DIVULGACIÓN de PROYECTOS I+D

Laboratorio Multidisciplinar de Ciencias Básicas

UTN Rosario







INTERACTIVO

DIVULGACIÓN de PROYECTOS I+D

Laboratorio Multidisciplinar de Ciencias Básicas

Directora:
Mg. Alicia María Tinnirello

Co-director:
Eduardo Alberto Gago

Integrantes:
Lucas D’Alessandro
Mónica Dádamo
Matías Romero
Paola Szekieta
Mariano Valentini

Rosario, 2018.

Índice

5

7

Modelización y Simulación
de Sistemas:
Matemática Computacional
y Tecnologías para la
Educación Multidisciplinar
en Ingeniería

Homologado por la Secretaría de Ciencia y Tecnología bajo la identificación Nº 25M/066

Fecha de inicio: 01/01/2013
Fecha de finalización: 31/12/2016

INTED 2012

INTEGRATION TECHNIQUES IN MATHEMATICAL APPLICATION PROJECTS FOR MECHANICAL ENGINEERING Descargar pdf

Tinnirello Alicia María, Dádamo Mónica Beatriz, Gago Eduardo Alberto

Laboratorio Multidisciplinar de Ciencias Básicas
Universidad Tecnológica Nacional, Facultad Regional Rosario (ARGENTINA)

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1 INTRODUCTION

In the introduction to physical systems modelling and control systems design, information technologies converge with advanced elements of mathematical calculation. Systemic and analytical paradigms, from incompatible philosophical roots, overlap; giving rise to a pedagogical and didactic problematic.
Feedback, is treated here in relation with mathematics associated with the study of linear timeinvariant systems (LTIS), being the primary focus of the proposed analysis. The feedback, as suitable concept of modelling systems, has been incorporated to current science and technology after the Second War World. [1] [2].
From multidisciplinary, cybernetic and systemic origin, feedback resists its incorporation to the analytical body of classic physics [3] [4]. The Authors, engineering professors, consider introducing in the early stage of the engineering curriculum the concept of feedback, in order to display the compatibility between the treatment of systems by traditional ways, by means of differential equations, and the one obtained through computational programs oriented to complex and

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These concepts will give to the student an optimal view of the current state on scientific-technological development linked to LTIS of continue data, with the exception of recent systemic studies that exceed the objectives of the advanced basic training [5].

4 BASEMENT

On the study of systems and simple phenomena, it can be considered that an efficient handling of the object under study is linked directly with the understanding of the topic. In complex systems, the relationship between operational expertise and explaining and justifying areas, is potentially more diffuse [6]. An example of the above is seen on the study of the LTIS, in which through its calculation, it is jointly introduced with the learning, the Laplace transform, the Fourier transform, the transfer matrix, the convolution theorem, etc.
One unexpected result of this combination of elements of LTIS would be to consider the respective transfer function (TF) as "obtained through Laplace transform, when the entry is a unit impulse" If focus is given on the inherent properties of linear

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interdisciplinary origin make difficult its incorporation into the basic educational curriculum, which has centuries-old roots.
- Cybernetic and systemic "feedback", though it complements the engineering analytical vision, refuses to be pigeonholed in the reductionism of the positive Sciences.

5 DEVELOPMENT OF METHODOLOGICAL APPROACH

A mass-spring damper system of mass one, with friction coefficient and constant elastic spring is considered. It is assumed that both are linear and invariant in time and space, and the system responds to an external force.
Analytical treatment based on Newton's laws or Lagrangian formalism leads to differential equation

If we assume a graphical interface proposal of the programs oriented to design control systems, guided by the program rules, the following figure summarizes, in a schematic way, how the same system is “thought”.

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Fig.2 Block integrator selection and feedback

so

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Fig.3 Second synthesis: series blocks

Fig. 4 shows the compacted second synthesis, which suggests a similar elaboration to the first one, changing the transfer function by and by , with

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This is equivalent to the transfer function obtained through modelling by means of ordinary differential equations. A mass-spring damper system as:

5.1 Analytical paradigm: a mass-spring damper system

Systems modelling, in the engineering basic training, are accessed by means of differential equations in accordance to the classic physic analytical view.
It is possible to use a wide variety of software (Matlab, Mathematica, Máxima, Scilab, Octave) for its resolution.
Fig.5 shows an example, where Mathematica software is used to solve differential equation stated on equation (8).

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their interaction, adding feedback loops. The latter has acquired special relevance as it is an essential part in any control system.
Using LabView software, simple and practically intuitively a graphic structure that model the system is developed, obtaining the answer to a particular entry, as shown in the Fig.6 and Fig.7.

Fig.6 Mass-spring damper system modelling by LabView software

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pedagogic and didactic publications that use this concept in their essence, bibliography that explore the concept of the feedback from a conceptual point of view could not be found. [7] [8] [9].
During engineering degree education, feedback is treated in a technical way, generally associated with programs that implement it to treat of complex systems, with little or no mention of the systemic approach.
In this paper, the concept of feedback is shown efficient and compatible to manage Newtonian mechanic, and independent of the analytical paradigm. Even though, its way of relating shows a breakage with the traditional severity of the analytic approach, it is consistent with the paradigm of complexity, where analogical reasoning, coupled with its verifiable effectiveness, is considered valid.
In a simple way, an early mention of the systemic approach is done, an approach for which there is currently little academic treatment and is intensively used in professional engineering life.
In the university education, the LTIS, objects on which the feedback concept is applied with significant success, have a

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REFERENCES

1. Rosnay J(1993). El macroscopio, hacia una visión global. Madrid: Editorial AC, 1993. Ch.2.

2. Pagels HR(1989). The dreams of reason, the computer and the rise of the sciences of complexity. New York: Bantam Books.

3. Arnoletto EJ(2007). Curso de Teoría Política. Available on http://www.eumed.net/libros/2007b/300/52.htm (Consulted 12/03/2011)

4. Shannon Claude E, Weaver W(1963). The Mathematical Theory of Communication (5th Ed). Chicago: University of Illinois Press.

5. Liu S, Lin Y(2010). Grey Systems. Theory and Application. Berlin: Springer-Verlag

6. Lilienfeld R(1984). Teoría de Sistemas. México DF: Trillas Ed.

7. Maldonado R, Eduardo C(2009). Sobre la retroalimentación o el feedback en la educación superior on line. Available on

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ICERI 2012

PROJECT LEARNING ENVIRONMENTS IN MECHANICAL ENGINEERING EDUCATION Descargar pdf

Tinnirello Alicia María, Gago Eduardo Alberto, Dádamo Mónica Beatriz

Universidad Tecnológica Nacional (ARGENTINA)
atinnirello@frro.utn.edu.ar, egago@frro.utn.edu.ar, mbdadamo@gmail.com.ar

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1 INTRODUCTION

New opportunities by using new technologies in teaching and learning mathematics in engineering community is becoming increasingly. To incorporate the opportunities offered by computer systems in increasing development and availability is necessary to design appropriate strategies. Learning strategies

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subject matter expert intervention. Nevertheless, to interpret the study of the operational conditions and the spectrums analysis of the report it is of utter importance for the decision making process to know the fundamentals of the signal analysis that corresponds to the study in question, as well as the history of the equipment behavior.
In the case of mechanical transmissions, vibration signal analysis has been proved to be one of the most effective techniques for detection and diagnosis failures. Power Spectral Density (PSD) estimation is performed predominantly using classical techniques based on the Fast Fourier Transform (FFT).The FFT is the favored methods for spectral analysis as it is well established. In recent years, there have been some researches who have applied to condition monitoring investigations parametric modeling technique.
Methodologies based on probabilistic concepts are presented, studying the signal added with disturbances by parametric models that establish the state of functioning which later are used as linear filters to process the future state, using the residual signal between the filtered modeled signal and the future signal in its original state.

