ORIGINAL ARTICLE

Effect of Vibrations and Operational Parameters in Frequency and Amplitude of a Vibratory Subsoiler

MSc. Luis Orlando Marín-Cabrera^{1  *}  , Dr.C. Armando Eloy García de la Figal-Costales¹  , Dr.Cs. Arturo Martínez-Rodríguez¹ 

¹Universidad Agraria de La Habana (UNAH), Facultad de Ciencias Técnicas, Centro de Mecanización Agropecuaria (CEMA), San José de las Lajas, Mayabeque, Cuba.

ABSTRACT

This paper aims to analyze the effect of free and forced vibrations and of working depth on the frequency and amplitude of a vibratory curved bent leg subsoiler plowing a silt loam soil (Rhodic Ferralsol) by the modal analysis of the soil-bent leg system. The finite elements method and the design software Solid Works with its complement Simulation were used to model and to simulate, the bent leg scarifier­ soil interaction. The soil was considered as homogeneous and elastoplastic of Drucker-Prager extended constitutive relation. The results showed the significant effect of the work depth on the frequencies and amplitude of the soil when the vibratory system works with free vibrations. When forced vibrations were used to different work depths, significant differences were not observed in frequencies neither of the bent leg nor of the soil. On the other hand, the resonant frequencies of the shank obtained corresponding to the two first vibration modes (2,20; 8,47 and 13,35 Hz) at a working depth of 200 mm allowed a better loosening of the soil.

Key words: Subsoiler; Free Vibrations; Forced Vibrations; Working Depth

INTRODUCTION

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., 2014}; ^{Prem et al., 2016}). 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, 1967}; ^{Smith et al., 1972}). The tillage tool vibrations were presented in 1955 by Gunn and Tramontini cited by ^{Rao et al. (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., 1999}), with more efficiency in its crumbling (^{Rao et al., 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, 2002}), 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, 2018}).

Investigations related with the use of vibratory tools have been developed by ^{Shkurenko (1966)}, ^{Sulatisky & Ukrainetz (1972)}, ^{Butson & MacIntyre (1981)}, ^{Zhang (1997)}, ^{Bandalan et al. (1999)}, ^{Karoonboonyanan et al. (2007)} and ^{Shahgoli et al. (2010)}. 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)} 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)} 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)} 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)} 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.

MATERIAL AND METHODS

Model for Soil

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. (2008a}, ^2008b), given the simplicity of it and the little quantity of necessary parameters for its implementation (^{González et al., 2014}).

FIGURE 1 Extended linear Drucker-Prager model: a) meridional plane b) Main stresses plane.  

Properties and Soil Parameters

The soil taken as study object was classified as Rhodic Ferralsol (^{Hernández et al. (2015)}, 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)}. The values of the soil properties required by the simulation model in finite elements (Table 1) were obtained from ^{García de la Figal (1978}, ¹⁹⁹¹⁾, ^{Herrera et al. (2008a}, ^2008b) and ^{De la Rosa et al. (2014)}.

The values of the properties and soil parameters required by the simulation model in finite elements are shown in the Table 1.

TABLE 1 Properties and soil parameters required by the FEM model 

Property or parameter	Symbol	Dimension
Friction internal angle	φ	5º
Modulus of elasticity	E	1575 kPa
Shear modulus	G	1793 kPa
Poisson's ratio	ν	0,22
Cohesion	c	15 kPa
Soil humidity	Ha	27%
Density	γ	1.05 g.cm^-3
Shear resistance	τ	190 kPa
Shear modulus	G	1 793 kPa
Traction limit of soil	σ t	20 kPa
Compression limit of soil	σ c	480 kPa
Elastic limit of soil	σ e	42 kPa
Soil-metal friction angle	δ	30.5º
Type of soil	Linear elastoplastic

Simulation Model of the Interaction Soil-Vibratory Farming Tool

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 a_e, 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., 2015}; ^{Marín & García de la Figal, 2019}).

FIGURE 2 Tridimensional model of the system. 

