Scientific Paper / Artículo Científico 



https://doi.org/10.17163/ings.n29.2023.03 


pISSN: 1390650X / eISSN: 1390860X 

CALCULATION OF ABRASIVE WEAR SPEED IN STRAIGHT AND
HELICAL GEARS WITH EVOLVING PROFILE, USING A MATLAB GUI 

CÁLCULO DE LA VELOCIDAD DE DESGASTE ABRASIVO EN ENGRANAJES DE DIENTES RECTOS Y HELICOIDALES CON PERFIL EVOLVENTE, UTILIZANDO UNA GUI DE MATLAB 
José Miguel Mena Chavarrea ^{1,*} , José Antonio Granizo^{2} , Eduardo Segundo Hernández Dávila ^{2 }, Mario Efraín Audelo Guevara ^{3}. 
Received: 11022022, Received after review:
20102022, Accepted: 09112022, Published: 01012023 
Abstract 
Resumen 
The
present investigation originated under the objective of designing a software
with Matlab that allows calculating the abrasive
wear rate of spur and helical gears with involute profile. Where, the
unstudied generalities about the design of gears have generated negative
consequences in functionality and have generated catastrophic failures that
have produced unexpected stoppages of mechanical processes, for which
understanding their causes becomes essential. Considering,
for the calculation of the abrasive wear of the tribological
pairs in heavy machinery, it will depend on the degree of grinding or size of
the abrasive particles, the hardness, the material of the base gear, the
lubrication regime and the geometric conditions that determine the nature of tribological contact, as well as the time and speed of
use of heavy machinery. Initially, the theoretical environment that
makes up the study based on the main components such as:
gears, lubrication, steel, heat treatments and finally Matlab
is established. Next, the mathematical models of Archard and Kraglesky were
established, which were adapted to the circumstances of the problem and based
on this information, a software capable of calculating the rate of abrasive
wear was formulated using a graphical Matlab user
tool; We proceed to execute the manual calculation with the determined
variables taking experimental data obtained from previous investigations and
that comply with the variables determined for the execution of the
mathematical process, given that the results generated through the software
can be tabulated, graphed in Excel and verified. with
manual mathematical development. Looking for the comparison, differentiation
and trend of the behavior of the wear rate, varying the parameters involved
in the process using the two mathematical models proposed. 
La presente investigación se originó con el objetivo de diseñar un software con Matlab que permita calcular la velocidad de desgaste abrasivo de engranajes rectos y helicoidales de perfil evolvente. Donde, las generalidades no estudiadas sobre el diseño de engranajes han generado consecuencias negativas en la funcionalidad y fallas catastróficas que han producido paras improvistas de procesos mecánicos, por lo cual entender las causas de los mismos se vuelve fundamental. El cálculo del desgaste abrasivo de los pares tribológicos en maquinaria pesada dependerá del grado de trituración o tamaño de las partículas abrasivas, la dureza, el material del engrane base, el régimen de lubricación y las condiciones geométricas que determinen la naturaleza del contacto tribológico, así como el tiempo y velocidad de uso de la maquinaria pesada. Inicialmente, se establece el entorno teórico que compone el estudio basado en los principales componentes como engranajes, lubricación, el acero, los tratamientos térmicos y finalmente Matlab. Seguidamente, se establecieron los modelos matemáticos de Archard y Kraglesky que fueron adaptadas a las circunstancias del problema y en función de esta información formular un software capaz de calcular la velocidad de desgaste abrasivo mediante una herramienta gráfica de usuario de Matlab. Se procede a ejecutar el cálculo manual con las variables determinadas, tomando datos experimentales obtenidos de investigaciones previas y que cumplan con las variables determinadas para la ejecución del proceso matemático, dado que los resultados generados a través del software puedan ser tabulados, graficados en Excel y comprobados con el desarrollo matemático manual. Buscando la comparación, diferenciación y tendencia del comportamiento de la velocidad de desgaste, variando los parámetros involucrados en el proceso utilizando los dos modelos matemáticos planteados. 
