveyed in front of each cutters. The feed rate of straws is constant from all the cutters. However, the numbers of straws are not equal in front of all the lugs which are under conveying the cut straws. Because the conveyor moves all the cut straws to one side. There is common difference (d) of straws between consecutive lugs. This implies that the distribution of grain is in arithmetic sequences. The numbers of cut straws conveyed between consecutive lugs can be defined by using the general formula of arithmetic sequences (mathplanet, n.d.). All the cutters feed same numbers of straws into the conveyor. Hence, the first term n 1 and common difference becomes same (n 1 = ). The numbers of straws n i conveyed in front of the i th cutter or knife defined using the general formula of the arithmetic sequences as follows.

Discussion
There So that the weight of cut straws depends on the fastness or slowness of the harvesting machine. When the forward speed of the reaper is fast, the weight of the cut straws becomes heavy in the conveyor by keeping other parameters constant.
 Spacing (Pitch) of lugs p: the spacing or pitch between two successive lugs also has direct relation with weight of the cut straws in the conveyor. When the spacing/pitch is too far, it takes much time to reach on the cut straws for the lug. Hence, too many cut straws will be collected in front of the cutter or knife waiting for the lug to be conveyed. Hence, the weight of straws becomes light in the conveyor when the diameter of conveyor pulley is large and the reverse is true.
 Diameter of straw : the last parameter (diameter of straw) has inverse relation with weight of cut straws in the conveyor. When the diameter is too large, it takes more to complete cutting of a single straw due to this reason the weight of cut straws becomes light.
In general. equation 8 helps manufacturers and researchers that are working in the vertical conveying reaper windrow to optimize the energy requirement to convey cut straws. The total numbers of cut straws conveyed at a time is defined by an arithmetic series see equation 7 which is the sum of arithmetic sequences of equation 6.
The arithmetic sequence in equation 6 has a positive common difference (d=n1) which indicated that the numbers of straws becomes increased when the numbers of cutters or knives are increased. It is obvious that the total weight of cut straws inside the conveyor has direct relation with total numbers straws see equations 7 and 8.

List of abbreviations
-common difference