TECHNICAL NOTE

Proposal for Redesign of a Winch for Pulling Loads

Dr.C. Alain Ariel de la Rosa-Andino^{I  *} 

Dr.C. Idalberto Macías-Socarrás^II 

Ing. Yoandrys Morales-Tamayo^III 

Lic. Danelys Pérez-Sutil^IV 

Dr.C. Ismael Rodríguez-Beltrán^I 

Ing. Jonathan Alexis Montaguano-Toaquiza^III 

^IUniversidad de Granma, Facultad de Ciencias Técnicas, Dpto. de Ingeniería Mecánica, Bayamo, Granma, Cuba.

^IIUniversidad Estatal Península de Santa Elena, La Libertad, Ecuador.

^IIIUniversidad Técnica de Cotopaxi. Extensión La Maná, Ecuador.

^IVUniversidad de Granma. Facultad Educación Media. Centro de Idiomas, Manzanillo, Granma, Cuba.

ABSTRACT

Winches are machines that are widely used in a wide variety of industrial and agricultural tasks, such as cement, metallurgical and mineral works, as well as in vehicles and tractors, specifically those with high capacity of passing or of traffic. Within their structure there are elements of force transmission and power (axles and sprocket wheels) that could fail due to superficial fatigue and pitting caused by the work. For these reasons, the objective of the present work was to determine the parameters to redesign the structural elements of a winch for load traction, since the chain transmission was affected by the wear phenomenon in the sprocket, keyway and setscrew. That caused strong vibrations that affected the fixings of the reducer in its chassis, as well as in the bearings. The replacement of the chain transmission by a speed reducer directly coupled to the motor is proposed, in addition to increasing the diameter of the drums that tighten the cable. For this, calculations were made for the new gear ratio. Among the results, the transmission ratio to the exit of the reducer stands out, as well as its efficiency 0.87.

Keywords: torque; gear reducer; reverse engineering; static analysis

INTRODUCTION

The winches were vertical axis winch type machines, widely used in mines to extract minerals and water, which initially had a drum at the top of the axis, and in its lower part the rod or rods to which the chivalry that moved it. Later they went on to use electrical energy to move a horizontal drum and to be on top of a tower. Today this name is used to refer to winches in many parts of Latin America (^{Carcamo, 1996}; ^{Menéndez, 2010} and ^{Rojas & Delgado, 2015}).

At present, a winch is a drum that contains a steel wire wound, supported by a base, which is fixed on a fixed surface, or on a vehicle. It is used to drag loads, or, in the case of vehicles and tractors, as an aid to cross terrain difficulties, or to move large weights in a controlled manner (^{Carcamo, 1996}; ^{McKee, 2011}; ^{Oltean et al., 2012}; ^{Paez and Gómez, 2017}).

The Manuel Fajardo Rivero factory provides casting and repair services for centrifugal pumps and mill masses for sugar mills. So it has a machining workshop divided into three main areas, where all parts that come out of the casting process are finished, but it is also used to maintain the equipment that is there.

The pieces that are subjected to the machining processes have large dimensions and weight and need to be moved through the different areas of the workshop, according to the technological sequence to be followed. This action is carried out by lifting, using hoisting equipment (bridge cranes) and towing through a trolley driven by a winch.

However, several of the structural elements that make up the winch have presented failures as a result of continued operation and the deficit of a maintenance plan for it, which is reduced to corrective maintenance, for presenting 40 years of exploitation. In addition to that, the transmission from the electric motor to the reducer is by chain, with a transmission ratio that is one to one, with high rotation frequencies (1,750 min^-1). Another factor that affects the occurrence of failures of some elements of the winch structure is corrosion. All this has caused superficial fatigue and pitting, as a source of failure in the elements of transmission of force and power (star wheel, chain, keyway and transmission shaft) of this machine tool. Because by working it without a casing, in a highly corrosive environment (dust, humidity, as well as the small particles that are lost as a result of the manufacture of parts), its deterioration is accelerated, threatening the longevity of the chain and the rest of the transmission elements. This has been referred to by authors as ^{Dobrovolki et al., 1968}; ^{Niemann, 1973} and ^{Reshetov, 1975}.

The aforementioned has also caused strong vibrations that affected the fixings of the reducer in its chassis and in the bearings. Another element that has been affected is the steel cable, which suffers from crushing causing fatigue. The fundamental causes of this failure are the diameter of the drum and the lack of grooves in it, for good winding.

