Mathematical and laboratory modeling of resonant impact on the spike for the purpose of grain selection

. Mathematical and computer finite element model in the ACELAN package of resonant impact on a spike was developed and a full-scale experiment was carried out. Two installations are considered, one based on a cantilever, the free end of which acts on the spike, and the second is a semi-passive round bimorph. Excitation of vibrations is carried out using an actuator based on piezoceramic elements. In the first installation, low-frequency vibrations of the stem with a spike are excited and the resonance frequency is determined at which only an spike with grain performs intense vibrations. The second installation is designed to excite high-frequency vibrations at which resonant movements of the grains themselves arise. The purpose of both installations is to separate the grain from the spike using resonance phenomena.


Introduction
The urgent task of today in the grain sector of agriculture is the task of ensuring lowenergy and low-traumatic forced extraction of grain from the spike. This requires not only the improvement of existing methods and methods of grain harvesting [1][2][3][4][5][6][7], but also the development of new highly effective physical and mechanical methods of its harvesting and, accordingly, the design of appropriate equipment [8][9][10][11][12].
The solution of this problem, on the one hand, requires the development and creation of adequate analytical and numerical physical and mechanical models and models based on CAD-CAE complexes capable of describing the interactions between the elements of the stem-spike-grain system, and at the same time, on the other hand physical and mechanical models of the process of impact on the stem-spike-grain system of the working bodies of grain harvesting equipment. The first stage of the work is the theoretical and experimental confirmation of the resonance modes of action on the spikes with grain. For this purpose, two laboratory installations have been developed, in which the power elements are piezo actuators, a mathematical and computer model has been developed in the ACELAN package of resonant action on a spike and a field experiment has been carried out.

Formulation of the problem
The model of the effect of a piezo actuator on a fragment of a grain mass is built within the framework of the mechanics of a deformable solid. A rigorous mathematical model of the dynamics of a spike and a piezo actuator can be constructed within the framework of a continuous medium and, in particular, the linear theory of elasticity and electroelasticity. Such a model is a boundary-value problem or an initial-boundary value problem and for complex geometry, its solution can only be constructed numerically. In this work, the finite element method and its implementation in the ACELAN package are used as a numerical solution method. Continuous and discrete models used in the ACELAN package are given below.

Continuous problem setting
In this section, linear equations of the theory of elasticity and electroelasticity are used taking into account the dissipation of energy adopted in the ACELAN and ANSYS packages [13], as well as the equations of motion of liquid and gaseous media in the acoustic approximation [14]. For piezoelectric media:  The homogeneous condition (1.11) is also posed on the axis of symmetry when solving the axisymmetric problem.
condition for determining the unknown electric potential, which is formulated in addition to condition (1.10) on an electrode connected to an external electric circuit, in particular, if the electrode is free, then the charge or current is zero (=0) where m Icurrent in the circuit, which in the case of a free electrode is zero.
Finite element approximation in ACELAN To solve dynamic problems of acoustoelectroelasticity, we will use the FEM in the classical Lagrangian formulation. A fairly complete and consistent presentation of the FEM for piezoelectrics and its implementation in the ACELAN package is presented in [13], [15]. Below, according to [13], the FE formulation of the problems considered in the work is given. When passing from the continual formulation to the FE, a coordinated discretization of geometric regions is carried out -partition into finite elements (triangulation) with a certain set of geometric points -nodes, as a result of which the boundary value problem (1.1) -(1.12) using the representation  It is essential that approximations (1.13) are specified in those regions where there are corresponding physical fields, and may differ for different finite elements. An approximation of the FEM (1.13) of generalized statements of dynamic problems (1.1) -(1.12), including the main principal and natural boundary conditions, is reduced to a system of ordinary differential equations with respect to the nodal unknowns , , where global matrices , , M C K , which, by analogy with the FEM for elastic media, can be called matrices of mass, damping, stiffness, and the column of the right-hand sides have a block structure of the form:  where j D and j Edamping factors; uuj M and uuj K structural FE matrices of mass and stiffness j -th submatrices, respectively, and the remaining submatrices in (1.15) are described in [16]. When considering ideal materials, the resonance region is inaccessible for investigation, since the amplitudes of various quantities tend to infinity. However, for using structures practice, the numerical values of these quantities are extremely important. For example, the amplitudes of mechanical stresses that determine the strength of structures at resonance. Energy losses of various kinds in real material lead to the final values of these quantities. In the region of low and medium frequencies, mechanical energy losses make the greatest contribution in solids. For a quantitative description of this type of energy loss, If we use real arithmetic, then problem (1.21) is a SLAE with respect to the vector of real and imaginary parts of the nodal degrees of freedom a with a matrix that does not have positive definiteness and does not have a saddle structure.
In conclusion of this subsection, let us note some features of modal analysis in ANSYS and ACELAN. This analysis is designed to find natural or resonant frequencies of elastic bodies or frequencies of electrical resonances and antiresonances for piezoelectric bodies.

Calculation of a system based on a piezoactive cantilever
The first piezo actuator is a cantilever-mounted plate with piezoelements glued at the point of attachment, to the electrodes of which an alternating voltage is applied (Fig. 2.1), which excites its bending vibrations.

Fig. 2.1. Cantilever-based piezo actuator
The free end of the piezo actuator acts on the ear and excites its bending vibrations. The calculation of the vibration frequency of the piezo actuator was carried out in the ACELAN package. Figures 2.2 and 2.3 show a diagram of the piezo actuator geometry and a finite element mesh, respectively. The frequency of the device changes due to the use of different designs (the length of the middle passive layer, the size of the piezoelectric elements) and the substrate material (fiberglass, duralumin), as well as by attaching an inertial mass.  Table 1 shows the first three resonant frequencies and vibration modes of a piezo actuator with a glass fiber laminate substrate PZT-4 piezoceramic for the following dimensions: substrate 200x16x2 (mm), piezoelectric elements 36x15x1 (mm), distance from the embedment 2 mm.   Table 2 shows the dependences of natural frequencies on the length of the substrate for the same material, and in Table 3 for a substrate made of duralumin

System calculation based on a round bimorph
The second laboratory setup for resonant mechanical action on fragments of the grain mass is a cylindrical container with a metal bottom, to which a round disc-shaped piezoelement is glued (Figure 2

Experimenting
Full-scale experiments were carried out on the installations shown in Figures 2.1 and 2.5. Figure 2.12 shows a diagram of the test bench.

Fig. 2.12. Test bench layout
Electronic equipment for excitation of mechanical vibrations consists of a Tektronix AFG 3022B signal generator, a signal amplifier based on PA 94 piezo drivers (Apex Co., USA) and TDK Lambda power supplies. The control signal is monitored using a LeCroy 422 digital oscilloscope.  Figure 2.15 shows a photograph of the cantilever-based setup. The conducted natural experiment showed that the first mode of vibrations of an ear with a stem (the first bending mode for the whole plant) is excited at a frequency of 2.6 Hz, and the second mode in which the ear vibrates intensively with a relative rest of the stem was 15 Hz, which is in good agreement with the calculations carried out earlier.  16 shows the second setup based on a round bimorph for high-frequency impact on the ear. The first three resonance frequencies obtained in the course of a fullscale experiment turned out to be 2.6 kHz, 9.8 kHz, and 21.5 kHz, which is in good agreement with the calculations performed in Section 2.2. Figure 2.17 shows a photograph with a highlighted grain.

Conclusion
Mathematical and computer model in the ACELAN package of the resonance effect on the ear was developed and a field experiment was carried out, the results of which showed good agreement between the computer model of the present and previous research.