Keywords: Gears, Abrasive, Wear, GUI, Matlab, Wear rate 
Palabras clave: engranajes,desgaste,abrasión, GUI, Matlab, velocidad de desgaste 
^{1,*}Tecnología Superior en Electromecánica,
Instituto Superior Tecnológico Cotopaxi, Ecuador. Corresponding
autor ✉: jmmenac@istx.edu.ec ^{2}Grupo de investigación Ciencia del Mantenimiento,
Escuela Superior Politécnica de Chimborazo, Ecuador. ^{3}Ingeniería Automotriz, Facultad de Mecánica, Escuela
Superior Politécnica de Chimborazo, Ecuador. Suggested
citation: Mena Chavarrea, J. M.; Granizo, J.A.; Hernández Dávila,
E.S. and Audelo Guevara, M.E.“Calculation of abrasive wear speed in straight and
helical gears with evolving profile,using
a Matlab GUI,” Ingenius,
Revista de Ciencia y Tecnología, N°. 29, pp. 3245, 2023, DOI: https://doi.org/10.17163/ings.n29.2023.03. 
1. Introduction At
present, gears have a great importance as transmission elements in mechanical
systems, since they are mechanical pieces found in any industry as main
machine elements that generate force and movement. The degree of wear of the
gears depends on the work carried out by the equipment, the grain size and
composition of the abrasive, the working power, the design load, and the
efficiency and reliability of the process. These are major
factors that should be considered to determine the costs of executing
corrective and preventive maintenance. Detecting failures
in a timely and efficient manner constitutes one of the most important
challenges associated to predictive maintenance. Unexpected failures may
affect the integrity and reliability of equipment through unscheduled stops,
reduction of useful life, high costs of corrective maintenance and low
quality of products [1]. Studying the wear
process of gears is increasingly important, because they are the mechanical
transmission elements mostly used in different fields and under different
working conditions, from tiny gears used in watches or as part of any machine
independent of its size. In industry, wear is
one of the most frequent problems that appear in systems that contain gears,
regardless of the work they perform. The abrasives are elements that are found in the environment and are typical of heavy
machinery elements, such as walking equipment, agricultural machinery,
construction machinery, mining industry machinery, etc. The abrasive
particles are a determining factor to significantly reduce
the useful life of the gear. The most important
aspects to be considered in the design of the gear teeth profile,
are related to the load capacity, transmission error, pressure angle, wear
and failure analysis [2]. The
lack of information and review of the analysis criteria for designers and for
people responsible for machinery maintenance, whose main object of design are
spur and helical gears with involute profile which are continuously subject
to abrasive elements, has generated that mechanisms get damaged with the
consequent unnecessary stops, and even the no implementation of laboratory
research processes that are extremely delayed and costly. The abrasive found in the
environment directly affects the performance of mechanical systems, since it
generates catastrophic failures that produce losses to the industrial
processes of companies. The parameters that have influence are grain size,
abrasive hardness, quantity of abrasive and work carried out, which reduce
the performance of industrial processes and the production. Hence, it is
necessary to study them to try to address the problems that may appear in
gears. Thus, the wear, 
the abrasive and the failures are
related aspects that will be the case study here, to determine the wear rate
depending on the abrasive and on the gear material. The substitution of 90%
of the gearwheels is due to an efficiency loss as a consequence of the wear
of the teeth; in general, the following mechanisms are used to tackle this phenomenon:
·
Optimization of geometrical parameters according to the design. ·
Improvement of the quality of the teeth surface and of the assembly of
the parts ·
Appropriate selection of the material and of the parameters of the
thermal treatment. ·
Appropriate selection of the lubricant and optimization of the
lubrication process [3]. Hence, it follows
that there will be a greater chance that a hydrodynamic film is formed in the
zone of the gear pole, and if lubricant is present in the tribological
process, this zone will have less wear in the
toothed transmissions. The present work
proposes the usage of a graphical user interface to carry out the theoretical
calculation of the wear rate in gears with involute profiles, to provide the
designer with tools that enable to predict the behavior of the tribological system, evaluate the validity of its design
theories and generate timely maintenance plans. Taking into account
the approximations made, it is considered the
process based on a scheme that uses vibrations to update a wear prediction
model. First, it is developed a dynamic model of a
system of spur gears to generate realistic vibrations, which enables a
quantitative study of the effects of wear on the surface of the gear teeth.