All this has caused continuous interruptions during the production process of the machining workshop. For this reason, the objective of the present work was to determine the redesigning parameters of the winch structural elements for load traction. Appropriate calculations were carried out to eliminate the chain transmission and connect, directly a gear reducer to the electric motor that generates the torque. That provides as advantages the high performance, great duration and reliability of operation, constancy in the transmission ratio by absence of skating (^{Dobrovolki et al., 1968} and ^{Reshetov, 1975}). The task was carried out using reverse engineering, which is a method that allows knowing the basic geometric and constructive parameters, for its subsequent reconstruction and / or evaluation of the load capacity (^{González, 1999} and ^{González & Marrero, 2008}).

METHODS

The research was carried out at Manuel Fajardo Rivero Factory in Manzanillo Municipality, Granma Province, Cuba. It is located on Paquito Rosales Avenue km 1 and belongs to the company of Industrial Technical Services (ZETI), of the AZCUBA Business Group.

Calculation Methodology

To determine the redesign parameters of the structural elements of the winch gear transmission for load traction, the methodological recommendations were followed for certain calculations presented by ^{Sokolov & Usov, 1976}; ^{Faires, 1980}; ^{Deutschman Aaron & Michels Walter, 1985}, ^{García de la Figal, 1985} and ^{Mott, 2006b}).

Determination of the Design Parameters of the Transmission Structural Elements

Electric Motor

Motor power ${(N}_1)=\mathrm{1,000} Nm s^{-1}$ . Number of revolutions ${(n}_1)=183.16 s^{-1}$

Torsional torque (${Mt}_1$): it was determined using Equation 1.

${Mt}_1=\mathrm{ }\frac{\mathrm{N}1}{{\mathrm{n}}_1}$ (1)

where:

N1	- is the motor power (Nm s^-1)
n1	- is the rotation speed (s^-1)

Gear Data

The gear data of the gear unit to be used are shown in Table 1.

TABLE 1 Data of the gears. Note: The output shaft has a spur gear attached, which would be Z7, and Z8 is the gear that is connected to the drum 

Number of shafts	Number of teeth	Type of tooth
Shaft I	Z1 = 16	Helical
Shaft II	Z2 = 58 Z3 = 18	Helical
Shaft III	Z4 = 61 Z5 = 12	Helical
Shaft IV	Z6 = 43	Helical
Shaft V	Z7 = 16	Straight
Shaft VI	Z8 = 60	Straight

Where: Z - is the tooth number of each of the gears.

Determination of the Torsional Moment in Shafts II, III, IV and V

Considering the transmission of the frequency of rotation of the motor to the reducer is direct; that is, by metallic coupling (flexible coupling), then the transmission ratio (i₁) and the efficiency (ɳ₁) between the metallic coupling of the electric motor and the reducer are equal to one. Taking this into account, the following parameters are determined.

Expression 2 was used to determine the torque in shafts II, III, IV and V.

${Mt}_x=\frac{N_x}{n_x}$ (2)

where:

Nx	- is the power in the shaft to be calculated (Nm s^-1)
nx	- is the frequency of rotation in the shaft to calculate (s^-1)

However, to obtain this parameter it is necessary to know the power (N_x) and the rotation frequency (n_x) in each shaft II. Then the power and frequency of rotation in each shaft were calculated using Equations 3 and 4.

Note: Hereinafter to determine the values of (Mt₃), (Mt₄) and (Mt₅), it will be necessary to determine the power and frequency of rotation in the corresponding shafts.

In the case of shaft II, the power (N₂) was determined using Equation 3.

$N_2=N_{motor}·{\mathrm{ɳ}}_{eng}·{\mathrm{ɳ}}_{coj}^2$ (3)

where:

Nmotor	- is the power of the electric motor
ηeng	- is the efficiency of the gears
η^2coj	- is the efficiency of the bearings

Shaft II rotation frequency (n₂): was determined by Equation 4.

$n_2=\frac{n_{motor}}{U_1}$ (4)

where:

nmotor	- is the motor rotation frequency (s^-1)
U1	- is the gear ratio in Shaft II

To determine this rotation frequency, the transmission ratio U₁ in Shaft II is presented as unknown. So it was determined through Equation 5.

$U_1=\frac{Z_2}{Z_1}$ (5)

where:

Z1	- is the number of teeth of the gear of the Shaft II
Z2	- is the number of teeth of the gear of the Shaft III

Note: to determine the values of the moments in the rest of the shafts, the same procedure described above will be used.