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the vibrations inside the bearings by measuring the magnetic field variation generating an electrical signal.

Fig.1 Low pressure turbine: 1) Housing 2) Rotor 3) Bearing 4) Proximitors 5) Amplifier 6) Shaft

2 LEARNING BY PROJECT

Laboratory work classes are an integral part of any educational program and their purpose is bringing the students closer to

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Fig.2 Proximity probe transducer

The orbit time base plots of the temporal signal from these proximitors are showed in Figure 3, obtained from different monitoring channels, captured at real time in determined radial and horizontal positions points.

Fig.3 Orbit timebase plots

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preceding values. This model is expressed by a linear regression on itself (ie. autoregression) plus an error series of approximation.

where e[n] is a Gaussian white noise series with zero mean and variance and the model order. The power spectral density (PSD) of the data sequence x[n] is:

Were represents the PSD of the AR coefficients, ie. a [n], n = 1, 2, ., p.

Since the estimation of AR parameters only involves linear equations, there are several wellestablished methods in estimating the AR coefficients, such as Levinson-Durbin recursion and Burg algorithm.
ARX model is an autoregressive model with exogenous inputs, these linear discrete time, single input/single output ARX model have the follow representation:

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Fig.4 Block diagram

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Fig.6 ARX Model

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6 FUTURE WORK: COMPARISON OF DIAGNOSTIC METHODS

This procedure for damage detection and localization within a mechanical system is solely based on the time series analysis of vibration signals. The standard deviation of the residual errors, which is the difference between the actual measurement and the prediction derived from ARX model, can be used as damage-sensitive feature to locate damage.
The premise of this approach relies on the fact that the residual error associated with ARX model, developed from data obtained when the structure is undamaged, will increase when this model is applied to data obtained from a damaged system.
To verify that one method is more useful than others, it will be necessary to examine several time records corresponding to a wide range of operational and environmental cases, a wide range of damaged and undamaged structures, as well as different damage scenarios.
The ability to perform damage detection in an unsupervised learning mode is very important as data from damaged structures are typically not available for most real-world

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strategy using simulations and other interactive forms, as simulations have always been seen as the most appropriate way to learn together with computers due to the high level of involvement that requires an action-oriented active learning.

ACKNOWLEDGMENTS

The authors would like to thank Apply Energy Support Company for the information submitted.

REFERENCES

1. Fernández, A., Bilbao J., Bediaga, I., Gastón A., Hernández, J.(2005). Feasibility study on diagnostic methods for detection of bearing faults at an early stage. WSEAS Int. conf. on DYNAMICAL SYSTEMS AND CONTROL, pp113-118.

2. Nour, A. , Chikh, N., Chevalier, Y., & Saci, R.(1989).Spectral Analysis and Autoregressive Model of a Rotor Vibratory Signal, ISMEP Paris.

3. Thanagasundram, S. & Soares Schlindwein, F.(2006). Autoregressive

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and bearings. British Non- Destructived Testing and Condition Monitoring . (Vol. 50, Issue 8 pp. 414-418).

10. Wang, W & Wong, A.(2002). Autoregressive Model-Based Gear Fault Diagnosis. Journal of Vibration and Acoustics. (Vol. 124, Issue 2, pp. 172 -180).

11. Chen, Z., Yang, Y. M., Hu, Z. & Shen, G.(2006). Detecting and Predicting Early Faults of Complex Rotating Machinery Based on Cyclostationary Time Series Model. Journal of Vibration and Acoustics. (Vol. 128 Issue 5 pp. 666-672).

12. Zhan Y., Makis, V., Jardine, A.(2003). Adaptive model for vibration monitoring of rotating machinery subject to random deterioration. Journal of Quality in Maintenance Engineering. (Vol. 9, Issue 4, pp. 351-375).

Descargar pdf

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EDULEARN 2013

INTERDISCIPLINARY ACTIVITIES TO IMPROVE THE LEARNING METHODOLOGY PERFORMED IN MECHANICAL ENGINEERING DEGREE STUDIES Descargar pdf

Alicia Tinnirello¹, Eduardo Gago², Mónica Dádamo<sup>2

1</sup> Universidad Tecnológica Nacional. Facultad Regional Rosario (ARGENTINA)
² Universidad Nacional de Rosario (ARGENTINA)
atinnirello@frro.utn.edu.ar, egago@frro.utn.edu.ar, mdadamo@frro.utn.edu.ar

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1 INTRODUCTION

In an age characterized by new dimensions of complexity, scale and uncertainty, many challenges require solutions that are beyond the reach of one thought discipline. More and more frequently, the advances in science and engineering that will have the greatest impact are those born at the frontiers of more than one engineering discipline. The benefits of multidisciplinary thinking - and the shortcomings of a world that has been “understood” primarily by specialization - have been apparent for several decades.
While the concept of multidisciplinary thinking, or “multidisciplinarity,” is not new, it has in recent years emerged as a pervasive term, gaining popularity both in science and in policy contexts. Multidisciplinarity traces its roots to the second

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increasingly powerful, allow making the modifications needed to achieve this purpose.
With the objective to carry out these changes we need to align the University curricula not only to the new work methods that allow intellectual development stimulation but also have a tendency to a multidisciplinary approach. The goal is not only to teach and learn only mathematics, is also doing it by facing with the stimulation of real cases and simple systems that lead the student to have a look at real situations.
The development of numerical methods and the advent of simulation platforms consider the need to guide the methodological approach of Higher Mathematics studies towards a new kind of teaching that modify the learning sequence. Driving the educational system approach to multidisciplinary models has an impact to the student’s cognitive process.
Multiphysics stimulation software create an engineering environment where learning opportunities and motivation increases exponentially. Moreover the comprehension improvement and ease of learning complex subjects are improved when trying to learn fluid flow processes with

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abstraction and generalization processes, or find it difficult to adapt to them. This is why the trend in contemporary education needs the implementation of a system that places the student in centred processes that identify him as an active and reflective learning subject.
To carry out these changes that aim at leading the acquisition of new ways of thinking and reasoning, it is necessary for the teacher to reformulate the methods implemented in the class and is adapted to work in other areas where available modelling and simulation are applied [5].

2 APPLIED METHODOLOGIES

This engineering analysis work is based on a design and planning computer aided system. It is important to create a learning space at the higher education system where developed a set of activities and communicational expressions as the fundamental line of the educational process. It will organize theoretical and practical activities where students perform technological applications with the topics developed in class, linking the themes of the subject, in different subjects of

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Sciences in Chemical, Mechanical, Civil and Electrical majors, conducting interdisciplinary workshops with the use of specific software and tailor-made training materials, training scholars in teaching and researches activities, developing teaching materials in electronic format to implement whether dual or distance education.
It is also intended to support planning and implementation of curricular activities to develop dual and distance learning through the institutional web site. This involves training teachers to develop activities suitable for distance learning and the ability to access materials and resources with the state of the arttechnology and bibliographical material as well as scientific publications throughout the working team[10].

