Strength analysis and modal analysis for the load-bearing structure of the equipment for opening and compartmentalizing watering furrows, using 1D

The paper proposes the analysis of the load-bearing structure of the equipment for opening and compartmentalizing watering furrows (EOCFW), using a 3D structural model, built with 1D finite element. Based on information from the experimental results of the EOCFW equipment, the load-bearing structure is supported and loaded. The results of the linear static analysis of the structure consist of the distributions of the relative displacement fields and the equivalent voltage in the structure. Also, the own frequencies of the structure and the deformed forms of the structure are obtained when it vibrates in its own ways with the lowest own frequencies. It shows how these results can be used. The field of relative displacements is used to assess the effects on the quality of the soil processing performed. The equivalent stress field is used to estimate the safety factor of the structure, by reference to the flow stress of the material from which the structure is built. The first four or five own frequencies are important for the prognosis of possible vibration regimes with resonance, their explanation and their amelioration. It emphasizes the ease with which the model can be modified to obtain improved or even optimized variants.


Introduction
Structural analysis is a modern tool, but already normal in the research of physical phenomena and the design of industrial, civil or other products. The tool is modern today because it constantly addresses new topics and deepens the already classic ones.
In the field of agricultural machinery, structural analysis has gained intense use in the last fifty years. In addition to the hesitant beginnings, in the last twenty years, there have been many works related to the design of equipment used in agriculture or the structural modelling of physical processes in agriculture, [1]. A review of the use of MEF in the field of agricultural mechanization is, for example [2,5]. Most of the literature of the last 20 years, in this field, proposes models with solid (three-dimensional) finite elements, [3,4,6,[7][8][9][10][11][12][13].
3D structural models, built with 1D bar-type finite elements, are older than those built with 3D finite elements. Mathematical models of the straight bar (Euler or Tymoshenko, for example) were used long before the appearance of the finite element method. The finite element method integrated these mathematical models. 3D structural models built with 1D finite elements are also well known in the literature in the field of agricultural machinery, [1,[14][15][16][17]. Structural models built with 1D finite elements are well suited to large structures formed with relatively thin bars, such as the load-bearing structures of bridges, vehicles or agricultural machinery [18][19][20]. The literature still abounds with examples of the use of 3D structural models, mashed with 1D or mixed finite elements, hybrid models that contain 1D elements (of the bar or beam type). This category also includes the model presented in this paper. The advantages and disadvantages of this model are presented in relation to the 3D models mashed with 3D elements. On the other hand, the usefulness of such a model in the research -design activity for certain structures, signalled above, is also argued, their structural models simplifying the calculation procedures and the interpretation of the results. Research on the testing of agricultural equipment in real field conditions was carried out by [27,29,33], simulation by finite element analysis (FEM) [24,28,31,32], on special installations/test stands in regime simulated and accelerated [26,30,31] or theoretical research [25], to identify and develop the most modern constructive solutions [34], similar to those carried out worldwide.

Material and method
The subject of the analysis, the results of which are presented in the article, is the loadbearing structure of EOCFW, fig. 1. The structural model (geometry, fixing conditions and loads, elements and nodes) of the load-bearing structure of EOCFW, is presented in fig. 2. There is a difference between the load-bearing structure of the EOCFW and the entire structure of this machine. The working organs (the double mouldboard and the palette) are represented in the model only by the forces with which they act on the load-bearing structure. Fixing the structure or the conditions on the border means cancelling the relative linear displacements in the articulation points to the tractor. The loading is done with the force of 4500 N from the double mouldboard side and 700 N from the blade side, in the direction of movement in the opposite direction to the direction of movement of the unit. A force of 100 N acts on the resistance frame (the load-bearing structure) in the same direction and in the same sense as the other two forces, as a result of the resistance of the copy wheel, fig.2. The material of the structure, for linear static analysis and modal analysis, is characterized by the linear elasticity modulus, E = 2.0·10 11 Pa, the Poisson's ratio ν = 0.29 and the density ρ = 7900 kg / m 3 . The model is mashed with ninety-eight nodes and ninety BEAM3D elements included in the finite element library of the COSMOS / M 1.75 program, [21]. The characteristics of the cross-sections of the bars used in the construction of the structural model of the load-bearing structure of the EOCFW are given in Table 1.