Calculation of the Total Gear Ratio of the Reducer

The gear ratio of the reducer (i_r): was calculated using Equation 6.

$i_r=\frac{Z_2}{Z1}·\frac{\mathrm{Z}4}{Z3}·\frac{\mathrm{Z}6}{\mathrm{Z}5}·\frac{\mathrm{Z}8}{Z7}$ (6)

where:

Z	- is the number of teeth of each gear.

Calculation of the Total Efficiency of the Gear Unit

Reducer efficiency (n_r): was calculated using Equation 7.

$n_{\mathrm{r}}=n_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{s}}^{(⋕\mathrm{e}\mathrm{s}\mathrm{c})}·n_{coj}^{(⋕esc+1)}$ (7)

where:

ntrans	- is the transmission efficiency
ncoj	- is the efficiency in the rolling bearings.

Note: The maximum value of the transmission efficiency n_trans (0.90-0.98) is taken for being in an oil bath.

Strength Calculation on Flexible Metal Coupling Bolts

The shear stress (τ) was determined by Equation 8.

$\tau =\frac{2·{Mt}_1}{D_{bc}·C(\pi ·\frac{d^2}{4})}$ (8)

where:

Mt1	- is the torque at the motor output (Nm)
d	- is the diameter of the bolts (mm)
C	- is the number of bolts in the coupling
Dbc	- is the radius of the bolt circle (mm)
π	- is the universal constant

The design shear stress $\left({\tau }_D\right)$ , was calculated using Equation 9.

${\tau }_D=\frac{0.577·\mathrm{S}\mathrm{y}}{FDS}$ (9)

where:

Sy	- is the material yield limit (MPa)
FDS	- is the safety factor

Drum Calculation

To perform the kinematic calculations related to the drum, such as translation speed (Vc), and rotation speed (Vr), it is necessary to know the drum's outer diameter (Dt = 600 mm) and its inner diameter (Di = 550 mm). So:

The carriage travel speed (Vc) was calculated using Equation 10.

$Vc=\frac{\mathrm{S}}{\mathrm{t}}$ (10)

where:

S	- is the displacement
t	- is the translation time of the car

The drum rotation speed (ɳ_t) was determined by clearing it from Equation 11.

$\mathrm{V}\mathrm{c}=\frac{\mathrm{\pi }\mathrm{ }·\mathrm{ }{\mathrm{ɳ}}_{\mathrm{t}}\mathrm{ }·\mathrm{D}\mathrm{t}}{60}$ (11)

where:

Dt	- is the external diameter of the drum
Vc	- the speed of translation of the car
π	- is the universal constant

So: solving for the rotation speed ɳ_t, we get Equation 12.

${\mathrm{ɳ}}_{\mathrm{t}}=\frac{\mathrm{V}\mathrm{t}·60}{\mathrm{\pi }·\mathrm{D}\mathrm{t}}$ (12)

During winch work, the steel cable is to be wound on the drum (Figure 1). So in order to wind up the cable on the drum satisfactorily, without being affected by crushing, the drum has to be grooved to accommodate and guide the cable, therefore:

Source: (^{SolidWork, 2014})

FIGURE 1 Drum to guide and accommodate the cable. 

The steel cable selected to pull the load on the proposed winch is of the hemp core type and its properties are shown in Table 2.

TABLE 2 Properties of the steel cable 

Cable Properties
Type of cable	Hemp core
Cable diameter	12 mm
Length	First section	60 m
	Second section	120 m

Since the cable is 12 mm, the depth of the groove will be equal to half the diameter of the cable, or 6 mm, and the pitch between grooves will be 13 mm.

Then, the perimeter of the drum (P) or the length of the cable to be wound or wrapped in one turn of the drum was determined by Expression 13.

$P=\pi ·Dm$ (13)

where:

Dm	- is the average diameter of the drum
π	- universal constant

The number of turns or grooves of the drum (Ke): was determined by Equation 14.

$Ke=\frac{\mathrm{L}}{\mathrm{P}}$ (14)

where:

L	- is the length of the cable to be wound
P	- is the perimeter of the drum

The length of the threaded part (l) was determined by Expression 15.

$l=Ke·Paso$ (15)

where:

Ke	- is the number of turns

As the drum must comply with the fact that, while winding one cable branch, the other branch must be unwrapped, the threaded part will be double to the right and the other to the left.

So, the total length of the drum (l_t) was determined by Expression 16.