3 TEACHING STRATEGIES

To improve their learning experience students need to have sufficient prior knowledge from which to address the content proposed, in order to establish more complex and rich relations. Therefore, it is initially convenient to help the student to remember, rearrange or assimilate their prior background

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theoretical and practical, so proceed from reflection to action and then the improvement of the action. The teacher acts as a facilitator of learning “, leading to questions, presenting situations, pointing out mistakes and avoiding address what the student can solve by themselves. The situations should be simpler in the first stage gradually increasing its complexity [10].

4 PROJECT PLANNING

Project based learning has proved to be a suitable method to demonstrate the need of mathematics in professional engineering. Students are confronted, complementary to their regular courses, with problems that are of a multidisciplinary nature and demand a certain degree of mathematical proficiency [2].
The curricula of the Mechanical Engineering programs at our university include Advance Calculus at the third year of the engineering career; the authors’ experience is that students increased their interest and their appreciation for the contents if they are involved by learning in an applied way.
The project proposal was a selection of contents related with

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• Helps design and prototype with quick solutions avoiding costly experiments.
• Process visualization and animation can be obtained in terms of the fluid variables [9].
The proposal presented is intended to explore potential theory examples and discuss some fluids that can be approximated using Computational Fluid Mechanic. When the potential flow presents complicated geometries or unusual current conditions, the conformal transformation based in complex variables cease being useful to generate forms of bodies. In this case the numerical analysis technique constitutes a more appropriate approach.
The finite difference method for the potential flow has the aim to approximate partial derivatives listed in a physical equation by “the difference” between the values of the solution in a number of modes spaced by some certain finite distance. The original equation in partial derivatives is replaced by a series of algebraic equations for the nodal values.
The system being studied is the two-dimensional flow of an incompressible fluid, non-viscous corresponding to an irrotational field that moves in a steady state for different values

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If these partial derivatives are continuous in R , then the above equations are sufficientconditions to be f(z) analytical in R. The functions that satisfy the above conditions are said conjugate and such functions satisfy the orthogonality property means that the type curves , and , are orthogonal.
(iii) In addition if the second partial derivatives p and q are continuous in R, and Laplace equation is meet for p and q:

(iv) The derivative of the function f can be calculated by the following expression:

4.2 2^nd Session: Modeling the system

The velocity complex potential F is an analytical function of complex variable composed of the ordered pair whose real part is the velocity potential and imaginary part is the current

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In this case the equipotential curves are those such that: , and are orthogonal to the streamline which are equation curves , (with k₁ y k₂ constants).
With the model accomplished and by Eq. 7 and Eq. 8 it could be concluded that the fluid velocity is a conjugated complex, derivate from the F function.

It expresses Eq. (9) also by means of Eq. (10):

In order to calculate the magnitude of the velocity was used Eq.(11).

4.3 3^rd Session: Analysis and discussion of different cases

4.3.1

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(iii) The circumference of radio a represents a path, and since there can’t be a flow through a path, it can be considered as a circular obstacle of radius a placed in the fluid path.
(iv) The complex velocity of the fluid has a variable value near the obstacle and its modulecorresponds to
(v) If we move away from the obstacle the velocity takes the value , i.e. the fluid is running on the positive x axis direction with constant velocity .
(vi) The stationary points of the system are those where the velocity is zero and are given by the values of and .

4.3.2

From the analysis of Fig. 2 we been able to visualize the behaviour of the fluid in this case, where the power lines are horizontal lines and the corresponding equipotential curves are vertical lines, indicating that the fluid flow is uniform and its direction is right, this is also interpreted as a uniform flow in the upper half plane bounded by the x axis, which is a streamline, or

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flow”. In this case the direction flow matches clockwise direction and the magnitude of the velocity is:

4.3.5

As shown in Fig. 5, we study the region in the first quadrant, in this case we observed that the streamlines for this region are bounded by the x axis and the line . The direction flow is downward and the value of the module of the velocity is .

4.3.6

In this case Fig. 6 shows that the streamlines are concentric circles around a point and that from that point equipotential curves are born. It appeared that the velocity module corresponding to the formula is:

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program allows adding to the proposed model, new physical parameters besides proposing, evaluating and exchanging boundary conditions. The potentiality that the program has, contributed to work with educational proposals that approximate to reality aside from the ideal conditions.

5.1 Simulated domain and physical conditions

By 6 cm diameter UNS C10100 bronze pipe, water flows in a laminar regime with a Reynolds number less than 2500 and a temperature of 20 ° C. In this pipe there is a blockage of a sphere of alloy steel 1006 (UNS G10060) of 0.3 cm radius. If the pressure ranges from 3 to 2 Pa in a section of 10 cm length, and it is considered that the fluid has viscosity on the outer walls and roughness in the sphere, it is requested to study the variations in the pressure and velocity caused by such obstruction.
Using conformal mapping techniques based on complex variable techniques there are no friction losses and the fluid is considered in ideal conditions, with COMSOL platform is not possible to consider the value of zero viscosity.

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Fig.7 Prototype Geometry Fig.8 Meshing

5.3 Results

Fig. 9, Fig. 10 and Fig. 11 shown the formation from the tube walls of a profile with increasing velocity called boundary layer. In the central part, where the sphere is positioned, zero velocity are noticed at the front and rear of it, indicating the absence of fluid flow in that area and as a result of this, significant pressure fluctuations.
Fig. 12, Fig. 13 and Fig. 14 show that at the side of the sphere there is a considerable pressure drop which results in an

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starting point for studies in nozzles, diffusers or in other engineering applications.

Fig.9 Velocity level Fig.10 Velocity level Fig.11 3D velocity level
surfaces curves surfaces

Fig.12 Level pressure Fig.13 Pressure level Fig.14 Level pressure
surface curves surfaces

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minimum diameter of the pipe, from any disturbance so as to achieveresults biased by them.

6 CONCLUSIONS

The use of analytical functions of complex variables, the study of idealized models and support of computer resources used appropriately lead to a contextualized analysis which achieve the goal of understanding the possibilities of integration between computational mathematics and fluid mechanics.
These laboratory experiences are a link between basic science and applied technologies that equip students an autonomous character to work with problem situations similar to their future professional work.
Through numerical simulation and basic knowledge of fluid flow simple problems are introduced using an environment such the one offered by COMSOL platform to assess the importance of the results obtained in an analytical and symbolic form, and the possibilities for future use of the virtual learning laboratory, in applied technologies studies, to support disciplinary integration activities.

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REFERENCES

1. Bird, R.; Stewart, W.; Ligthfoot, E. (2007). Transport Phenomena. Wiley & Sons Inc. 2nd. Ed.

2. Bischof; G., Bratschitsch, E.; Casey, A.; Rubesa, D. (2007). Facilitating Engineering Mathematics Education by Multidisciplinary Projects American Society for Engineering Education.

3. Cengel, J.; Cimbala, J. (2006). Mecánica de Fluidos: Fundamentos y aplicaciones. México. 2ª Ed. Edit. McGraw-Hill.México.