The sliding speed and the contact forces of the model are
used in combination with the known Archard
wear model, to calculate the wear depth at each contact point in the mesh.
Since the wear coefficient in the model is not constant during the wear
process (and in any case it is difficult to estimate initially), the
vibrations measured are compared with the vibrations generated by the model,
to update the coefficient when it is detected a deviation in the predictions
[4]. 1.1. Tribology The term tribology comes from the
Greek terms «tribos» and «logos», which mean
friction and study, respectively. Hence, this term is used
to designate the science that studies the surfaces with relative movements
between each other. For this reason,
terminologies such as friction, wear of the different surfaces and the
presence or absence of lubricant between the parts in contact, are essential
to obtain machines and processes with less energy loss, 
avoid long stop
times that limit the efficiency of the production activity, improve the life
cycle of machines and, above all, have available a reliable tool to generate
good repair and maintenance practices [5]. 1.2. Mechanical contact Figure 1
shows the apparent area corresponding to the entire surface of the parts in contact,
and the real area, which considers that all surfaces have rough points that
cause that the contact occurs only at the points of coincidence of the
corresponding crests of each of the surfaces involved in the movement [6]. Figure 1.
Apparent contact area [6] 1.3. Wear Is the process of
destruction and detachment of material that occurs between the surfaces of
the bodies, revealed as accumulation of deformations and variation of the
initial dimensions of the corresponding object [7]. 1.3.1.
Abrasive wear This wear mechanism is characterized by the presence of hard particles that
interact with surfaces that slip against each other. As it is determined in
Figure 2, under this system it is important to characterize that this type of
wear causes imperfections and microbreaks, due to
the action of extremely hard and small particles, compared to the base
surface [8]. Figure 2. Wear due to abrasion [8] Figure
3 indicates that, according to its nature, abrasive wear may
be classified in two types, namely, of two or three bodies. The
abrasive wear of two bodies is generally used as a machining mechanism, to
obtain specific results in a particular surface, whereas the 
Abrasive wear of three bodies is
due to the contamination of the interface between two surfaces.
Figure 3.
Abrasive wear a) 2 bodies y b) 3 bodies [9] 1.4. Failure analysis in gears The American Society for Metals has
created four subgroups to classify failure modes, namely, wear, surface
fatigue, deformations and crack. Table 1 shows the percentages of the common
failures, which take into account the parameters that are
established due to the abrasive wear of gears, analyzing each failure
mode and consequence. 1.5. Matlab
Matlab is a very powerful and adaptable
tool for mathematical calculation, with graphical capabilities that improve
the data presentation experience. As a consequence
of these features, it has become popular as an option for making calculations
in science and research [10]. Table 1.
Causes of failures in gears [11]

One
of the most important Matlab features in the
interactive user interface, which enables a fast numerical calculation and an
efficient data processing. In addition, it has various functionalities, such
as the presentation of graphical tools that enable that user’s experience is
simpler, pleasant and efficient enough to fulfill all needs, without
requiring many software [12]. 1.5.1.
GUI Tool A Graphical User
Interface (GUI) is a software package within Matlab, that uses a set of
preprogrammed images and action boxes to synthetize the need of the user for
managing data and tasks. Figure
4 determines the main functionality of this tool, which is to facilitate the
communication between the user and the software, so that it is not necessary
to consider programming processes and the language required to modify and
create the graphical interface [13].
Figure 4. Interface “GuideMatlab” 2.
Materials and methods The aim
is to obtain the necessary theoreticalconceptual
tools, evaluate the formulas presented in the selected bibliography and adapt
them to the circumstances of the problem, and appropriately arrange,
interpret and use the data obtained.
Figure 5.
Flow diagram of the methodology [14] 
For
this purpose, dependent and independent variables are defined and, based on
this information, it is formulated a software capable of calculating the
abrasive wear rate of gearwheels through a Matlab
graphical user tool. The flow diagram of the methodology is
illustrated in Figure 5. 2.1. Techniques for collecting information The techniques used
during this research work will focus on two main aspects, as described below. 2.1.1.
DocumentaryBibliographic It has been
collected books, certified journals, scientific papers and user manuals about
the abrasive wear of gearwheels and the use of Matlab.
