The mechanical manipulation of the soil is made by the using of farming tools or implements, which make to soil appropriate for the growth and development of plants (Ani et al., 2014ANI, A.; UZOEJINWA, B.; EZEAMA, O.; UGWU, S.; OHAGWU, C.; ODIGBOH, E.: “Soil bin facility for soil-machine interaction studies”, En: International Soil Tillage Research Organization (ISTRO) Nigeria Symposium, Akure, November 3 - 6, Ed. International Soil Tillage Research Organization (ISTRO), Department of Agricultural and Bioresources Engineering, University of Nigeria, Nsukka, pp. 110 - 124, 2014.; Prem et al., 2016PREM, M.; SWARNKAR, R.; KANTILAL, V.D.K.; JEETSINH, P.S.K.; CHITHARBHAI, K.B.: “Combined tillage tools-a review”, Current Agriculture Research Journal, 4(2): 179-185, 2016, ISSN: 2347-4688.). It is well known that the vibrations of tractive farming tools (knives, chisels, etc.), reduce the necessary force for their movement through the soil, which is highly desirable for the implements that require to diminish draft force like subsoiler and produce better break of the soil, although the total requirements of power cannot be reduced (Larson, 1967LARSON, L.W.: “The future of vibratory tillage tools”, Transactions of the ASAE, 10(1): 78-79, 1967, ISSN: 2151-0032.; Smith et al., 1972SMITH, J.L.; DAIS, J.L.; FLIKKE, A.M.: “Theoretical analysis of vibratory tillage”, Transactions of the ASAE, 15(5): 831-0833, 1972, ISSN: 2151-0032.). The tillage tool vibrations were presented in 1955 by Gunn and Tramontini cited by Rao et al. (2018)RAO, G.; CHAUDHARY, H.; SHARMA, A.: “Design and analysis of vibratory mechanism for tillage application”, Open Agriculture, 3(1): 437-443, 2018.. With the draft force reduction by means of the use of vibratory tools, it is possible to carry out operations of deep farming like subsoiling, with tractors of little tractive class and to achieve smaller compaction of the soil (Bandalan et al., 1999BANDALAN, E.; SALOKHE, V.; GUPTA, C.; NIYAMAPA, T.: “Performance of an oscillating subsoiler in breaking a hardpan”, Journal of Terramechanics, 36(2): 117-125, 1999, ISSN: 0022-4898.), with more efficiency in its crumbling (Rao et al., 2018RAO, G.; CHAUDHARY, H.; SHARMA, A.: “Design and analysis of vibratory mechanism for tillage application”, Open Agriculture, 3(1): 437-443, 2018.). These tools oscillate longitudinally or transversely, with frequencies of 2 to 14 Hz and amplitudes of 1,6 to 9,6 mm (Luna & González, 2002LUNA, L.A.; GONZÁLEZ, I.J.A.: “Estudio de la influencia de las vibraciónes verticales en los requerimientos energéticos del laboreo profundo del suelo”, Revista Ciencias Técnicas Agropecuarias, 11(3): 39-41, 2002, ISSN: 1010-2760, e-ISSN: 2071-0054.), along the direction of movement advance, that can be linear or curve, regarding the reference system of the implement, and the vibration way can be longitudinal or transverse. The oscillation plane can be vertical, horizontal or to have some inclination in the three-dimensional space (Rao & Chaudhary, 2018RAO, G.; CHAUDHARY, H.: “A review on effect of vibration in tillage application”, En: IOP Conference Series: Materials Science and Engineering, ser. Mater. Sci. Eng. . 377 012030, Ed. IOP Publishing, vol. 377, 2018, ISBN: 1757-899X.).