4. Churchill, R. (2004). Variable compleja y Aplicaciones. McGraw Hill, 7ª Ed.

5. Gago, E.; Dádamo, M. et al. (2010). La Importancia del Enfoque Multidisciplinar en la Enseñanza en Ingeniería Mecánica. II CAIM - Segundo Congreso Argentino de Ingeniería Mecánica. San Juan, Argentina.

6. James, G. (2002). Matemáticas Avanzadas para Ingeniería, Prentice Hall, 2ª Ed.

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DESIGN, SIMULATION AND ANALYSIS OF A FLUID FLOW SYSTEM THROUGH MULTIPHYSICS PLATFORM Descargar pdf

Alicia Tinnirello, Eduardo Gago, Mónica Dádamo, Mariano Valentini

Universidad Tecnológica Nacional (ARGENTINA)

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1 INTRODUCTION

The technological resources provide a broad possibility of representation and calculation allowing to incorporate fundamental changes in how to teach. For more than two decades, in the academic field, the potentialities offered by the technology affects the activities that are derived to the classroom’s area. Under the experiences which have been developed in engineering careers, are then installed the need to develop methodologies for teaching and implementing the interaction between the different areas to enhance the student's ability in the acquisition of concepts and its further use in the models required for the resolution of engineering problems.

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that integrate the knowledge of the compartmentalized disciplines.
The developments that have experienced the mathematical software and the affinity that the students have to be linked with the technologies, imposes on the university teachers makes the effort to transform the teaching-learning process in the process of learning investigating.
In the areas of engineering education have emerged a potential alternative to the traditional laboratories, the so-called virtual labs. This new system of learning is possible given the capacity of the recent development of technologies that allows you to resolve each system symbolic and numerically, which it can be represented by a mathematical model and designed using simulation models. [1]

2 OBJECTIVE

The proposal aims to show the consequences of the provision of a current of air stream than enters a fluidized bed dryer using the theory of the Computational Fluid Mechanics.
Through the analysis of the proposed system is intended to

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3 DESCRIPTION OF THE PROCESS

The fluidized bed dryer consists of a system where the particles are partially suspended in a gas stream ascending. The particles are lifted and then fall at random so that the resulting mixture between solid and gas acts as a boiling liquid. The contact is established between the solid and gas is the adequate to make it a better mass transfer between the two.
The fluidized bed dryer works with the condition that the air’s velocity is greater than the speed of sedimentation of the particles that float partially suspended in the flow of gas (The resulting mixture of solid-gas behaves like a fluid, which is why it is said that the solid are fluidized).
This technique is very efficient for drying granular solids, because each particle is completely surrounded by gas. This equipment is projected to analyze the output of the fluid under different conditions of entry of the hot air, determining the relationships between the input and output of the fluid’s parameters. The implementation of fluidized beds to a dryer of solid particles represents an approach to the problem of increasing the speed of heat transfer between the walls and currents of the process.

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4 APPLICATION OF CFD TECHNOLOGY

The simulation is performed with the Creeping Flow module of COMSOL Multiphysics. The working environment of the program allows adding to the proposed model, new physical parameters besides proposing, evaluating and exchanging boundary conditions. The potentiality that the program has, contributed to work with educational proposals that approximate to reality aside from the ideal conditions.[4]
Using CFD is possible to build a computational model that represents a system to study by specifying the physical and chemical conditions of the fluid to the prototype virtual software delivery and the prediction of the dynamics of the fluid, Therefore, it is a design technique and analysis implemented in a computer. The main benefits of using this technology are: the prediction of the properties of the fluid with great detail in the domain studied the design and prototyping of the computer while avoiding costly experiments, and the visualization and animation of the process in terms of the variables. [3]
Using the simulation with the COMSOL software will play that happens with the flow of fluid within the grain dryer. It is

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Using CFD is possible to construct a computational model representing the grain dryer to study the physicochemical specifying air conditions assumed in the virtual prototype and the software supplied predicting fluid dynamics, therefore, it is a design technique and analysis implemented in a computer. The main advantages in using the CFD technique are that fluid properties are predicted in great detail within the domain of the system studied, collaborates with the design and prototyping through the possibility of quick solutions avoiding costly and risky experiments, and the possibility of obtaining the display and animation of the process in terms of the variables in the fluid.
By means of the simulation with the software COMSOL will reproduce that it happens to the flow of the fluid inside the dryer of grains. Constant density within grains entering the dryer at a constant temperature is estimated. Different temperatures of heating element are then evaluated, keeping the velocity of incoming air.

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flow which is considered incompressible. Furthermore, the pressure drops and temperature variations are neglected. Constant density within grains entering the dryer temperature constant is estimated 293 K (20°C), and considering the thermal conductivity of the heating element
The values of the physical conditions of the air selected to make the experience are set forth in the following table:

Table 1. Physical properties of air.

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Fig.3 Navier Stokes equations.

9 ANALYSIS AND RESULTS

COMSOL provides the results is by the appearance of an image showing the model solution based on each requested parameter.
At this stage certain subfolders that help the user to interpret the results from different representations, ie, a graph showing a figure with a legend or just a table with the values corresponding to the desired result is. The software allows the

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Fig.5 Temperature gradients.

In the Fig. 6, it is seen that as the velocity of air entering the dryer is increased, the outlet temperature of the air is increased first in a linear fashion, to about 1 m / s, and then begins to stabilize at a constant value, meaning that even if the input speed is increased not achieved that the outlet temperature behaves linearly. [5],[6]

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Fig.7 Outlet temperature based on the temperature of the heating element.

In the Fig. 7, considering the rate of air intake of 0.5 m / s, and the heating element temperature is varied, the outlet temperature has a linear relationship with the temperature of the heating element, i.e, it is constant if the input speed is achieved to adjust the cyclone outlet temperature by adjusting the temperature of the heating element.
After this analysis, it was decided to reduce the temperature of the heating element to fifty percent to see what happens to the

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Fig.9 Temperature gradients.

In the Fig. 10, it is noted that as the velocity of air entering the dryer is increased, the outlet air temperature is always increases linearly shaped but with two straight sections that have different slopes. Accordingly, if the input speed to be increased is achieved that the outlet temperature behaves linearly, but with temperatures lower than those obtained with a heating temperature to twice its output value.
By reducing the temperature of the heating element slowing dryer process is obtained, as temperatures exiting the cyclone

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occur in the drying process, as the temperature is maintained at the same values. However, the risk of increasing the pressure of the circulating air is run, which could alter other parameters of the dryer. The Fig. 10 and 11 show the temperature within the fluidized bed when the number of slots through which the air is changed.

Fig.10 Temperature within the fluidized bed.

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such as analyzing the fluid outlet when changes occur in the grain size study are planned in future research.
The use of COMSOL in the analysis of parameters of a unit shows the importance of the implementation of technologies for the study of major phenomena that occur in the nature of fluid flow and heat transfer in a given context.
From the results obtained in the simulation can effectively analyze the behavior of the fluid under different conditions and comprehensively analyze the behiavor of its path. It follows that the use of the virtual platform streamlines the combined study of physical phenomena, contrasting the methods used to verify the experimental results.
It is of great importance to computer application in solving fluid flow systems, where due to the geometry of the model is very complex resolution by conventional analytical methods.

REFERENCES

1. Tinnirello, A.; Gago, E.; et al. (2013). Interdisciplinary activities to improve the learning methodology performed in mechanical engineering degree studies. EDULEARN 2013 (pp. 5407-5416).