$l_t=2l·3\left(\mathrm{T}\mathrm{h}\mathrm{i}\mathrm{c}\mathrm{k}\mathrm{n}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{ }\mathrm{o}\mathrm{f}\mathrm{ }\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{ }\mathrm{g}\mathrm{u}\mathrm{a}\mathrm{l}\mathrm{d}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{s}\mathrm{ }\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{ }\mathrm{s}\mathrm{e}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{r}\right)$ (16)

where:

l	- is the length of the threaded part

Bearing Selection

For the selection of bearings, the procedure referred by ^{Mott (2006a)} was used.

The duration of the design (Ld) was calculated by Equation 17.

$Ld=\left(h\right)\left(n_5\right)\left(60\min h^{-1}\right)$ (17)

where:

h	- design duration (h)
n5	- speed of rotation in the shaft V (min^-1)

The basic dynamic load capacity (C) was determined using Equation 18.

$C=Pd{\left(\frac{Ld}{{10}^6}\right)}^{\frac{1}{k}}$ (18)

where:

Pd	- is the given design load in pounds
Ld	- design duration in hours
k	- is equal to 3 for being the ball bearings

Once the above parameters have been determined, the bearing with the most appropriate dimensions is selected, identified already a set of bearings with the referred basic dynamic load capacity.

Selecting the ball bearing to be used, the bearing with the data of the ball bearing is chosen. For this, the catalog proposed by ^{Mott (2006a)} will be used

Static Analysis of Drum and Chassis

The static analysis of the drum and the chassis has been carried out with computer-aided engineering techniques, by means of the finite element analysis software ^{SolidWork (2014)}, performing the following operations: pre-processing, allocation of materials, establishment of applied forces, meshing and obtaining results regarding deformations, displacements (mm) and von Mises stress (MPa).

RESULTS AND DISCUSSION

Determined the parameters for the design of the structural elements of the winch transmission for load traction, the following results were obtained. Table 3 shows the values ​​of the torques, powers, rotation frequency as well as the transmission ratio in the five reduction gear shafts.

TABLE 3 Results of calculations related to the gear reducer 

Torque on drive shafts $\left(Nm\right)$	Powers $\left(kW\right)$	Rotation frequency (s^-1)	Transmission ratio between conjugated gear pairs (Dimensionless)
${\text{Mt}}_1$ =5, 459	$N_1$ =10	${\text{n}}_1=$ 183,16	-
${\text{Mt}}_2$ =19,257	$N_2$ =0,97	${\text{n}}_2$ =50,879	$U_1$ =3,6
${\text{Mt}}_3$ =57,082	$N_3$ =0,96	${\text{n}}_3$ =16,818	$U_2$ =3,3
${\text{Mt}}_4$ =195,74	$N_4$ =0,94	${\text{n}}_4$ =4,805	$U_3$ =3,5
${\text{Mt}}_5$ =669,10	$N_5$ =0,92	${\text{n}}_5$ =1,377	$U_4$ =3,7

Total Gear Ratio of the Reducer

The total transmission ratio of the reducer (i_r), was determined through mathematical expression 6. Obtaining a result of 153.846.

Total Reducer Efficiency

The efficiency of the reducer (n_r) is equal to 0.87 and was determined using Equation 7.

Bolt Strength of Flexible Metal Coupling

The values of real shear stress (τ) and design (τ_D) were calculated using Equations 8 and 9. The magnitudes obtained are equal to 5.1·10-5 MPa and 67.7 Mpa, respectively. With this result the condition is fulfilled that the real shear stress is less than the design shear stress τ ≤ τ_D, therefore, the bolts in the flexible coupling resist.

Drum

Table 4 shows the parameter values for the drum design.

TABLE 4 Drum Design Parameters 

Parameters	Results
Car travel speed (Vc)	0,333 m s^-1
Drum rotation speed ($n_t$)	10,60 min^-1
Drum perimeter (P)	1,885 m
Number of drum grooves (Ke)	32 turns
Threaded part length (l)	0,4 m
Total length of the drum (lt)	1 m

Selection of Ball Bearing

For the selection of the ball bearing to be used, it was necessary to determine the design duration (Ld) with a value of 15,792,000 min^-1 and the basic dynamic load capacity (C) which is 19,845.87 lb.

Static Analysis of Drum and Chassis

Material Allocation

When presenting the model as a rigid solid, a material was assigned to the part, so that it would give its results as close to reality as possible, such as ASTM A36 Steel, and its physical properties were as follows (Table 5).