Investigations related with the use of vibratory tools have been developed by Shkurenko (1966)SHKURENKO, N.S.: Experimental data on the effect of oscillation on the cutting resistance of soil, Ed. National Institute of Agricultural Engineering, USA, 1966., Sulatisky & Ukrainetz (1972)SULATISKY, M.T.; UKRAINETZ, P.R.: “Draft reduction by vibratory soil cutting”, Transactions of the Canadian Society for Mechanical Engineering, 1(4): 175-181, 1972., Butson & MacIntyre (1981)BUTSON, M.; MACINTYRE, D.: “Vibratory soil cutting: I. Soil tank studies of draught and power requirements”, Journal of Agricultural Engineering Research, 26(5): 409-418, 1981, ISSN: 0021-8634., Zhang (1997)ZHANG, J.: Vibratory analysis of tillage operation, University of Saskatchewan, Department of Agricultural and Bioresource Engineering, Doctoral Thesis, Saskatchewan, Canada, 1997., Bandalan et al. (1999)BANDALAN, E.; SALOKHE, V.; GUPTA, C.; NIYAMAPA, T.: “Performance of an oscillating subsoiler in breaking a hardpan”, Journal of Terramechanics, 36(2): 117-125, 1999, ISSN: 0022-4898., Karoonboonyanan et al. (2007)KAROONBOONYANAN, R.; SALOKHE, V.; NIYAMAPA, T.; NAKASHIMA, H.: “Vibration effects on the performance of a single-shank subsoiler”, Agricultural Engineering International: CIGR Journal E journal. Manuscript PM 07 018, 9, 2007, ISSN: 1682-1130. and Shahgoli et al. (2010)SHAHGOLI, G.; FIELKE, J.; DESBIOLLES, J.; SAUNDERS, C.: “Optimising oscillation frequency in oscillatory tillage”, Soil and tillage research, 106(2): 202-210, 2010, ISSN: 0167-1987.. All these studies had the objective of determining the optimum vibration modes, operational and geometric parameters, as well as the required power and their effect in the magnitude of the necessary draft forces for breaking the soil.
Shkurenko (1960) carried out experiments with the bent leg oscillations in horizontal and vertical direction, frequencies of 100 and 210 Hz and 0.3 m. s-1 of forward speed. The draft force diminished from 50 to 60% when the width increased from 0 to 10 mm. Butson & MacIntyre (1981)BUTSON, M.; MACINTYRE, D.: “Vibratory soil cutting: I. Soil tank studies of draught and power requirements”, Journal of Agricultural Engineering Research, 26(5): 409-418, 1981, ISSN: 0021-8634. carried out experiments to oscillation frequencies bigger than 50 Hz and widths of 8 mm, with forward speeds from 0.54 to 1.98 km. h-1. The draft force diminished above 50%, but the total consumption of power increased. However, Sulatisky & Ukrainetz (1972)SULATISKY, M.T.; UKRAINETZ, P.R.: “Draft reduction by vibratory soil cutting”, Transactions of the Canadian Society for Mechanical Engineering, 1(4): 175-181, 1972. reported that, reduction of the draft force as high as 80%, was achieved when the tool vibrated to frequencies higher than 30 Hz and widths bigger than 12 mm.
Bandalan et al. (1999)BANDALAN, E.; SALOKHE, V.; GUPTA, C.; NIYAMAPA, T.: “Performance of an oscillating subsoiler in breaking a hardpan”, Journal of Terramechanics, 36(2): 117-125, 1999, ISSN: 0022-4898. carried out experiments in a vibratory subsoiler of vertical right arm and plough share with lift angle of 30° and working width of 70 mm, tilling a compacted soil, with oscillation frequencies of 3,7; 5,67; 7,85; 9,48 and 11,45 Hz; widths of 18; 21; 23,5; 34 and 36,5 mm and forward speeds of 1,85; 2,20 and 3,42 km.h-1. The vibratory system diminished the traction force 0,33% and the consumption energy increased 1,24% regarding the system without vibrating. The subsoiler could not work to frequencies smaller than 5 Hz (resonance of the tool). However, Shahgoli et al. (2010)SHAHGOLI, G.; FIELKE, J.; DESBIOLLES, J.; SAUNDERS, C.: “Optimising oscillation frequency in oscillatory tillage”, Soil and tillage research, 106(2): 202-210, 2010, ISSN: 0167-1987. carried out experiments with vibratory subsoiler of two arms and cam mechanism, with right and curved plough share in loam-sandy soil oscillating with amplitude of ± 69 mm; oscillation angle 27º; forward speed of 3 km.h-1 and oscillation frequency of 1,9 to 8,8 Hz. They concluded that with frequencies near 3,3 Hz and forward speed of 1,5 km.h-1, the draft force diminished 26% compared with the rigid one.
The general objective of this study was to carry out a modal analysis of the soil-vibratory tool interaction by means of a simulation model with the finite elements method to determine the vibration modes and their specific frequencies (resonant) and to select the most appropriate ones for the operation of the system, as well as the effect of the work depth in the frequency and amplitude of vibrations.