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INTEGRACIÓN DE CONCEPTOS EN EL ANÁLISIS DE LOS MÉTODOS NUMÉRICOS: VINCULACIÓN MATEMÁTICA-INFORMÁTICA Descargar pdf

Mónica Dádamo¹, AliciaTinnirello²

Laboratorio Multidisciplinar de Ciencias Básicas
^1,2 Universidad Tecnológica Nacional, Facultad Regional Rosario.
Zeballos 1341. Rosario. Argentina.
¹mdadamo@frro.utn.edu.ar; ²atinnirello@frro.utn.edu.ar

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este contexto proponemos hallar el núcleo matemático esencial en la integración numérica, articular éste con las herramientas informáticas acordes, de modo de producir un conocimiento significativo en lo conceptual y operativo en lo matemático. Así también el conocimiento debe ser lo suficientemente crítico para conocer las limitaciones y permitir auto-gestionar su perfeccionamiento, por ello abandonamos el tipo de enseñanza que cultiva la ficción de un rigor eterno y perfecto de las matemáticas [1].

2 ENFOQUE DE LA INVESTIGACIÓN

Buscando una integración entre las disciplinas matemática e informática que sea mayor que la suma de las partes, comenzamos una investigación bibliográfica que planteamos en tres escenarios: Didáctico, Matemático e Informático. Investigación didáctica: El marco de la teoría de transposición didáctica (TTD), nos lleva a considerar la existencia de un “invariante matemático” o “núcleo matemático” que constituye la esencia de un saber independiente de la forma en que se lo presenta, las transformaciones que sufra este a causa de las

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particular debido a la aguda crítica del obispo y filósofo Berkeley [3]. Los infinitesimales retornan al universo matemático en 1960, reconstruidos con fuerte base lógica-matemática por Robinson en el llamado análisis no estándar.
En el período próximo al 1900, los trabajos de Runge, Heun y Kutta continúan en cierta medida el camino comenzado por Euler, en particular el enfoque de Kutta que basado en el trabajo de los dos anteriores generaliza el hecho de utilizar los polinomios de Taylor de grado mayor a uno, sin derivar. En este caso modifica el método numérico de modo que replique el orden de convergencia del polinomio de Taylor. Se crean los métodos Runge-Kutta. En las décadas del 60 y 70 del siglo pasado J.C. Butcher introduce la teoría de árboles en el estudio de los métodos Runge-Kutta, los cuales presentan el inconveniente de la complejidad en la notación al aumentar el número de pasos.
La teoría moderna de estos métodos se atribuye a los trabajos realizados por Dahlquist en 1956 y divulgada en los textos de Henrici [4].
En la actualidad, los trabajos de J.C. Butcher permiten evitar el cálculo de derivadas de orden superior al aplicar Taylor

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diferencias. Por problemas técnicos y financieros se abandonó el proyecto. En 1833 retoma la construcción de un proyecto más ambicioso, la Máquina Analítica, que incluye la tarjeta perforada del telar de Joseph Jacquard, el cual usaba estas para determinar cómo un tejido debía ser realizado y que Babbage adaptó su diseño para conseguir como indicar a su máquina el cálculo de funciones analíticas. Como consecuencia, esta máquina "programable" ofrecía dos nuevas ventajas: i) por primera vez, una máquina sería capaz de utilizar durante un cálculo los resultados de otro anterior sin necesidad de reconfigurar la máquina, lo cual permitiría llevar a cabo cálculos iterativos, y ii) habría la posibilidad de que la computadora siguiese instrucciones alternas, dependiendo de los resultados de una etapa anterior del cálculo. Babbage describió esta máquina como "la máquina que se muerde la cola" [5].
A su trabajo se une en carácter de colaboradora Ada Lovelace, matemática inglesa, que entre sus notas sobre la máquina, se encuentra lo que se reconoce hoy como el primer algoritmo destinado a ser procesado por una máquina, por lo tanto se la conoce como la primera programadora. "La Máquina Analítica

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Turing, y demostró que existían problemas que una máquina no podía resolver. Los trabajos de Alonso Church sobre cálculo lambda, (funciones recursivas) son coincidentes con los de Alan Turing y se dice que ambos tienen el mismo poder de expresión.
En 1960, John McCarthy publica “Recursive functions of symbolic expressions and their computation by machine”. En este artículo se sintetiza el trabajo desarrollado en el MIT, sobre finales de la década del 50, donde se publica la creación del lenguaje LISP, el cual Steve Russell, estudiante y colaborador de MsCarthy logra hacer correr en ordenador. Este lenguaje permitió el desarrollo de los primeros entornos computacionales matemáticos con capacidad de cálculo simbólico. La elección del medio informático para ejecutar estos algoritmos es, a menudo, a nuestro entender, crítica. Una opción interesante es utilizar entornos informáticos relacionados con el futuro campo laboral del ingeniero como el Scilab, Matlab, Comsol, etc. Otro camino es tomar entornos matemáticos como Mathematica, Maxima, etc.
En este trabajo volcamos la experiencia con Maxima, entorno que permite el cálculo simbólico y una programación sencilla.

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Utilizando terminología actual y partiendo de la definición de derivada de Cauchy (1823) ensayamos una construcción equivalente.

Con los nuevos valores construimos y repetimos el proceso hasta llegar a los puntos distantes en el entorno que nos interesa, tal como lo exhibe la siguiente tabla extraído del original de Euler.

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El anterior programa solo ejecuta de manera transparente un proceso rutinario. Los métodos numéricos comienzan tras la elaboración del algoritmo de Euler e independiente del medio en que este se ejecute.

Babbage y la matemática Ada Lovelace, tenemos el siguiente programa para la ecuación diferencial de primer grado , con valor inicial

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Runge-Kutta2, en clara vinculación con los polinomios de Taylor.
A fin de reafirmar el concepto, recurrimos al software para generar gráficas en las que se observa que los polinomios de Taylor nos sirven de comparación, dentro de nuestros parámetros visuales, del grado de convergencia del método. (Fig. 2.)

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4 CONCLUSIONES

Podemos resumir nuestra investigación diciendo que matemática aporta para acotar y explicar los errores, determinar ámbitos de aplicación, mejorar algoritmos, interpretar resultados y supervisar la aplicación de las herramientas computacionales, siendo importante la integración de matemática e informática para la conceptualización de los métodos. Cambiar el entrenamiento en resolver manualmente algoritmos, por el de volcarlos a un determinado sistema informático ayuda a la formación del pensamiento crítico.
La matemática que emerge como consecuencia del uso de los medios informáticos permite incursionar en los sistemas no lineales, de complicada resolución en forma analítica. La transparencia del proceso informático, la problemática de la transposición didáctica y el aprendizaje conceptual de los conceptos directrices, es la tarea fundamental del trabajo docente.

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7. Sanz-Serna, J. M.: Stability and convergence in numerical analysis I: Linear problems-A simple, comprehensive account. En Nonlinear Differential Equations, Boston, 1985, 64–113; citado en Sanz Serna J.M.: Discurso leído en el acto de su recepción como Académico de Número por el Excmo. Sr. D. Jesús María Sanz Serna. Integración Geométrica (2007)

8. L. Héctor Juárez V. Análisis Numérico. Departamento de Matemáticas, Universidad Autó-noma Metropolitana

9. Caligaris M., Rodriguez, G. , Laugero, L.: La visualización en la compresión de errores que influyen en la solución numérica de un problema de valor inicial. III Jornadas de Enseñanza de Ingeniería. 2013. ISBN.2313-9056.