The equation of the displacement (damped forced vibrations) is:

$x(t)=X\times sen\left(\omega t-\theta \right)$ (1)

where: X -amplitude of vibrations, mm; ω- frequency of vibrations, Hz

The speed is given by:

$\dot{x}=X\times \omega \times \cos \left(\omega t-\theta \right)$ (2)

The period of the vibration (T) is calculated by:

$T=\frac{2\pi }{\omega }$ (3)

The frequency of the vibrations is given by:

$f=\frac{1}{T}=\frac{\omega }{2\pi }$ (4)

The natural frequency is calculated by:

${\omega }_n=\sqrt{\frac{k}{m}}$ (5)

being: k - elastic constant of spring; m - spring mass;

The equation of displacement in the non-damped free vibratory movement is:

$x(t)=Xsen\left({\omega }_{n}t+\varphi \right)$ (6)

The speed equation is:

$\dot{x}(t)={\omega }_{n}X\cos \left({\omega }_{n}t+\varphi \right)$ (7)

For the modal analysis of the simulation model, three working depths (a_e) 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) a_e = 6 mm, with mesh control of the surfaces in contact, both the plough shares and the soil prism e _rp = 4 (^{Marín et al., 2020}).

RESULTS AND DISCUSSION

Bent leg Modal Analysis

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)} 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.

TABLE 2 Bent leg resonant frequencies: a) free non-damped vibrations b) Forced damped vibrations 

a)				b)
Mode	Frequency (rads)	Frequency (Hz)	Period (seg)	Frequency (rads)	Frequency (Hz)	Period (seg)
1	13.884	2.2097	0.45255	2.8767	0.45784	2.1842
2	83.924	13.357	0.074867	53.26	8.4767	0.11797
3	387.18	61.622	0.016228	184.46	29.357	0.034063
4	961.29	152.99	0.0065362	839.61	133.63	0.0074835
5	2078	330.72	0.0030237	1758.6	279.9	0.0035727
6	2181.5	347.19	0.0028802	1952.6	310.77	0.0032178
7	2526.2	402.06	0.0024872	2390.2	380.41	0.0026287
8	3844.1	611.82	0.0016345	3444.6	548.23	0.001824
9	4303.7	684.96	0.0014599	3730.6	593.74	0.0016842
10	4699.3	747.92	0.0013371	4234.1	673.89	0.0014839
11	5091.5	810.34	0.0012341	4437.2	706.19	0.001416
12	6006.5	955.97	0.0010461	5812.1	925.03	0.001081
13	6760.6	1076	0.0009293	6058.2	964.19	0.0010371
14	7542.5	1200.4	0.000833	7159.4	1139.5	0.00087762
15	8325.3	1325	0.0007547	7994.3	1272.3	0.00078596

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)}, 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)} 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.

FIGURE 3 Resonant frequencies in the first two vibration modes: a) bent leg with free non-damped vibrations b) bent leg with forced damped vibrations. 

Modal Analysis of the Soil

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 a_e=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).

TABLE 3 Resonant frequencies of the soil model (a_e= 200 mm)  

a)				b)
Mode	Frequency (rads)	Frequency (Hz)	Period (seg)	Frequency (rads)	Frequency (Hz)	Period (seg)
1	0.021701	0.0034538	289.54	0.013164	0.0020951	477.31
2	579.97	92.305	0.010834	0.019076	0.0030361	329.37
3	637.24	101.42	0.00986	6.5912	1.049	0.95327
4	649.29	103.34	0.009677	16.546	2.6334	0.37974
5	701.2	111.6	0.0089606	26.004	4.1386	0.24163
6	777.49	123.74	0.0080814	51.241	8.1552	0.12262
7	790.38	125.79	0.0079496	69.592	11.076	0.090286
8	817.58	130.12	0.0076851	101.86	16.212	0.061684
9	828.63	131.88	0.0075826	129.7	20.642	0.048445
10	850.64	135.38	0.0073865	142.17	22.627	0.044196
11	867.48	138.06	0.007243	190.51	30.321	0.032981
12	873.4	139.01	0.0071939	234.96	37.395	0.026742
13	925.1	147.23	0.0067919	261.13	41.56	0.024062
14	935.75	148.93	0.0067146	298.42	47.495	0.021055
15	941.92	149.91	0.0066706	352.19	56.052	0.01784