TABLE 5 Properties of the material. Source: ^{SolidWork (2014)}  

	Name:	ASTM A36 Steel
	Model type:	Linear elastic isotropic
	Default error criteria:	von Mises maximum tension
	Elastic limit:	250 N mm^-2
	Traction limit:	40 N mm^-2
	Elastic module:	200 000 N mm^-2
	Poisson coefficient:	0,26
	Density:	7 850 kg m^-3
	Shear modulus:	79 300 N mm^-2

Applied Loads

Fixed holdings were applied to the drum in each section where the cable is to be wound, simulating that it is wound on it, in addition, bearing-type supports were applied to each end of the shaft where the drum is supported (Figure 1).

FIGURE 1 Fixings on the drum. 

In the case of the chassis, a remote mass load was applied to simulate the conditions of the torque. It was where the coupling between the motor and the reducer is located, where the torque has a value of 5.4597 N·m and the loads of the weight of the system were also taken into account. These are the most critical conditions of it. This has fixed restrictions, which are located where the chassis will be embedded (Figure 2).

FIGURE 2 Loads applied to the chassis. 

Von Mises Voltages for Drum and Chassis Stress State

This is the most suitable failure theory for ductile and uniform materials (tensile strength approximately equal to compressive strength), and whose shear strength is less than tensile strength. This theory basically consists of determining the so-called effective von Mises stress, after having determined the stress state of the worst hit point (^{Norton, 2011}).

As shown in Figure 3, the maximum stress values (9.46 MPa) will be accumulated in the entire section of the shaft where the gear and bearing is coupled, and the minimum stress values in the sections where the cable is wound. So in that place is where the breakage of the piece can occur. The maximum values of stresses do not exceed the elastic limit of the material, therefore, the deformations will not be permanent. This can be corroborated when analyzing the safety factor; it has a value of 26, which can be considered acceptable.

FIGURE 3 Von Mises stresses in the drum. 

Applying the finite element method to the structure by determining the von Mises stresses, loads were applied to the sections where the motor, reducer and drum are located. It was obtained that the maximum stresses to which the structure will be subjected are equal to 75.6 MPa, located in one of the embedment of the structure, which is not a significant value compared to the elastic limit of the material (250,000 MPa), therefore, the structure will withstand the loads to which it is subjected (Figure 4).

FIGURE 4 Von Mises stresses on the chassis 

Displacement

In Figure 5 it can be seen that the maximum displacements are in the area where the gear is coupled and then, the bearing with a maximum displacement of 0.008513 mm. In that area is where the greatest amount of stresses are found, for which the displacements will be maximum due to the torsional torque generated there, which is 669,104 Nm. The point where the minimum stress values are located is in the sections of the drum where the cable will be wound.

FIGURE 5 Displacements in the drum 

Security Factor

As shown in Figure 6 the safety factor is 26 for the drum and 3.6 for the chassis. This value obtained by the software takes into account the ratio of the value of the limit voltage (in this case the value of the yield stress of the material) and the von Mises stresses. As this value is greater than 1, it can be expressed that the maximum internal stresses in the parts resulting from the acting loads do not exceed the elastic limit, so the deformations will not be permanent.

FIGURE 6 Safety factor in the drum (a) and in the chassis (b). 

CONCLUSIONS

• The parameters for the redesign of the winch structural elements for load traction were determined.

• With the new gear transmission, better efficiency (0.87) was achieved in the winch transmission ratio.

• The results of the analysis using the finite element method showed that the maximum stress values did not exceed the elastic limit of the material (250 N mm^-2), so there will be no permanent deformations in the assembly, with a safety factor of 3.6 for the chassis and 26 for the drum.

• They managed to eliminate vibrations and also the environmental noise caused by it.

REFERENCES

CARCAMO, J. ¿Qué esconden los enigmáticos malacates? [en línea] Disponible en: http://wikipediacmz.udg.co.cu/wikipedia_es_all_2015-09/A/Cabrestante.html [Consulta: 20 de diciembre 2017 ].

DEUTSCHMAN, A. D.; W. J. MICHELS & C. E. WILSON: Diseño de máquinas. Teoría y práctica, Ed. Continetal S. A., Mexico D.F., 1987.

DOBROVOLKI, V.; K. ZABLONSKI; A. RADCHIK y L. ERLIJ: Elementos de máquinas, Ed. Editorial Mir, Moscu, 1968.