The soil was modeled as continuous, homogeneous and elastoplastic, using the linear form of the extended Drucker-Prager model (Figure 1), utilized with success by Herrera et al. (2008aHERRERA, S.M.; IGLESIAS, C.C.E.; GONZÁLEZ, C.O.; LÓPEZ, B.E.; SÁNCHEZ, I.Á.: “Propiedades mecánicas de un Rhodic Ferralsol requeridas para la simulación de la interacción suelo implemento de labranza mediante el Método de Elementos Finitos: Parte I”, Revista Ciencias Técnicas Agropecuarias, 17(3): 31-38, 2008a, ISSN: 1010-2760, e-ISSN: 2071-0054., 2008b)HERRERA, S.M.; IGLESIAS, C.C.E.; GONZÁLEZ, C.O.; LÓPEZ, B.E.; SÁNCHEZ, I.Á.: “Propiedades mecánicas de un Rhodic Ferralsol requeridas para la simulación de la interacción suelo implemento de labranza mediante el Método de Elementos Finitos: Parte II. Interfase suelo-herramienta”, Revista Ciencias Técnicas Agropecuarias, 17(4): 50-54, 2008b, ISSN: 1010-2760, e-ISSN: 2071-0054., given the simplicity of it and the little quantity of necessary parameters for its implementation (González et al., 2014GONZÁLEZ, C.O.; HERRERA, S.M.; IGLESIAS, C.C.E.; LÓPEZ, B.E.: “Modelos constitutivos drucker prager extendido y drucker prager modificado para suelos rhodic ferralsol”, Terra Latinoamericana, 32(4): 283-290, 2014, ISSN: 0187-5779.).
The soil taken as study object was classified as Rhodic Ferralsol (Hernández et al. (2015)HERNÁNDEZ, J.A.; PÉREZ, J.J.M.; BOSCH, I.D.; CASTRO, S.: Clasificación de los suelos de Cuba, Ed. INCA, Ediciones INCA ed., San Jose de las Lajas, Mayabeque. Cuba, 93 p., 2015, ISBN: 978-959-7023-77-7., with density of 1050 kg·m-3, plasticity index of 36.1% and matter content of 2.8%. The elasticity module (E) was determined as the slope of a tangent straight line to the curve effort-deformation in its right tract, obtained for this type of soil by De la Rosa et al. (2014)DE LA ROSA, A.A.A.; HERRERA, S.M.; GONZÁLEZ, C.O.: “Los Modelos Constitutivos para la Simulación de la Respuesta Mecánica de los Suelos Agrícolas mediante el Métodos de Elementos Finito (MEF).”, Revista de Investigaciones de la Universidad Le Cordon Bleu, 1(1): 49-59, 2014, ISSN: 2409-1537.. The values of the soil properties required by the simulation model in finite elements (Table 1) were obtained from García de la Figal (1978GARCÍA DE LA FIGAL, C.: “Estudio de la fricción suelo-metal y suelo-plástico para dos suelos cañeros cubanos”, Ciencias Técnicas, Ingeniería en la Construcción de Maquinaria, 3(1): 107-122, 1978., 1991)GARCÍA DE LA FIGAL, C.A.: “Estudio de las propiedades tecnológicas más importantes de los suelos cubanos”, Revista Ciencias Técnicas Agropecuarias, 3(2): 61-77, 1991, ISSN: 1010-2760, e-ISSN: 2071-0054., Herrera et al. (2008aHERRERA, S.M.; IGLESIAS, C.C.E.; GONZÁLEZ, C.O.; LÓPEZ, B.E.; SÁNCHEZ, I.Á.: “Propiedades mecánicas de un Rhodic Ferralsol requeridas para la simulación de la interacción suelo implemento de labranza mediante el Método de Elementos Finitos: Parte I”, Revista Ciencias Técnicas Agropecuarias, 17(3): 31-38, 2008a, ISSN: 1010-2760, e-ISSN: 2071-0054., 2008b)HERRERA, S.M.; IGLESIAS, C.C.E.; GONZÁLEZ, C.O.; LÓPEZ, B.E.; SÁNCHEZ, I.Á.: “Propiedades mecánicas de un Rhodic Ferralsol requeridas para la simulación de la interacción suelo implemento de labranza mediante el Método de Elementos Finitos: Parte II. Interfase suelo-herramienta”, Revista Ciencias Técnicas Agropecuarias, 17(4): 50-54, 2008b, ISSN: 1010-2760, e-ISSN: 2071-0054. and De la Rosa et al. (2014)DE LA ROSA, A.A.A.; HERRERA, S.M.; GONZÁLEZ, C.O.: “Los Modelos Constitutivos para la Simulación de la Respuesta Mecánica de los Suelos Agrícolas mediante el Métodos de Elementos Finito (MEF).”, Revista de Investigaciones de la Universidad Le Cordon Bleu, 1(1): 49-59, 2014, ISSN: 2409-1537..