TABLE 4 Resonant frequencies of the soil model (a_e= 300 mm) 

a)				b)
Mode	Frequency (rads)	Frequency (Hz)	Period (seg)	Frequency (rads)	Frequency (Hz)	Period (seg)
1	516.25	82.163	0.012171	468.47	74.56	0.013412
2	583.26	92.829	0.010773	561.69	89.396	0.011186
3	606.84	96.581	0.010354	585.99	93.263	0.010722
4	691.38	110.04	0.0090879	667.4	106.22	0.0094144
5	749.66	119.31	0.0083813	720.82	114.72	0.0087167
6	762.34	121.33	0.008242	747.5	118.97	0.0084056
7	765.03	121.76	0.008213	758.99	120.8	0.0082784
8	776.87	123.64	0.0080878	766.79	122.04	0.0081941
9	789.33	125.63	0.0079601	785.55	125.02	0.0079985
10	839.71	133.64	0.0074826	836.73	133.17	0.0075093
11	857.04	136.4	0.0073313	837.85	133.35	0.0074992
12	859.91	136.86	0.0073068	850.02	135.28	0.0073918
13	870.26	138.51	0.0072199	863.44	137.42	0.0072769
14	919.53	146.35	0.006833	912.67	145.26	0.0068844
15	924.87	147.2	0.0067936	919.12	146.28	0.0068361

TABLE 5 Resonant frequencies of the soil model (ae= 400 mm) 

a) Free non-damped vibrations				b)Forced damped vibrations
Mode	Frequency (rads)	Frequency (Hz)	Period (seg)	Frequency (rads)	Frequency (Hz)	Period (seg)
1	551.09	87.709	0.011401	477.05	75.925	0.013171
2	619.25	98.557	0.010146	583.77	92.91	0.010763
3	646.92	102.96	0.0097125	615.04	97.886	0.010216
4	734.29	116.87	0.0085569	688.95	109.65	0.00912
5	797.82	126.98	0.0078755	732.64	116.6	0.0085761
6	807.18	128.47	0.0077841	762.12	121.3	0.0082443
7	817.29	130.08	0.0076879	771.35	122.76	0.0081457
8	827.74	131.74	0.0075908	773.12	123.05	0.0081271
9	837.04	133.22	0.0075064	794.03	126.37	0.0079131
10	878.84	139.87	0.0071494	844.92	134.47	0.0074364
11	883.54	140.62	0.0071114	857.07	136.41	0.007331
12	910.39	144.89	0.0069016	865.95	137.82	0.0072558
13	928.34	147.75	0.0067682	869.82	138.44	0.0072236
14	968.19	154.09	0.0064896	928.76	147.82	0.0067652
15	986.92	157.07	0.0063665	928.97	147.85	0.0067636

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.

FIGURE 4 Modal forms of soil prism corresponding to vibration modes 3,4,5,6,7 and 8 with forced damped vibrations to a depth of 200 mm. 

Modal Analysis of the Bent Leg-Soil System

The statistical analysis (Table 6) included variance analysis, Scheffé (posteriori test for differences) and simple linear regression, for both free and forced vibrations.