Results
The main results of the two analyzes are relative displacement field (values in nodes), the tensor field of Cauchy stresses (average values per element), reaction values, deformed shape of the structure (from static linear elastic analysis), respectively: the lower first five eigenfrequencies and the deformed shapes of the structure when it vibrates in the corresponding eigenmodes.

The main results of the linear elastic analysis
The vector field of relative displacement or the deformation field as it is called in engineering language has three components, corresponding to the three axes of the absolute reference system. Therefore there are three scalar fields of relative displacements in the directions of the three coordinate axes. A scalar field of the relative deformation, which reflects the overall displacement, is that of the resultant relative displacement, whose distribution in the structural model is given in fig. 3.
The stress field has in this model two main components, the axial stress field and the equivalent stress field (Von Mises), [21]. In fig. 4 represents the map of the average equivalent stress field distribution for each element of the model. The maximum value of the relative displacement is 4.96 mm and corresponds to the component along the vertical axis at the ground plane (Oy). The maximum value of the equivalent stress in the structure is 123 MPa, is located at the upper part of the double mouldboard support and at the lower part of the beam with which the structure is coupled to the upper link of the tractor. Reaction values at the tractor points of linked to the load-bearing structure is given in Table  2.

The main results of the modal analysis
The first five eigenfrequencies of the structure, in Hz, have, in ascending order, the values: 17.57, 24.60, 42.44, 107.48, and 126.62. The deformed shapes of the structure when it vibrates on one of its own frequencies, help to determine the components of the assembly, which most likely causes a possible movement with high amplitude or even resonance.

Discussion
In addition to the results presented in the previous chapter, the FEA program that solves the analysis of the structural model, COSMOS / M 1.75, [21], also provides other results. These results include the inertial characteristics of the entire structure, for example, the mass of 70.07 kg, the length, the area and the volume of the model bars, the geometric and mass moments of inertia usable in calculations of the aggregate dynamics, the coordinates of the mass centre and radius of gyration. The mass given by the program is smaller than the mass of the EOCFW because it does not include the working organs and the additional elements for their support or guidance (in total, around 110 kg). The presented model is the simplest possible with natural loads in forces derived from experimental data, [22]. Obviously, the model can be complicated by the introduction of working organs and additional guiding elements. For ordinary steels, the yield strength (at the end of which irreversible deformations begin to occur), has the value of about 220 -230 MPa. Taking into account the maximum value of the equivalent stress in the structure ( fig. 4), a safety factor of 1.8, [23], that required in the standards for agricultural machinery and equipment, results. If the structure is built with steel whose minimum yield stress value is 351,571 MPa, then the minimum safety factor of the structure is 2.86.

Conclusion
The results presented in the paper show that a mashed 3D structural model with 1D finite elements (Tymoshenko bars), is particularly useful in the rapid design of a load-bearing structure if sufficiently precise loading assumptions and boundary conditions are known. Changing the bars of the structure is very simple because the design does not change but only the numerical characteristics of their cross sections change. The changes, purely numerical, are made simply in the database. In this way, the bars are replaced by probing until a safety factor is reached in accordance with the standards and a negligible deformation for the working process. In the same way, some vibrations that can affect the work process can be largely removed. This optimization process can be also operated on the experimental model after detecting some deficiencies in the tests carried out in the field. The use of mashed 3D structural models with 1D type elements is useful at a first start design, it is simple and fast compared to the direct calculation on mashed structural models with 3D finite elements (SOLID). 3D CAD-CAM models are models prepared especially for execution documentation. Their use in the structural analysis must go through the stage of transformation into a CAD -CAE model (elimination of gaps and interferences, mashing). Modifications by probing beams or other components are difficult, often involving redesign. For this reason, a mashed 3D structural model with 1D elements (possibly also 2D) is useful, the pre-design made with its help greatly limiting the calculation on models with 3D finite elements or even eliminating it.