This information will be the base to discretize and adapt the calculation
equations and the mathematical models to the problem stated, and will guide
the process of constructing the Matlab graphical
tool. 2.1.2.
Theoretical
and experimental Once the software has been developed, the validity of equations will be
evaluated based on the results obtained, the similarity between the
mathematical models and the trends of the wear rate. A value of error will be obtained when the parameters of the equations are
varied, and it will be intended to adjust the constants of the equations to
get an appropriate fit to real values, that enable to obtain the initial
approximations of the design. The data obtained in the real approximations
carried out in the operation tests is taken into
account [4]. 2.2. Fundamentals
of gears Since gears are key
for this work, it should be remarked the importance of the fundamental tools
and knowledge to determine the geometry, type, materials and manufacturing
processes of the most common gears, to correctly do the calculation and
dimensioning process. 2.2.1.
Terminology To introduce the
analysis and study of gears, it is necessary to define the terminology shown
in Figure 6. The INEN 1143 standard about gears states that: ·
Teeth of a gear. Elements that carry out the thrust work,
transmit power and have a characteristic profile according to their
arrangement. ·
Outer circumference. Part of the circumference of the gear shape that limits it on its
outmost part. ·
Inner circumference. Part that limits the base of the teeth, also known as root. ·
Primitive circumference. Circumference formed due to the
rotation of the contact points of the gear teeth involved in the process. 
·
Addendum. Perpendicular
distance between the pitch circle or primitive circumference and the highest
point of the teeth. ·
Helix angle. Angle formed by the base of the cylinder and the teeth of a helical or
srew gear with involute profile. ·
Gear or crown. It refers to the largest gear in an arrangement of gears. ·
Pinion. Smallest
gear, typically in charge of transmitting movement. ·
Eccentricity. Is the offset between the common centers of two circumferences. ·
Face width. Length
of the tooth at the plane located at 90 degrees of the gear formation plane. ·
Gear ratio. Ratio
between the greatest and the smallest number of teeth in meshed gears. ·
Module. Ratio
between the pitch circle diameter (in millimeters) and the number of teeth. ·
Pitch. Distance
between a point of a tooth and the same point in the adjacent tooth. It is an
indication of the tooth size. ·
Reference line. Imaginary flat surface tangent to the pitch surfaces of two gears; it is basically the plane that limits the contact points
between the gears. ·
Preassure angle. Angle between the pressure line of
the tooth and the flat tangent to the pitch surface. It is
basically the direction normal to a gear tooth [15]. Figure 6. Elements that constitute a gear [16] 2.3.
Spur gear Consider
that the total force in a gear is given by Equation
(1).

Where:
F is the total contact force in gears, P is the power of the
machine given in KW, Do is the outer diameter and n is the
angular speed. The outer diameter is given by
Equation (2).
Where
z is the number of teeth and m is the gear module. For
the spur gear, the normal force applied in the process is
given by Equation (3).
With
θ equal to the
pressure angle of the gear [17]. 2.4. Helical gear The normal force
applied for helical gears is given by Equation (4).
Where:
ψ is the
helix angle and ϕt is the
angle of transverse pressure. To
incorporate time as a variable in the base Archard formula
of Equation (5), it is necessary to relate the distance traveled in the
abrasion process with the linear displacement of the gear surface when the
machine is in operation.
Where
n is the angular speed, t is the time variable, and Dp the primitive diameter of the gear, given by
Equation (6):
Where
Do is the outer diameter, and z is the number of teeth in the gear. 
2.5. Calculation of abrasive wear According to Equation (7), the volume loss (W)
in a piece is directly proportional to the probability (z) that an abrasive
particle removes material when it finds a crest of the surface in its
trajectory, and to the normal force (N) that acts between the sliding
surfaces and the abrasive particles, and is inversely proportional to their
hardness measured in the Brinell scale (HB)
[18], i.e. Equation (8) shows
the abrasive particle that has semispherical shapes with a radius given by
the radius of the contact point between the surfaces.