TABLE 6 Modal analysis of bent leg-soil system 

Depth (mm)	Vibration	Bent leg frequency (Hz)				Soil frequency (Hz)
		Mean	Deviation standard	Minimum	Maximum	Mean	Deviation standard	Minimum	Maximum
200	Forced	581,504	437,078	2,210	1325,000	19,958	18,677	0,002	56,050
	Free	797,372	514,733	45,966	1583,500	118,582	37,338	0,003	149,910
300	Forced	581,504	437,078	2,210	1325,000	122,150	19,484	82,163	147,200
	Free	797,372	514,733	45,966	1583,500	119,717	21,012	74,560	146,280
400	Forced	581,504	437,078	2,210	1325,000	129,392	20,284	87,709	157,070
	Free	797,372	514,733	45,966	1583,500	129,364	20,303	87,718	157,190
Depth (mm)	Vibration	Bent leg amplitude (mm)				Soil amplitude (mm)
		Mean	Deviation standard	Minimum	Maximum	Mean	Deviation standard	Minimum	Maximum
200	Free	15,807	20,082	2,330	68,200	0,335	0,116	0,120	0,530
	Forced	7,835	4,592	3,300	16,200	0,076	0,011	0,055	0,101
300	Free	15,807	20,082	2,330	68,200	0,095	0,028	0,060	0,130
	Forced	7,835	4,592	3,300	16,200	0,100	0,026	0,060	0,140
400	Free	15,807	20,082	2,330	68,200	0,114	0,041	0,064	0,208
	Forced	0,837	0,459	0,330	1,620	0,130	0,050	0,064	0,231

Variance Analysis

It is shown in Table 7. Significant differences exist in the soil frequencies and amplitudes, for both free and forced vibrations.

TABLE 7 ANOVA 

Diferences		Sum of squares	Degree of freedom	Quadratic mean	Fisher	Significance
Bent leg frequency (Hz)	Between groups	0,000	2	0,000	0,000	1,000
	Inside groups	8 023 566,900	42	191 037,307
	Total	8 023 566,900	44
Soil frequency (Hz)	Between groups	112 355,799	2	56 177,900	147,853	0,000
	Inside groups	15 958,263	42	379,959
	Total	128 314,062	44
Bent leg amplitude (mm)	Between groups	0,000	2	0,000	0,000	1,000
	Inside groups	16 938,412	42	403,296
	Total	16 938,412	44
Soil width (mm)	Between groups	0,531	2	0,266	49,770	0,000
	Inside groups	0,224	42	0,005
	Total	0,756	44

Frequency Analysis (free 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).

TABLE 8 Multiple comparisons (free vibrations) 

Dependent variable		(I) Depth (mm)	(J) Depth (mm)	Means differences (I-J)	Standard error	Signif.	95% confidence interval
							Upper limit	Lower limit
Bent leg frequency (Hz)	Scheffé	200,0	300,0	0,000	159,598	1,000	-405,011	405,011
			400,0	0,000	159,598	1,000	-405,011	405,011
		300,0	200,0	0,000	159,598	1,000	-405,011	405,011
			400,0	0,000	159,598	1,000	-405,011	405,011
		400,0	200,0	0,000	159,598	1,000	-405,011	405,011
			300,0	0,000	159,598	1,000	-405,011	405,011
Soil frequency (Hz)	Scheffé	200,0	300,0	-102,191*	7,117	0,000	-120,253	-84,128
			400,0	-109,433*	7,117	0,000	-127,495	-91,370
		300,0	200,0	102,191*	7,117	0,000	84,128	120,253
			400,0	-7,242	7,117	0,600	-25,304	10,820
		400,0	200,0	109,433*	7,117	0,000	91,370	127,495
			300,0	7,242	7,117	0,600	-10,820	25,304
Bent leg amplitude (mm)	Scheffé	200,0	300,0	0,000	7,332 990	1,000	-18,608	18,608
			400,0	0,000	7,332 990	1,000	-18,608	18,608
		300,0	200,0	0,000	7,332 990	1,000	-18,608	18,608
			400,0	0,000	7,332 990	1,000	-18,608	18,608
		400,0	200,0	0,000	7,332 990	1,000	-18,608	18,608
			300,0	0,000	7,332 990	1,000	-18,608	18,608
Soil amplitude (mm)	Scheffé	200,0	300,0	0,239*	0,026 678	0,000	0,171	0,307
			400,0	0,220*	0,026 678	0,000	0,152	0,287
		300,0	200,0	-0,239*	0,026 678	0,000	-0,307	-0,171
			400,0	-0,019	0,026 678	0,772	-0,086	0,048
		400,0	200,0	-0,220*	0,026 678	0,000	-0,287	-0,152
			300,0	0,019	0,026 678	0,772	-0,048	0,086

*. The differences of means are significant in lever 0,05.