The values of the properties and soil parameters required by the simulation model in finite elements are shown in the Table 1.
The model is composed by the subsoiler (with curved bent leg and logarithmic profile), the soil block, the vibrant mechanism and the interaction surfaces between both (Figure 2). The bent leg moves in the direction of the X axis to constant speed and working depth ae, vibrations frequency of the vibrating mechanism of 0.1 Hz and amplitude of 4 mm. It has angular movement freedoms in the vertex of the phase angle (θ) and linear in the X and Y axes. The lift angle (α) is 25° and the amplitude is 78 mm. The soil block has movement restrictions in lateral, posterior and inferior surfaces. Its dimensions are: length L (2 m), height H (0.9 m) and width B (1 m). The area of the tip surface is 0,0017 m2 and of the attack surface 0,018 m2. The width of the cut soil prism (b0) coincides with the rake width. An increase of the dimensions of the soil prism, beyond those assigned, as a result of the interaction with the bent leg, can be rejected (Ibrahmi et al., 2015IBRAHMI, A.; BENTAHER, H.; HBAIEB, M.; MAALEJ, A.; MOUAZEN, M.A.: “Study the effect of tool geometry and operational conditions on mouldboard plough forces and energy requirement: Part 1. Finite element simulation”, Computers and Electronics in Agriculture, 117: 258-267, 2015, ISSN: 0168-1699.; Marín & García de la Figal, 2019MARÍN, C.L.O.; GARCÍA DE LA FIGAL, C.A.: “Model of Soil-TillageTool Interaction Using Finite Element Method”, Revista Ciencias Técnicas Agropecuarias, 28(4), 2019, ISSN: 1010-2760, e-ISSN: 2071-0054.).
The equation of the displacement (damped forced vibrations) is:
where: X -amplitude of vibrations, mm; ω- frequency of vibrations, Hz
The speed is given by:
The period of the vibration (T) is calculated by:
The frequency of the vibrations is given by:
The natural frequency is calculated by:
being: k - elastic constant of spring; m - spring mass;
The equation of displacement in the non-damped free vibratory movement is:
The speed equation is:
For the modal analysis of the simulation model, three working depths (ae) were used: 200, 300 and 400 mm and two vibration modes: free non-damped and forced damped. The forward speed was kept constant Vm = 0,6 m.s-1, the mesh density (size of elements) ae = 6 mm, with mesh control of the surfaces in contact, both the plough shares and the soil prism e rp = 4 (Marín et al., 2020MARÍN, C.L.O.; GARCÍA DE LA FIGAL, A.E.; MARTÍNEZ, R.A.: “Effect of the Geometry and Operational Conditions in the Draft Forces of the Arm of a Vibratory Scarifier”, Revista Ciencias Técnicas Agropecuarias, 29(2), 2020, ISSN: 1010-2760, e-ISSN: 2071-0054.).
The free non-damped and forced damped vibration modes were simulated. The first fifteen modal forms for both and their corresponding resonant frequencies (f nb ) were obtained and the two first vibration modes were the most appropriate for the operation of the bent leg (Table 2). With the natural frequencies obtained with free non-damped vibrations f nbl = 2,21; 13,35 Hz and forced damped vibrations f nbf = 8,48 Hz, bigger soil crumbling was achieved as well as a diminishing of the draft force and power requirements. Similar frequencies of: 3,7, 5,67, 7,85 and 9,48 Hz, were employed by Bandalan et al. (1999)BANDALAN, E.; SALOKHE, V.; GUPTA, C.; NIYAMAPA, T.: “Performance of an oscillating subsoiler in breaking a hardpan”, Journal of Terramechanics, 36(2): 117-125, 1999, ISSN: 0022-4898. in field experiments with a vibratory subsoiler of simple arm and they obtained the highest values in reduction of the draft force in the longitudinal plane (0.33%) and power requirements (1.24%), with a frequency of 9,48 Hz, vibration amplitude of 36,5 mm and forward speed of 0,61 m. s-1.