10. Chapra S., Canale R.: Métodos numéricos para ingenieros McGraw-Hill, 2007

Descargar pdf

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MÉTODOS DE VARIABLE COMPLEJA EN EL ESTUDIO DE LA DINÁMICA DEL FLUJO DE CALOR Descargar pdf

Alicia Tinnirello¹, y Eduardo Gago<sup>1

1</sup> Laboratorio Multidisciplinar de Ciencias Básicas,
Universidad Tecnológica Nacional, Facultad Regional Rosario,
Zeballos 1341, 2000 Rosario, Argentina
Alicia Tinnirello, Eduardo Gago, egago@frro.utn.edu.ar

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la aplicación de los recursos informáticos.
Las recomendaciones realizadas por los organismos de acreditación instan a que el alumno desarrolle actividades de aprendizaje con herramienta computacional y emigren del contexto de la clase magistral para proponer una estrategia metodológica facilitada por el enfoque multidisciplinar. Se pretende que el alumno adquiera el conocimiento mediante un proceso de experimentación, utilizando la simulación y la visualización para ello debe plantear hipótesis de trabajo, inducir respuestas, cuestionar predicciones y validarlas para darle significado a los aprendizajes.
Es posible resaltar un conjunto de contenidos mínimos indispensables, para la formación básica del ingeniero que vienen siendo trabajadas en la Cátedra de Cálculo Avanzado en la búsqueda de integrar las disciplinas en la carrera de Ingeniería Mecánica, lo que implica por parte del docente el conocimiento de los programas de enseñanza de las demás materias paralelas, no solo las del Tronco Integrador, sino de materias básicas y de la especialidad, a fin de organizar dichas “prácticas coordinadas” o de coordinar los problemas básicos de su materia con las situaciones problemáticas que se

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técnica en términos matemáticos, para lo cual se presenta una situación simplificada, se traduce dicha situación en terminología matemática, y se trabaja con dicho modelo. Esta metodología permite estimular el interés por el descubrimiento y adquirir confianza en la utilización de los aspectos formativos de la Matemática, relacionadas con otras áreas de conocimiento, como en este caso en el análisis de la dinámica de un determinado fluido.
Los objetivos pedagógicos quedan planteados en las siguientes fases:
· Presentación de los sistemas a analizar
· Investigación teórica
· Modelización y análisis de los parámetros
· Interpretación de los resultados en términos técnicos

3 DESARROLLO DE LA EXPERIENCIA

Los sistemas que se analizan corresponden al flujo de calor en una región del plano complejo de un fluido incompresible en estado estacionario, no viscoso correspondiente a un campo irrotacional. En primer lugar se propone analizar el flujo de calor

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dice analítica en , si existe la derivada en todo punto de .
La temperatura compleja está compuesta por el par ordenado cuya parte real es el potencial de temperatura y la parte imaginaria es la función de corriente .

Una condición necesaria para que sea analítica en una región , es que, y en R , satisfacen las ecuaciones de Cauchy-Riemann

si estas derivadas parciales son continuas en , entonces las ecuaciones (3) son condiciones suficientes para que sea analítica en . Las funciones que satisfacen dichas condiciones se dicen conjugadas y armónicas.
Las funciones conjugadas y armónicas cumplen con las siguientes propiedades:
i) Propiedad de ortogonalidad, las curvas de las ecuaciones (4), son ortogonales

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Suponiendo que no hay fuentes o sumideros dentro de C . La ec. (6) resulta

Lo cual llega a resultar que la componente es armónica

Las ecs. (10) representan las líneas isotérmicas (o simplemente isotermas), y líneas de flujo de calor respectivamente.

3.2 Modelización, análisis y resultados.

1º Caso: Análisis del flujo de calor alrededor de un cilindro de sección transersal circular
Se propone analizar el flujo de calor alrededor de un cilindro de paredes adiabáticas, para una temperatura compleja cuya la ley

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tuviera un sumidero en el punto

Fig.1 Curvas de flujo de calor e isotermas

Las funciones y , cumplen con las ecuaciones de Cauchy-Riemann, y consecuentemente con la ecuación de Laplace, probando así que ambas funciones son conjugadas y armónicas.
Las ecuaciones de las líneas de flujo de calor y las isotermas son, respectivamente:

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por los valores de y

Fig.2 Curvas de flujo de calor e isotermas

En la Fig.2 se observó la simulación de situaciones alternativas a la dada originalmente. Se visualizó que cuando el flujo de calor se enfrenta con un cilindro de sección transversal elíptica, las líneas de flujo tienen una separación más amplia, mientras que las isotermas están más próximas. También, realizaron cambios en el sistema, la nueva simulación consistió en el suprimir el cilindro, y considerar un sumidero en el origen de

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f) ¿Cuál es la ecuación que mide la cantidad de calor en todos los puntos del plano ?
El potencial de temperatura es:

Fig.3 Curvas de flujo de calor e isotermas

En el flujo de calor que analizaron, la fuente de intensidad A situada en el plano complejo en , más el sumidero de intensidad -A situado en el plano complejo en . Las Ecs.

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enseñanza aprendizaje innovador.
Para lograr la conceptualización teórica de los contenidos abstractos de variable compleja fue fundamental poder analizar matemáticamente un sistema de transmisión de calor, siendo la propuesta muy motivadora y el disparador de futuras investigaciones., tales como la inquietud planteada por los propios alumnos de completar el análisis con herramientas específicas para el tratamiento multifísico del sistema.

REFERENCIAS

1. Cengel, J.; Cimbala, J.: Mecánica de Fluidos: Fundamentos y aplicaciones. México. 2ª Edición. Editorial McGraw-Hill. México. (2006).

2. Bird, R., Stewart, W., Ligthfoot, E.: Fenómenos de Transporte, Reverté, México. (2011)

3. James, G.: Matemáticas Avanzadas para Ingeniería, Prentice Hall, 2ª Edición, (2002)

4. Tinnirello, A., Gago, E., Dádamo, M.: Designing Interdisciplinary

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GESTIÓN DEL CONOCIMIENTO MATEMÁTICO-COMPUTACIONAL: EXPLORACIONES PARA EL ABORDAJE DE LA COMPLEJIDAD Descargar pdf

Mónica Beatriz Dádamo, Alicia María Tinnirello

Laboratorio Informático Departamento Ciencias Básicas
Facultad Regional Rosario, Universidad Tecnológica Nacional
Zeballos 1341 (2000) Rosario, Argentina
mdadamo@frro.utn.edu.ar - atinnirello@frro.utn.edu.ar

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conduce a un cambio en la gestión del conocimiento a través de una planificada experimentación, que muestra como algoritmos computacionales sugieren interesantes problemas matemáticos. Este campo es el de la matemática computacional, aquella cuyo objetivo es desarrollar algoritmos que permitan resolver problemas matemáticos tratados con el computador, particularmente aquellos problemas del dominio complejo donde emergen en forma natural lazos de realimentación como en los problemas de dinámica no-lineal. Siendo el computador el medio necesario para diversos desarrollos, existen formas habituales de utilizarlo, mediante un Sistema Algebraico Computacional (SAC), práctica más extendida en el ámbito universitario ingenieril, y/o mediante la programación en un determinado lenguaje o la programación en un SAC.
Los inconvenientes de mixturar el contenido matemático con el medio informático, más allá de las distintas epistemes que validan los conocimientos disciplinares, son: la rápida obsolescencia de los medios informáticos (caso del SAC DERIVE y del lenguaje BASIC) y las limitaciones propias del SAC o del lenguaje de programación.

