Modeling the Digging Process of Tree Root System by the Mechanism with Hydropulse Drive

A mathematical model of the mechanism for digging large-sized seedlings or removing stumps on the undergrowth is developed, which is based on the idea of the root system and surrounding soil, as a combination of plurality of spherical elements, interacting with each other, and the knife mechanism, as a set of elementary surfaces that interact with elements of the stump and soil. The effect of blade vibration on the stump removal efficiency with the help of the model is studied.


Introduction
When digging or removing large-sized seedlings or stumps clearance on the undergrowth, roadside and shelter belts, machines of manipulator-type with replaceable technological equipment of discrete action have high efficiency. We proposed a mechanism with hydropulse drive as removable processing equipment for excavators and forestry cranes.
The purpose of this article is to develop a mathematical model of digging large-sized seedlings or stumps clearance in the undergrowth by mechanism with hydropulse drive, which will further assess the effectiveness of the proposed design and parameters of blade vibration in various modes. In the simulation of mechanism, shape and physical characteristics of the stump, forest soil, as well as the interactions of these environments with a knife must be properly represented in the model.

2.
Justification and implementation of the process by the mechanism with hydropulse drive. Timber and forest soil are extremely difficult to simulate by objects because of their typological diversity and large number of physical and mathematical parameters describing it, in particular, the type of soil or wood, humidity, anisotropy, friability, crispness, topography. Model of tree and soil is based on the method of discrete elements; environment is represented as the collection of large number of spherical elements (about 10.000) of small size, which are capable of reacting both between themselves and with the working surfaces of the mechanism. Figure 1. Scheme of mechanism for removing tree stumps: 1 -bearing beams; 2 -frame of mechanism; 3 -vertical rods; 4 -working body; 5 -semi-basket; 6 -spiked bumpers; 7 -hydraulic cylinder; 8 -hinges for mounting vertical rods  3. Modeling the digging process of tree root system The interaction of mechanism working surfaces with wood and soil, from the geometrical point of view, is the problem of finding the distance rв from a plane (the elementary plane of the working surfaces of the mechanism) to the surface of the any spherical element of environment. At this viscoelastic force ВУ ij F  , acting on the i-th element of the soil from the j-th element is given by where n  and v  -direction and speed of the interaction of a ball element and a given surface, calculated for each spherical element of the soil methods of analytic geometry; c and k -the coefficients of stiffness and viscosity of interaction.
The elastic component of interaction between the elements ensures both the repulsion of elements (distance between centers rij of i-th and j-th elements is less than the element diameter dE) and attraction of element over a narrow range of distances (dE > rij > rк) (  where i -element number; m i -mass of the i-th element; x i , y i , z i -Cartesian coordinates of the element; t -time; N E -quantity of elements; j -element number, possibly contacting with the i-th element; с ij and d ij -coefficients of stiffness and viscosity of the interaction of elements i and j; r ijthe distance between the centers of the elements i and j; v xi , v zi -the Cartesian components of the velocity of i-th element; d I -distance of interaction limitations between the elements; g -gravitational acceleration; k -the number of elementary surface of the mechanism acting on the environment; N Sthe number of elementary surfaces; F S xik , F S yik , F S zik -Cartesian components of the force exerted by the k-th elementary surface on the i-th element.
The distance r ij between the centers of the elements is calculated by center coordinates by When simulating in the model the process of removing the stump pivotal movement of the working body and the vibration transmitted from the hydropulsator to the cylinder takes place. Turning the working body and its vibration are described in the following coordinate transformation of reference points of the elementary surfaces: where i -index of reference point of elementary surface, which the working body consists of; (x i , y i , z i ) and (x i ', y i ', z i ') -coordinates of the i-th reference point before and after the coordinate transformation (in the initial and the current time); (x h , z h ) -coordinates of axis of the working body passing through the hinges; r i and φ i -the polar coordinates of the i-th reference point in the coordinate system associated with the axis of the working body; ω w -the angular velocity of rotation of the working body; φ v -vibration amplitude (it is a maximum angle through which the operating element is deflected from the guideline in vibrations); f v -the frequency of vibration of the working body.
