Forced Vibrations of the Hopper of Fertilizer Applying Machine

To date, soil application of mineral fertilizers is one of the key factors in increasing the yield [1-3]. This leads to higher requirements to the mineral fertilizers, which are aimed at obtaining the maximum number of agricultural products, as well as compliance with environmental safety [4, 5]. For soils prone to erosion, almost the only way to carry out intensive farming will be tilter-free plowing. This practice significantly reduces the risk of soil erosion, but there are problems with other agricultural technicians, as it is fertilizing, sowing material, etc. historically optimized for widely used dump plowing and in the case of application of soilless lead to unproductive fertilizer consumption and sowing material. One of the advanced technology in modern agriculture is the intra soil differentiated (precisely) fertilization the introduction of exact amounts of fertilizer directly to the roots of plants (or in the furrow at seeding). A common feature of the machines for this type of fertilizer is that the fertilizer is fed from a mine hopper to multiple tines (working bodies), which put it into the soil. Studies have shown that to realize maximal effect of this method it will be necessary that all the tines uniformly feed the same amount of fertilizer. Improving the uniformity of fertilizing increases yields of cereals to 15 – 20% [6, 7]. Among the existing systems for feeding (sowing) the material (fertilizer, seeds) in the soil most attention deserve the pneumatic ones, in which a flexible pipe takes the material to the working bodies [8-10]. The advantages of this apparatus are: the high frequency oscillations make to the seeding system resistant to external factors, created a major force capable to destroy any connection between the individual elements of the body [11-13]. The main problem with them is uniform submission of the material to any pipeline of the system. The basis of the solution of this problem is to design a dosing device which provides uniform delivery of fertilizer to all working bodies. There are different designs of dosing devices for granular bulk material, and as a rule they do not provide a uniform density of the material flow in the whole section because it was not necessary in the cases for which they were designed [14-15]. The object of this study is a dosing device which is being developed by young scientists from Kazakhstan [16]. The purpose of this research is to build a dynamic model to describe the behaviour of a hopper under forced vibration. Since a dosing hopper operates under forced oscillations (vibration) to achieve a uniform distribution of the sown material, oscillations that take place in particular units and in the mechanisms cannot be neglected. This is why this work – a model describing the behaviour of a dynamic system, and a system in general, with vibrations at different frequencies – is needed. One of the goals set was to investigate unit performance at frequencies of vibration that are close to resonant frequency. If the system does not enter into resonance in any of operating modes that give a uniform fertilizer/seed application, dynamic system design will be assumed successful.


Introduction
To date, soil application of mineral fertilizers is one of the key factors in increasing the yield [1][2][3].This leads to higher requirements to the mineral fertilizers, which are aimed at obtaining the maximum number of agricultural products, as well as compliance with environmental safety [4,5].For soils prone to erosion, almost the only way to carry out intensive farming will be tilter-free plowing.This practice significantly reduces the risk of soil erosion, but there are problems with other agricultural technicians, as it is fertilizing, sowing material, etc. historically optimized for widely used dump plowing and in the case of application of soilless -lead to unproductive fertilizer consumption and sowing material.
One of the advanced technology in modern agriculture is the intra soil differentiated (precisely) fertilization -the introduction of exact amounts of fertilizer directly to the roots of plants (or in the furrow at seeding).A common feature of the machines for this type of fertilizer is that the fertilizer is fed from a mine hopper to multiple tines (working bodies), which put it into the soil.Studies have shown that to realize maximal effect of this method it will be necessary that all the tines uniformly feed the same amount of fertilizer.Improving the uniformity of fertilizing increases yields of cereals to 15 -20% [6,7].Among the existing systems for feeding (sowing) the material (fertilizer, seeds) in the soil most attention deserve the pneumatic ones, in which a flexible pipe takes the material to the working bodies [8][9][10].The advantages of this apparatus are: the high frequency oscillations make to the seeding system resistant to external factors, created a major force capable to destroy any connection between the individual elements of the body [11][12][13].The main problem with them is uniform submission of the material to any pipeline of the system.The basis of the solution of this problem is to design a dosing device which provides uniform delivery of fertilizer to all working bodies.
There are different designs of dosing devices for granular bulk material, and as a rule they do not provide a uniform density of the material flow in the whole section because it was not necessary in the cases for which they were designed [14][15].
The object of this study is a dosing device which is being developed by young scientists from Kazakhstan [16].
The purpose of this research is to build a dynamic model to describe the behaviour of a hopper under forced vibration.Since a dosing hopper operates under forced oscillations (vibration) to achieve a uniform distribution of the sown material, oscillations that take place in particular units and in the mechanisms cannot be neglected.This is why this worka model describing the behaviour of a dynamic system, and a system in general, with vibrations at different frequenciesis needed.One of the goals set was to investigate unit performance at frequencies of vibration that are close to resonant frequency.If the system does not enter into resonance in any of operating modes that give a uniform fertilizer/seed application, dynamic system design will be assumed successful.