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giran en torno a la Matemática del continuo [3].

2 CONSIDERACIONES HISTÓRICAS-EPISTEMOLÓGICAS DEL DESARROLLO DE LA COMPUTACIÓN

2.1 Antecedentes históricos

Si bien la introducción de dispositivos mecánicos para realizar cálculos es milenaria, como por ejemplo el ábaco, centramos la atención a partir de los trabajos de Charles Babbage y Ada Lovelace. Se trata del diseño de la máquina diferencial, la máquina analítica y la creación del primer programa computacional.Si bien la introducción de dispositivos mecánicos para realizar cálculos es milenaria, como por ejemplo el ábaco, centramos la atención a partir de los trabajos de Charles Babbage y Ada Lovelace. Se trata del diseño de la máquina diferencial, la máquina analítica y la creación del primer programa computacional.
Entre 1847 y 1849 Babbage se encaminó a diseñar la Máquina Analítica, basada en el mecanismo del telar que utilizaba tarjetas perforadas para controlar de manera automática el

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modelado matemático tradicional no se percibe. Las notas de Ada Lovelace sobre la máquina analítica de Babbage, son reconocidas hoy en día como el software de dicha máquina y la máquina analítica como modelo temprano del ordenador, motivo por el que Ada Lovelace es reconocida como el primer programador de la historia.
En el año 1920 el matemático David Hilbert presentó la exigencia de establecer la matemática sobre la base de un sistema axiomático completo y libre de contradicciones. Un sistema tal debe tener las características de Consistencia, Decibilidad y Completitud.
Un sistema tiene la propiedad de ser consistente cuando no es posible deducir una contradicción dentro del sistema, es decir, dado un lenguaje formal con un conjunto de axiomas, y un aparato deductivo (reglas de inferencia), no es posible llegar a una contradicción. Se dice de un sistema que es decidible cuando, para cualquier fórmula dada en el lenguaje del sistema, existe un método efectivo para determinar si esa fórmula pertenece o no al conjunto de las verdades del sistema. Cuando una fórmula no puede ser probada verdadera ni falsa, se dice que la fórmula es independiente, y que por lo tanto el

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matemática no es. Nos centramos en el problema de la decibilidad, llamado Entscheidungsproblem.
Para resolver este problema Alan Turing ideo una máquina hipotética, hoy en día denominada Máquina de Turing, podemos concebir la naturaleza de esta máquina en forma dual: un objeto mecánico y un objeto matemático. De modo informal se puede decir que fue creada intelectualmente como un objeto matemático para leer una proposición matemática y dar un veredicto acerca de si esa afirmación es o no es demostrable, en un dado sistema. Hoy en día se toma como modelo general de una computadora.
En este programa propuesto por David Hilbert concurrieron entre otros, Kurt Gödel, Alonso Church, Stephen Kleene y Alan Turing todos ellos matemáticos de profesión y de diversas nacionalidades. La colaboración académica fue cortada durante la segunda guerra mundial y Alan Turing pasa a liderar la construcción de una máquina, destinada a descifrar los mensajes encriptados por la máquina alemana Enigma, lo cual es logrado, pero acontece bajo el secreto impuesto en el marco de una guerra a gran escala y prosigue luego en gran parte debido a las tensiones de la llamada guerra fría. De igual modo

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recordar que interactuamos con la interfaz de usuario, el problema matemático-computacional fue resuelto por quien elaboró el software y lo que realmente tenemos es un entrenamiento en el manejo de esa interfaz. Si el software es cerrado la matemática implicada queda cerrada, si el software es abierto y hay conocimiento del lenguaje de programación con que fue escrito, los algoritmos implicados se hacen visibles, del mismo modo que si trabajásemos programando en ese lenguaje. Sin embargo, sin conocer los límites teóricos y tecnológicos de la máquina, el conocimiento no será completo; conoceremos el cómo y no el porqué.
- ¿Utilizamos la computadora para hacer la misma matemática de siempre, más rápido y con menor esfuerzo o estamos lo suficientemente capacitados para llevar las matemáticas al nivel en que es capaz de generar algoritmos netamente computacionales?
Si continuamos con la matemática tradicional solamente, el problema consistiría en traducir los algoritmos manuales a un lenguaje de programación y esto sería todo, como de alguna manera se puede inferir.
En cierta medida, si aceptamos las soluciones pre-armadas de

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problema, se introduce en el modelo a formar tipos de solución que requieren algoritmos que serán ejecutados por máquinas [6, 7].

2.3 Reflexiones sobre las consideraciones

La contribución de la comunidad matemática al desarrollo de las ciencias de la computación es fundamental. Considerando que el modelo teórico actual del computador se desprende de la demostración de un teorema matemático, resulta cuasi-paradójica la temática como introducir el computador en la matemática, ya que éste es un producto de tal actividad. Esta conexión entre las computadoras y las matemáticas se utilizó más tarde para desarrollar las bases matemáticas para la ciencia informática. La pregunta: ¿Es posible utilizar la misma conexión para incorporar los fundamentos informáticos a la matemática-computacional?
Actualmente es posible construir el concepto de Máquina de Turing, partiendo del desarrollo de Autómatas Finitos (AF), siguiendo luego por el de Autómatas con Pila se llega a modelar con una Máquina de Turing, como se explícita en el

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meta-problema de Ingeniería Mecánica. En este caso en particular comenzamos con una pregunta fuertemente vinculada a la ingeniería: ¿Qué es una máquina? Tratando de llegar a un concepto operativo respecto de la palabra máquina recurriremos a la matematización de este concepto, recordando las características clásicas de esta forma de proceder.
1. Presentarlo en forma abstracta, es decir independiente de su función, de los elementos que la componen, de la energía que utilizan, etc.
2. Debe ser tan general como se pueda. No interesan los casos particulares.
3. Debe admitir un tratamiento lógico-formal riguroso.
Comenzaremos con un ejemplo clásico que representa una máquina y a partir de ella observamos los elementos que conforman el modelo de Máquina de Estado Finito (FSM). El ejemplo concreto es una máquina expendedora de gaseosas [9] que entrega bebida gaseosa previo un pago que se realiza depositando monedas en la misma.
La máquina acepta monedas de $1.00 y $2.00. Cada lata cuesta $5.00 y si se deposita dinero de más lo retorna. Lo modelamos gráficamente en la Fig. 1.