For the bent leg with free non-damped vibrations and the modal forms1 and 2 (Figure 3a), the bent leg can work the soil without risks of the resonance effect, because the frequencies obtained in both modal forms allow its appropriate work. For the bent leg with forced damped vibrations (Figure 3b), the modal form 1 is near a resonant condition (f nbf = 0,45 Hz), and that is why it is not the most appropriate for the operation of the vibratory system. The modal form 2 (f nbf = 8,47 Hz) is the optimum. Similar results were obtained by Shahgoli et al. (2010)SHAHGOLI, G.; FIELKE, J.; DESBIOLLES, J.; SAUNDERS, C.: “Optimising oscillation frequency in oscillatory tillage”, Soil and tillage research, 106(2): 202-210, 2010, ISSN: 0167-1987., when they reached a reduction of the draft force of 26% to a frequency of 8,8 Hz in the longitudinal plane, amplitude of ± 69 mm, oscillation angle of 27° and forward speed of 0,83 m.s-1. However, Luna & González (2002)LUNA, L.A.; GONZÁLEZ, I.J.A.: “Estudio de la influencia de las vibraciónes verticales en los requerimientos energéticos del laboreo profundo del suelo”, Revista Ciencias Técnicas Agropecuarias, 11(3): 39-41, 2002, ISSN: 1010-2760, e-ISSN: 2071-0054. affirm that the best results for vibratory subsoilers are obtained for frequencies of 80-100 rad. s-1 (12-16 Hz) and amplitudes greater than 8 mm in a plane of vibrations (vertical), working depth between 300-400 mm and forward speeds between 0,56 and 1,4 m. s-1.
The results of the frequency study carried out to the soil model to different working depths (Tables 3, 4 and 5) show that, to a depth ae=200 mm and the bent leg subsoiler with damped forced vibrations (Table 3b), the most appropriate values of soil natural frequencies are obtained for its loosening (2,63; 4,13; 8,15; 11,07 and 16,21 Hz).
The Figure 4 shows the modal forms of the soil prism that correspond to the modal forms 3,4,5,6,7 and 8 with forced damped vibrations to depth of 200 mm.
The statistical analysis (Table 6) included variance analysis, Scheffé (posteriori test for differences) and simple linear regression, for both free and forced vibrations.
It is shown in Table 7. Significant differences exist in the soil frequencies and amplitudes, for both free and forced vibrations.
With free vibrations, to different working depths, the frequencies of the bent leg were not different (p=1); but in the frequencies of the soil significant differences were observed (p=0,000) between the depths 200 mm with 300 mm and 400 mm, respectively, but they were not observed between 300 mm and 400 mm (Table 8).
The changes in the magnitudes of the soil frequency are explained in 69,3% by the changes in the levels of the working depth (Figure 5). For each mm of depth increased or diminished, soil frequencies were increased or decreased 0,837 Hz. The changes of the soil frequency that are explained by other factors (residuals) are almost null (0,00).
*. The differences of means are significant in lever 0,05.
The changes in the magnitudes of the soil amplitude are explained in 47% by the changes in the levels of the working depth. For each mm of the working depth increased or diminished, it increased or it diminished 0,694 mm the soil width (Figure 6). The changes in the soil width due to other factors (residuals) they are almost null (0,00).
With forced vibrations (Table 9), at different work depths, soil and bent leg frequencies were not different (p>0,05); but the amplitudes were different significantly (p=0,000) for both with evidence of the differences in the width of the bent leg, between the depths 400 mm with 200 mm and 300 mm respectively, but don't between 200 mm and 300 mm, as well as in the widths of the soil, between the depths 200 mm and 400 mm, but don't between 300 mm and 400 mm.
*. The differences of means are significant in lever 0,05
The changes in the bent leg amplitude are explained in 32,7% by the changes in the depth levels. For each mm of depth increased, its amplitude diminished 0,585 mm (Figure 7). In the case of the soil amplitude, 30,7% of the changes is due to changes in the depth levels and, for variations of depth per mm, the soil amplitude varied 0,568 mm (Figure 8).
Of the modal analysis by finite elements carried out to the subsoiler bent leg and to the soil, the first fifteen vibration modes and their modal forms were obtained, as well as the corresponding natural frequencies, for both, free non-damped vibrations and damped forced vibrations.
The modes 1 and 2 of vibration of the bent leg are the most appropriate for the simulation. The modal forms1 and 2, corresponding to the first vibration mode, as well as the modal form 2 in the second mode, have the most appropriate resonant frequencies for loosening of soil.
The study of frequency carried out to the soil model to different work depths shows that, using the bent leg with damped forced vibrations, to a work depth of 200 mm, resonant frequencies are obtained that allow better crumbling of soil.
The statistical analysis showed significant effect of the work depth in the frequencies and soil amplitude, when the vibratory system works with free vibrations. With forced vibrations, to different work depths, the frequencies, for both, the bent leg and the soil were not different, but the widths were significantly different for both.