Where k is
the probability of finding an abrasive particle of the contact point between
the surfaces, and varies between 10^{−2} and 10^{−7}
[19]. The methods of Equation (9) to calculate the wear due to abrasion in
various machines elements were tested experimentally and widely by the
scientific community, recognizing the nature of wear due to fatigue,
and finally, Kraglesky equation will be taken as
reference.
Where (V )is
the wear rate measured in [um/h], (A) is the parameter that
characterizes the abrasive material, (K) is the characterization of
the geometrical conditions of the contact point of the sliding surfaces and (M)
depends on the properties of the material of the surface [3]. The base equation to
measure the wear rate in spur and helical gears is obtained
from these considerations. ·
Archard Equation: Wear rate in a spur gear [3] (10).
·
Archard Equation: Wear rate in a helical gear [3]
(11).
· Kraglesky abrasive wear rate (12)

The variables
necessary to characterize Kraglesky equation
(Equation (12)) are divided in three parameters, the term corresponding to
the abrasive particle, where the mechanical properties, size and composition
of the abrasive material modify its characteristics according to Equation
(13) [20]. ·
Properties of Kragelsky abrasive particle.
In this same
context, factor (M) of Equation (14) is related to
the mechanical properties of the base material, and is directly proportional
to the hardness and the stretching percentage of the material under analysis,
taking the abrasive data from the values in Table 2 [20]. Table 2.
Size of the types of abrasive particles ·
Kragelsky parameter of mechanical properties of the
material (14).
Where
ϵ t 0
corresponds to the stretching percentage of the material before the crack, t
is a nondimensional parameter associated to the
contact between bodies, and HB represents the hardness of the
materials that constitute the tribological pair of
the process, measured in Brinell scale. Finally,
factor (K) involves all the geometrical conditions that impact
the variation of the contact between the surfaces, such as the type and size
of pieces, lubrication conditions and distribution of the contact forces
between the elements of the tribological pair. For
this reason, Equations (15) and (16) are established
for spur and helical gears, respectively [20]. · Spur gear
· Helical gear

·
Kraglesky Equation: Wear rate spur gear ·
Kraglesky Equation: Wear rate helical gear
2.6. Designation of variables for calculating
the abrasive wear rate Archard equation was
used as a first approximation for calculating the abrasive wear rate
in gearwheels. On the other hand, the mathematical model formulated by Kragelsky, given by Equation 3, was
used to determine the result closest to the real value. Afterwards,
both methods were programmed in a beta version of
the software designed with the Matlab graphical
tool. Then, results were generated to test the
calculation application, aspect issues were improved and the software mas
made as friendly as possible for the user. At last, the final
version of the application was developed and compiled to make it independent
of Matlab. Then, the variables and the parameters
of the equations are changed, and the data obtained is used
to make plots in Excel. The final step involved preparing a user manual and
presenting the results. 2.7. Declaration of variables Based on the data required by the equations
and to clearly state the names of the variables used
in the programming process, this section presents. ϑ (theta) = pressure angle ψ (psi) = helix angle z1 = number of teeth of gear 1 z2 = number of teeth of gear 2 m = module of the gears P = power of the machine 
H1 = hardness of material 1 (Brinell)
H2 = hardness of material 2 (Brinell)
Karch = process constant, Archard equation
n = angular speed rg = average grain size of the
abrasive particle cv = concentration in volume of the abrasive
particle Eo1 = stretching percentage of
the material of gear Eo2 = stretching percentage of the material of
gear 3.
Results and discussion 3.1. Results 3.1.1.
App version When entering the
app (see Figure 7), it should be selected the gear type and the known
parameters. The interface enables a clear view of the requirements, such as
number of teeth of the gear, gear diameter, pressure angle, helix angle, gear
type, power required, angular speed, among the most important factors for
recording the data necessary for the calculation.
Figure 7.
App for calculating the wear rate Afterwards, the
necessary inputs and outputs of the system are evaluated
for each equation, as shown in Figure 8. Then, a second test version of the
model was generated taking into account details that
improve user’s experience, such as a help button, a table for unit
conversion, as well as other factors to avoid unnecessary formulation errors.
In other words, it is sought to avoid repetitive
data, or to minimize the data that should be entered when the values
necessary can be calculated from the ones already obtained. 
Figure 8.