FAIRES, V. M.: Diseño de elemento de máquinas, Ed. Montaner y Simón S. A., Barcelona. España, 1980.

GARCÍA DE LA FIGAL, C. J.: Equipos de elevación: grúas, elecadores y montacargas, Ed. Dpto. Ediciones ISPAJE, CUJAE, Marianao, La Habana, 1985.

GONZÁLEZ, R. G.: "Procedimiento para la obtención de los parámetros geométricos básicos de un engranaje cónico de dientes rectos", Revista Ingeniería Mecánica, 2(1): 23-31, 1999.

GONZÁLEZ, R. G. y O. S. A. MARRERO: "Reingeniería de la geometría desconocida de engranajes cónicos con dientes rectos y curvilíneos", Revista Ingeniería Mecánica, 11(3): 13-20, 2008.

MCKEE, J.: Desingn, constrution, and evaluation of a hydraulic powered pto winch, 28pp., BioResource and Agricultural Engineering, BioResource and Agricultural Engineering Department, California Polytechnic State University, California, 2011.

MENÉNDEZ, S. C.: "Los castilletes mineros: Una aproximación a su tipología", Energía y Minas. Revista profesional, técnica y cultural de los ingenieros técnicos de minas, 8: 46-53, 2010.

MOTT, L. R.: Diseño de Elemento de Máquinas, Ed. Pearson Educación, Mexico D.F., 2006.

NIEMANN, G.: Tratado teórico práctico de elementos de máquinas. Cálculo, diseño y construcción, Ed. Labor S. A., Volumen II, Barcelona. España, 1973.

NORTON, R. L.: Diseño de Máquinas, Ed. Pearson Educación, Mexico D.F., 2011.

OLTEAN, I. A.; M. S. POPA y S. VASILE: "Mechanical winch chacteristic calculation", Transfer of Innovations - An Internet magazine about industry innovations, 22: 31-33, 2012.

PAEZ, A. L. y S. D. A. GÓMEZ : "Design and development of an electric winch for the extraction of vagonetes in a carbon underground mine", Revista de la Universidad Distrital Francisco José de Caldas, 14(1): 13-26, 2017.

RESHETOV, D.: Elementos de Máquinas, Ed. Pueblo y Educación, Ciudad de La Habana, 1975.

ROJAS, S. J. I. y J. M. DELGADO: "3D Parametric design and static analysis of the first Spanish winch used to drain water from mines", Revista DYNA, 82(190): 237-245, 2015.

SOKOLOV, F. y P. USOV: Mecánica Industrial, Moscú, 1976.

SOLIDWORK, 22.0.0.5018. EE.UU, 2014.

⁸The mention of trademarks of specific equipment, instruments or materials is for identification purposes, there being no promotional commitment in relation to them, neither by the authors nor by the publisher.

Received: October 10, 2019; Accepted: June 14, 2020

^*Author for correspondence: Alain Ariel de la Rosa-Andino, e-mail: [e-mail]

Alain Ariel de la Rosa-Andino, Prof. Auxilia, Universidad de Granma. Facultad de Ciencias Técnicas, Dpto. de Ingeniería Mecánica, Carretera a Manzanillo, km 17 ½, Peralejo, Apartado 21, Bayamo, M. N. Código Postal 85149. Provincia Granma, Cuba, e-mail: [e-mail]

Idalberto Macías-Socarrás, Profesor, Universidad Estatal Península de Santa Elena, Avenida Principal La Libertad-Santa Elena, La Libertad, Ecuador, e-mail: [e-mail]

Yoandrys Morales-Tamayo, Prof. Universidad Técnica de Cotopaxi, Extensión La Maná, Ecuador, e-mail: [e-mail]

Danelys Pérez-Sutil, Prof. Instructora, Universidad de Granma, Facultad Educación Media, Centro de Idiomas. Manzanillo, Código Postal 87510, Provincia Granma, Cuba, e-mail: [e-mail]

Ismael Rodríguez-Beltrán, Prof. Titular, Universidad de Granma, Facultad de Ciencias Técnicas, Dpto. de Ingeniería Mecánica, Carretera a Manzanillo, km 17 ½, Peralejo, Apartado 21, Bayamo, M. N. Código Postal 85149, Provincia Granma, Cuba, e-mail: [e-mail]

Jonathan Alexis Montaguano-Toaquiza, Ing., Universidad Técnica de Cotopaxi, Extensión La Maná, Ecuador, e-mail: [e-mail]

The authors of this work declare no conflict of interests.