In the proposed version of the model the vibration of the working body is considered only by addition of the term of sum φ v cos(2πf v t) into the expression for angular position of the working body, however, that already allows to study the influence of the amplitude and frequency of vibration on the stump clearance efficiency.
Blade and limit stop of mechanism are represented in the model as a set of a plurality of elementary triangles (figure 3). Before you start the computer experiment, location of reference points P1 ... P35 is performed in space according to the following formulas, which include the geometric parameters of the blade. After determining the current coordinates of the base points (for this step -time integration τ i ) the construction of the elementary triangles is made that define the working surfaces of the blade and the stop limit. So, for the blade we have the following 54 triangles: Т 1 = P 1 P 2 P 15 , Т 2 = P 2 P 15 P 16 , Т 3 = P 2 P 3 P 16 , Т 4 = P 3 P 16 P 17 , Т 5 = P 3 P 4 P 17 , Т 6 = P 4 P 17 P 18 , Т 7 = P 4 P 5 P 18 , Т 8 = P 5 P 18 P 19 , Т 9 = P 5 P 6 P 19 , Т 10 = P 6 P 19 P 20 , Т 11 = P 6 P 7 P 20 , Т 12 = P 7 P 20 P 21 , Т 13 = P 8 P 9 P 22 , Т 14 = P 9 P 22 P 23 , Т 15 = P 9 P 10 P 23 , Т 16 = P 10 P 23 P 24 , Т 17 = P 10 P 11 P 24 , Т 18 = P 11 P 24 P 25 , Т 19 = P 11 P 12 P 25 , Т 20 = P 12 P 25 P 26 , Т 21 = P 12 P 1 (P 8 ) P 2 (P 9 ) P 3 (P 10 ) P 5 (P 12 ) P 6 (P 13 ) P 7 (P 14 ) where E -bulk modulus of the working fluid. If pressures in two hydraulically interconnected cavities i and j are different, there is a flow of the where i and j -indexes of cavities; k ij -throttling coefficient; sign(x) -function that returns the sign of the variable x.
This formula is used both for throttles (throttling coefficient is sufficiently small), and piping (large throttling coefficient).
In the model it is assumed that all throttling elements have circular cross section, and then the throttling coefficient is determined through orifice diameter d ij by the formula: where μ -discharge coefficient; g -gravitational acceleration; γ -relative density of the working fluid.
Ability of pipelines to expand elastically under the influence of pressure in the model is not directly considered, but the elasticity of working fluid is taken into account indirectly, i.e., through the ratio E.
The rotation of the spool and movement of the piston of the hydraulic cylinder causes a change in the volume of considered cavities, so at each step of integration recalculated of volume of cavities takes place.
To solve the system of differential and algebraic equations, which is laid in the basis for the model, a computer program "Program to simulate the operation of the mechanism with hydropulse drive for stumps clearance" is developed. The program is developed in Borland Delphi 7.0 environment, programming language Object Pascal. Before you start the computer experiment the required parameters of blade, geometric and physical and mechanical parameters of the stump, the parameters of the soil, parameters of hydropulsator and hydraulic system can be set. In the process, the program continuously displays a schematic representation of the mechanism of the stump in three dimensions, and performance indicators in numerical and graphical form (figure 4). Estimated time of a computer experiment is about 5 minutes (at processor speed 3 GHz). The first computer experiments have confirmed the effectiveness of the proposed design of the mechanism and efficiency of vibration to improve properties of cutting characteristics of the working body. The model provides a wide range of mechanical properties of the cutting process. In particular, the time dependence of modulus of resistance to the blade deepening was obtained (figure 5) which allows proving optimum shape and thickness of the blade at which the blade has sufficient strength and high cutting properties, but has small thickness and low metal content.  Thus, the mathematical model of the mechanism with hydropulse drive for digging large-sized seedlings or removing stumps on the undergrowth, with its high spatial detail is developed, that allows studying the influence of parameters of blade, vibration, tree root systems and soil on the efficiency of work processes.