Version 2 of the App for calculating the wear rate Based on the
necessary variables and on the requirements for the correct operation of the
software for calculating the abrasive wear rate of gearwheels, using the Archard and Kraglesky methods,
Figure 9 shows the final interface developed so that it is user friendly and
has a pleasant look.
Figure 9. Versión final APP 
3.1.2.
Collection of
final data At this point, it is sought a comparison
of the mathematical models to point out their differences, and compare the
results for the cases of plots with spur and helical gears. In this manner,
it is possible to see a trend in the models to assure that there are no data
with significant errors, and verify the validity of the model in each case.
Hence, isolated cases are limited, and it is understood
when and under which conditions it is relevant to use each of the models
stated. For this purpose,
two simple analyses are performed. In the first one,
which is shown in Table 3, the number of teeth of
one of the gears is changed, and it is intended to calculate the wear for
both spur and helical gears, for each of the gearwheels involved in the
process. The hardness and the
stretching percent of the base material of both wheels are kept constant at
250 HB and 18%, respectively; it is also kept constant at 1430 rpm the
frequency of gearwheel one. The module of the system is set at 4, and the pressure angles at 20 and 21 degrees,
respectively. The average power of
the machine to be evaluated was set at 200 HP for Archard equation. Regarding the particle characterization
for Kraglesky equation, it was used quartz sand
with an average grain size of 0.05 mm and 4% of concentration in volume in
the surrounding medium [20]. 
Table 3.
Wear rate with variable number of teeth [14]

The
wear values for spur gears under the aforementioned variable specifications
are determined in Figures 10 and 11; these values are changed to obtain the
data plotted in Excel, to be able to appropriately manage the parameters of
the approximations, taking into account that Kraglesky
mathematical equation is the appropriate one for analyzing the closeness to
the values under data variability.
Figure 10.
Wear rate 1 of spur gears with Z2
Figure 11.
Wear rate 1 of helical gears with Z2 The results of both Archard and Kraglesky models are plotted in an Excel sheet for spur gears (Figure 12)
and helical gears (Figure 13). 
Figure 12. Wear rate 2 of spur gears with Z2
Figure 13. Wear rate 2 of helical gears with Z2 The
previously stated data will be considered for the
second case of analysis, with the difference that the number of teeth of the
gears involved will be kept constant at 43, and the surface hardness of the
second gear will be varied. Similarly, it will be calculated the wear rate
for each gear, either spur or helical. In this manner, the data given by the
final version of the App will be presented in
detail, where another comparison is made according to the resistance HB2
(Table 4). 
Table 4. Velocidad de desgaste con dientes variables [14]

3.2. Discussion Considering the real approximations performed,
it is taken into account the scheme based on vibrations proposed in [4] for
updating a wear prediction model. First, it is developed
a dynamic model of a system of spur gears to generate realistic vibrations,
which enables performing a quantitative study of the effects of wear on the
surface of the gear teeth. The sliding speed and the contact
forces are used in combination with Archard wear model, to calculate the depth at each
contact point of the mesh. The profile of the teeth of the wornout gear is fed back in the dynamic model as a new geometrical
transmission error, which represents the offset of the profile from an ideal
involute curve; hence, it is zero for perfect gears. Since the wear coefficient of the
model is not constant during the wear process (and difficult to estimate
initially in any case), the vibrations measured are compared with the ones
generated by the model, to update the coefficient when offsets in the
predictions are detected. 3.2.1.
Data for the
calculation The following data obtained from a study are
considered to determine the evaluation process: Acero dulce
AISI 1045 Module: 4 Pressure angle: 20 Power: 4 kW (5,3641 HP) Hardness: 163 HB Stretching percentage: 16 Number of teeth: pinion 19 gear 52
Frequency: 6000 rpm Grain average: 0.05 Concentration of the volume: 1.85
[4] Figure 14 shows the
values determined in the paper under analysis, which specifies the values of
the experimental model and Archard model,
calculated by the authors; the data obtained from this paper is used for the
comparative analysis between the experimental process and the calculations
carried out by the Matlab App developed in this
work. 
Figure 14.