Fig. 1 Design of the dosing device
The design of the dosing device is shown in Fig. 1.The material at the bottom of the hopper 1 is loosened by 2 and entering the chamber 4, wherein the vibrating plate 5 provides to fill the entire cross section and uniformly distributes of the material on the conveyor belt 6.An equalizer 7 and relief of the belt 8 contribute to improving the uniformity of the flow.

Method
A dynamic model of the dosing device is shown in Fig. 2.
Body 1 (hopper) is set to 4 elastic damping supports (shown on corners of lower part of hopper) -makes small-amplitude movements along the axes of the coordinate system O1x1y1z1 and small-amplitude rotations about the same axis.It has 6 degrees of freedom.Body 2 (rod) makes small-amplitude movements along the axis O2x2 -translation relative to body 1 due to the prismatic connection between the two bodies.

Fig. 2 Dynamic model of the dosing device
There is a linear-elastic damping element between body 1 and body 2. Forced vibrations are excited by electromagnet, acting on the axis O2x2 on body 2. The mechanical system is with 7 degrees of freedom.

Kinematics
To determine the kinetic and potential energy, energy of dissipation, and generalized forces, kinematics of the mechanical system must first be studied -positions of points (mass centers) and speed -angular of the bodies and linear of the points [14].
A vector of generalized coordinates that define the position of the mechanical system of bodies in space is: Matrices of transformation through which the vectors are projected in reference coordinate system are: -for body 1: ..
where: ., where: Vectors of the position of the mass centers of the bodies projected in the reference coordinate system are: -for body 1: -for body 2: Vectors of absolute linear velocity of the mass centers projected in reference coordinate system are: -for body 1: -for body 2: Vectors of absolute angular velocity of bodies projected in local coordinate systems (they are necessary to calculate the kinetic energy of the mechanical system) are: -for body 1: where:

Dynamics
The kinetic energy of the mechanical system, the potential energy of deformation of the elastic elements are calculated as well as from the weights of the bodies the energy of dispersion of the damping elements was calculated [14].From kinetic energy through differentiation in generalized speeds are obtained matrix of mass-inertial properties of the mechanical system.From potential energy by differentiating a generalized coordinate are obtained matrix of elastic properties of the mechanical system.From the energy of dispersion by differentiation in generalized speeds are obtained matrix of the damping properties of the mechanical system.Generalized forces are calculated taking into account the forces that excite vibrations and their applied points.
Using the Lagrange's equation of 2 nd kind, a system differential equations is compiled which describes the forced small-amplitude oscillations of the mechanical system.