FIGURE 5 Soil frequency summary 

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).

FIGURE 6 Soil width summary. 

Frequencies Analysis (forced vibrations)

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.

TABLE 9 Multiple comparisons (forced vibrations) 

Dependent variable		(I) Depth (mm)	(J) Depth (mm)	Mean differences (I-J)	Standard error	Signif.	95% confidence interval
							Upper limit	Lower limit
Bent leg amplitude (mm)	Scheffé	200,00	300,00	0,00000	1,37259	1,000	-3,4832	3,4832
			400,00	7,05120*	1,37259	0,000	3,5680	10,5344
		300,00	200,00	0,00000	1,37259	1,000	-3,4832	3,4832
			400,00	7,05120*	1,37259	0,000	3,5680	10,5344
		400,00	200,00	-7,05120*	1,37259	0,000	-10,5344	-3,5680
			300,00	-7,05120*	1,37259	0,000	-10,5344	-3,5680
Soil amplitude (mm)	Scheffé	200,00	300,00	-0,02393	0,01217	0,157	-0,0548	0,0069
			400,00	-0,05453*	0,01217	0,000	-0,0854	-0,0237
		300,00	200,00	0,02393	0,01217	0,157	-0,0069	0,0548
			400,00	-0,03060	0,01217	0,053	-0,0615	0,0003
		400,00	200,00	0,05453*	0,01217	0,000	0,0237	0,0854
			300,00	0,03060	0,01217	0,053	-0,0003	0,0615
Bent leg Frequency (Hz) Scheffé		200,00	300,00	0,00000	187,95374	1,000	-476,9688	476,9688
			400,00	0,00000	187,95374	1,000	-476,9688	476,9688
		300,00	200,00	0,00000	187,95374	1,000	-476,9688	476,9688
			400,00	0,00000	187,95374	1,000	-476,9688	476,9688
		400,00	200,00	0,00000	187,95374	1,000	-476,9688	476,9688
			300,00	0,00000	187,95374	1,000	-476,9688	476,9688
Soil Scheffé Frequency (Hz)		200,00	300,00	-1,13507	9,99518	0,994	-26,4998	24,2296
			400,00	-10,78207	9,99518	0,563	-36,1468	14,5826
		300,00	200,00	1,13507	9,99518	0,994	-24,2296	26,4998
			400,00	-9,64700	9,99518	0,631	-35,0117	15,7177
		400,00	200,00	10,78207	9,99518	0,563	-14,5826	36,1468
			300,00	9,64700	9,99518	0,631	-15,7177	35,0117

*. 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).

FIGURE 7 Bent leg amplitude summary 

FIGURE 8 Soil amplitude summary 

CONCLUSIONS

• 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.

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Received: September 15, 2020; Accepted: March 01, 2021

^*Author for correspondence: Luis Orlando Marín-Cabrera, e-mail: [e-mail]

Luis Orlando Marín-Cabrera, Especialista, Universidad Agraria de La Habana (UNAH), Facultad de. Ciencias Técnicas, Centro de Mecanización Agropecuaria (CEMA), San José de las Lajas, Mayabeque, Cuba, e-mail: [e-mail]

Armando Eloy García de la Figal-Costales, Profesor Titular, Universidad Agraria de La Habana (UNAH), Facultad de. Ciencias Técnicas, Centro de Mecanización Agropecuaria (CEMA), San José de las Lajas, Mayabeque, Cuba, e-mail: [e-mail]

Arturo Martínez-Rodríguez, Universidad Agraria de La Habana (UNAH), Facultad de. Ciencias Técnicas, Centro de Mecanización Agropecuaria (CEMA), San José de las Lajas, Mayabeque, Cuba, e-mail: [e-mail]

The authors of this work declare no conflict of interests.