Results for the comparison of the maximum wear depth: experiment and model
[4] 3.2.2. Evaluation of the experimental
model from the paper and Kraglesky model in the Matlab App The error percentage
is calculated in this subsection, to verify the
coincidence between the data of the application from Kragelsky
equation and the data of the experimental model, to plot the analysis model
of the abrasive wear prediction with the use of vibrations. A
primary analysis is performed between the
mathematical model (blue line) and the experimental model (orange dots).
Figure 15 shows the data obtained from the calculation with Kragelsky mathematical model (gray line), used in this
research work, and Archard equation (yellow line). Figure 15.
Calculation of proximities between the values given by the App and the
experimental paper [14] Based
on the above, it will be verified if the data
calculated through the Matlab application are
within the permissible ranges to consider them as appropriate for comparing
the two situations, namely, Kraglesky equation and
the data obtained from the experiment. The mean quadratic error is considered for the statistical calculation, since it enables to compare the difference between the estimator and what is being estimated. This function is a risk evaluator corresponding to the expected value of the quadratic loss. The 
difference is due to the randomness or
because the estimator does not take into account the information that could
produce a more precise estimation (ecuation (19)).
The
mean square error (MSE) given by Equation (19) is used to calculate the
difference of the randomness process to produce a more precise estimation.
The data chosen corresponds to the data that will be used
to determine the expected loss. Table
5 shows the values calculated, which reflect the effectiveness of the
proximity with the data of the experiment. In particular, Krageslsky
mathematical model is effective for predicting the wear index, with a 48.56
%, which indicates that the data calculated are within the range of the
experimental values of wear. It is convenient to specify that the mathematical
model of the paper yields an approximation of 49.06
% with respect to the experimental values. Table 5. Data of the mean square error [14] On
the other hand, Archard equation is
analyzed for the identification of the values calculated; Figure 15
depicts that the data show a significant offset, with an error close to 90 %. It
is important to remark that the analysis shows that Kragelsky
equation is more effective than Archard equation,
since it has data that encompasses the abrasive size and percentage; for this
reason, it is feasible for determining the wear index in mechanical elements. 4. Conclusions Archard and Kraglesky
mathematical models have been selected, and it has been correctly identified
the necessary variables and the usage of each of these models for calculating
the abrasive wear rate. Based on the analysis with the paper under study and
the analysis of the quadratic error between the experimental values and the
values obtained with the equations, it was determined that Kragelsky model shows a better performance, since it
yields a quadratic error of 48.56 % with respect to the experimental data,
compared to a quadratic error of 49.06 % given by the mathematical 
model of the paper. The limitations of
this research work are that the data necessary for the calculations should be
obtained after the element has been designed and it is also
known the construction material. Regarding
the graphical tool created using the Matlab user
graphical application (GUIDE), it fulfilled the requirements to calculate the
abrasive wear rate in gearwheels, needing prior knowledge of the variables
that the user should enter as well as the data of the type of abrasive to be analyzed. The programming was made
according to the mathematical model under study, avoiding unintentional
errors, and providing options to help the user to determine the unknown
variables. The results obtained with the software were
tabulated and plotted in Excel to compare their trends, as shown in the
results section, concluding that the best decision for the design of a gear
with specific features can be made after the data has been varied; in
addition, the designer has available data that may improve the functionality
of the mechanical element designed, taking into account the abrasive found in
the environment. Based
on the verification of the data obtained from the app and the experiment,
besides the explanation provided, it is concluded
that the software developed in this research work enables to calculate the
abrasive wear rate on spur and helical gearwheels with involute profile. It
also establishes prediction data for decision making regarding the execution
of maintenance programs, and the prevention of catastrophic failures that may
harm the working area and the personnel in charge of the machinery. The
contribution of the software is to become a tool to determine values that
enable the mechanical designer to perform a quick evaluation of the
construction of a mechanical element, based on variables that may be found in the gear construction process. Future
works The perfect
companion for the software developed is a shortterm database that contains
experiments with gears and different abrasives, to verify the data and
similarly calibrate the constants of the processes for different working
conditions under lubrication regime and controlled operation. In addition, to
assess the data with statistical tools for improving the error percentage,
and that the data calculated with the Matlab app
are as close as possible to the data obtained when executing an experimental
process. References
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