Kinetic energy
The kinetic energy of the system is the sum of kinetic energies of the two bodies: where: i i Ω is vector of absolute angular velocity of the respective body, projected in the local coordinate system; 0 Ci V is vector of absolute velocity of mass center of the respective body, projected in the reference coordinate system; Matrix of mass and inertial properties:

Potential energy
The potential energy of the mechanical system is the sum of the potential energies of deformation of any flexible elements and the weights of the two bodies: where: is vector that defines the gravitational acceleration at the reference coordinate system; 0 Ci R is vector of a mass center position of the body i, defined in the reference coordinate system.
A matrix of elastic properties is:

Energy of dissipation
The energy dissipation of the mechanical system is the sum of the energies of dissipation of all damping elements:   A matrix of damping properties is: Generalized forces.A vector of generalized forces is:

Differential equations describing the forced vibrations of the mechanical system.
A system of differential equations which describes the forced small-amplitude oscillations of the mechanical system is: Solution of the differential equations.
Finding general solutions of the system ( 21) is related to the determination of the initial conditions of motion q(0) and q (0), which depend on the movement of the system.
In harmonic kind of disturbing and initial conditions     00 0, 0 , 0 t q q q q    , general system solutions of differential equations written in matrix form are: q t q q e t gh q q e t gh ik where: Q is the vector of generalized external forces; and other symbols are: As an example, a small model of a real fertilizer hopper is solved, developed in Kazakhstan Agro Technical University, Astana, Republic of Kazakhstan (Fig. 3).

Discussion
As an example, for the calculation, there is considered a mechanical system with investigated coulter scatterers of the fertilizer and accepted real constructive parameters.
In Table 1, parameters of the small model of the fertilizer hopper are given as Input data (verification was performed on a reduced large-scale copy of a real hopper; calculations were made using data obtained with a scaled model to compare the results): masses; mass inertial moments, geometrical parameters, elastic and damping characteristics of elastic-damping elements and exciting forces.The reference coordinate system coincides with the coordinate system of body 1.
Fig. 4 shows 3D graphic of generalized coordinates, natural frequencies and natural forms of the mechanical system.
In Fig. 5 the diagrams of free damped vibrations of the system at all generalized coordinates are given.
In Fig. 6 the diagrams of forced vibrations of the system at all generalized coordinates are given.
In Fig. 7 the diagrams of amplitude-frequency characteristics of the system at all generalized coordinates are given.The obtained results show that the study design of the small model of fertilizer hopper will work outside the resonance frequencies (7.96 Hz).The properties of the elastic-damping elements are selected so that free vibrations are damped within 1 s.Figs. 5, 6 and 7 depict the major achievement of this work.These graphs show that free oscillations decrease over a limited period of time, which is about 1 s.This really facilitates the avoiding of resonant zones.Forced uniform oscillations are reproduced in time, and do not experience the influence of free oscillations.Such a result falls within the set goal.These computational results are in tune with the results of modelling that showed a scaled model not entering into resonance at any frequency from the broad range of frequencies of the electromagnetic actuator.In order to improve the operation of the hopper elasticity coefficients of elastic elements may be changed [16].Thus resonant zones of operation are avoided should any change in operating frequency of the electromagnet become necessary.

Conclusion
A methodology for dynamic analysis of the hopper of a fertilizer applying machine is proposed.
After the design development of the hopper, specific values for the masses, mass moments of inertia, geometry, elastic and damping characteristics of elastic-damping elements exciting forces will be received.
With these input data using the developed methodology numerical and graphical results for forced vibrations of the device can be obtained.These will provide for the optimization of the design in order to avoid resonance of the system.
Thus, the wear of the apparatus will decrease while improving the quality of his work.The results of this article can be used to further improve the quality of the point of fertilization in the application of forced vibration of the hopper car.
tensor of the relevant body in the local coordinate system.
deformation of the elastic element.

Fig. 3
Fig. 3 Small model of a real fertilizer hopper 1 together with chamber 4 from Fig. 1


Numerical and graphical results for free, damped and forced vibrations of the mechanical system (mentioned mechanical system shown on Fig.2):Natural frequencies, Hz:

Fig. 4 Fig. 6
Fig. 4 Generalized coordinates, natural frequencies and natural forms of the mechanical system

Table 1
Parameters of the small model of studied fertilizer hopper