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entrada solo hay una sola arista emergente. Tenemos a los Autómatas Finitos como un formalismo matemático para modelar todo tipo de máquinas, independiente de su construcción, finalidad, material, etc.
El poder expresivo de los autómatas finitos es poderoso. Incorporar esta forma de analizar las máquinas otorga una perspectiva distinta a la tradicional en las carreras de ingeniería y permite un interactuar más fluido en la incorporación de la informática o robótica a la ingeniería. La expendedora entregará la gaseosa solicitada si se cumplen previamente dos requisitos: Primero se depositaron monedas de $ 1.00 y $ 2.00, Segundo la combinación de esas monedas suma la cantidad exacta de $ 5.00. Ésta es la forma correcta de comunicarse con la máquina, el lenguaje que ella entiende.
¿Todo lenguaje puede ser reconocido por un autómata finito determinista? La importancia de esta pregunta es radical y no solo en ingeniería, tomando en cuenta que todas las máquinas construidas hasta la fecha se pueden modelar como autómatas finitos deterministas y que todo problema se puede plantear como el reconocimiento de cierto lenguaje, la pregunta y su respuesta equivale a decir: Dado un problema ¿Es posible

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Fig.2 Autómata finito modelando la máquina expendedora de gaseosa

Fig.3 Autómata finito de pila determinista

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3.1 Algunos ejemplos desarrollados en el laboratorio

El modelado de la máquina expendedora, utilizado como problema introductorio pasa a segundo plano, quedando los autómatas como una perspectiva para interpretar o reinterpretar las máquinas que nos rodean y lo utilizaremos para un segundo problema, introducir el computador como máquina matemática.
Para ello, mostraremos las funciones recursivas como la conexión entre la Máquina de Turing y las matemáticas que fuera señalada en la tesis de Turing-Church.
Nos referimos a éstas como “funciones definidas en forma recursiva”, en lugar de la sintética y poco exacta funciones recursivas.
En una primera aproximación esto sucede cuando en la definición de la función aparece la misma función, sin embargo, a fin de no ingresar en un bucle infinito nos debe conducir hacia una interpretación más simple de ella y que sea calculable, condición de parada.
Se presenta como ejemplo de función definida recursivamente la función factorial y la función exponencial. Advertimos que el

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finitos con pila, deterministas y no deterministas. Una máquina de Turing puede simular un autómata de pila actuando sobre dos cintas. La primera cinta es la estándar de entrada, la segunda simula la pila. Asociaremos la pila a la memoria del computador y recordamos que la administración de la memoria y el tiempo forman parte de la problemática central de la computación real. Supongamos que al programa se le pide calcular el factorial de cuatro f (4), el primer paso será f (4) = 4*f (3), luego f (3) = 3*f (2) y así sucesivamente, como reflejamos en la siguiente tabla.

Tabla 1: La pila como recurso de algoritmo

Disponemos de herramientas computacionales que hacen transparente esta forma de proceder en el ordenador, tal es el caso del lenguaje de programación LISP, el cual ejecutamos en

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Cuando el algoritmo es diseñado para utilizar la estructura de pila acumula una sucesión de pasos hasta llegar al caso base, en ese punto regresa utilizando en cada paso la salida anterior de la función.
La segunda función es la exponencial que al estar definida en reales, no se amolda al tratamiento discreto propio del computador tradicional. En el tratamiento de la misma nos proponemos hacer visible el vínculo entre los autómatas finitos deterministas con pila y las funciones definidas en forma recursiva.
Emplearemos para aproximar esta función el Método de Euler para problemas de valor inicial (PVI), nuestro interés es revelar la estructura recursiva de este método y cómo un autómata

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efectuar las gráficas de las derivadas antes de resolver el problema diferencial que admite una resolución aproximada siguiendo el método de Euler y es factible de ser ejecutado en una máquina tal como la proyectada por el matemático Charles Babbage y la matemática Ada Lovelace.
El hecho de discretizar nos permite que tome el resultado de una etapa anterior y se “realimente” de su salida, lo cual nos sitúa en la cibernética de primer orden. Tomamos la función exponencial ya que su derivada coincide con la función y podemos escribir situándonos a través de un ejemplo simple y conocido en el terreno de lo complejo. El hecho de aproximar no es lo sustancial del método de Euler y los siguientes [11, 12], de hecho la aproximación a través de la discretización en métodos como el Runge- Kutta4 (RK4), nos permite incursionar en el caos [13, 14]. Señalamos como sustancial en estos métodos, su capacidad de traspasar los límites del cálculo.
La rigurosidad de la matemática tradicional no se ve reflejada en este tipo de enfoque, porque lo que se busca es un sustento válido que permita pensar en las interacciones entre matemática y el computador.
Algoritmo 2. Ejemplo de la función exponencial f(x)= e^0.5x,

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Fig.5 Aproximación de la función exponencial por el método de Euler

4 CONCLUSIONES

En una época como la actual, que se caracteriza por la escala, la incertidumbre y las nuevas dimensiones de la complejidad, muchos desafíos requieren soluciones que están fuera del alcance de un único pensamiento disciplinar [15].
Frente a los desafíos que debemos afrontar los docentes en la enseñanza de la Matemática en carreras de Ingenierías no informática, es necesario un cambio de perspectiva que nos

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investigación (2007)

4. Instituto Latinoamericano de la Comunicación Educativa: La ciencia para todos. http://bibliotecadigital.ilce.edu.mx/sites/ciencia/volumen2/ciencia3/088/html/sec_5.html Accedido 2 de Julio de 2015

5. Martínez Dopico, F.: Matemática Computacional: Un nuevo pilar para el desarrollo científico y tecnológico en Matemáticas en la frontera: Nuevas infraestructuras matemáticas en la comunidad de Madrid. Ed. Comunidad de Madrid, Consejería de Educación, Dirección General de Universidades e Investigación. http://www.madrid.org/bvirtual/BVCM001757.pdf Accedido 29 de Junio de 2015

6. Wolfram S.: Programación en ciencias y en Matemáticas Investigaciones y Ciencias N° 98. pp 124-138 (1984)

7. Calegaris, M.; Rodriguez, G.: Simulaciones Computacionales: Autómatas Celulares. Primer congreso sobre Los Métodos numéricos en la enseñanza, la ingeniería y las ciencias- (EMNUS 2010)

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15. Van Der Linde, G.: ¿Por qué es importante la interdisciplinariedad en la educación superior? Cuaderno de pedagogía Universitaria, Año 4. N°8: 11-13. (2007).

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ANÁLISIS DINÁMICO PARA LA MODELIZACIÓN DE SISTEMAS CON FUNCIONES COMPLEJAS Descargar pdf

Alicia M. Tinnirello, Eduardo A. Gago

Laboratorio Informático y Multidisplinar de Ciencias Básicas,
Universidad Tecnológica Nacional, Facultad Regional Rosario
Zeballos 1341
{atinnirello, egago}@frro.utn.edu.ar

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vez más complejos que requieren el uso de software específicos, evidenciando la conexión de conceptos en ciencias básicas, con las herramientas necesarias para el modelado y la simulación en problemas de ingeniería [1].
Una nueva definición del perfil profesional está basada en las competencias que desarrolla el estudiante en el transcurso de su carrera universitaria, situación fuertemente influenciada por las incumbencias requeridas por los organismos responsables de acreditación de los programas de ingeniería que proponen continuas innovaciones [2].
Con el objetivo de llevar adelante estos cambios necesitamos adecuar la currícula universitaria no sólo a nuevas metodologías de trabajo que permitan la estimulación del desarrollo intelectual, sino también que tengan un enfoque multidisciplinar. El objetivo no es enseñar y aprender solamente matemáticas, sino hacerlo enfrentándose con casos reales y sistemas sencillos que conduzcan al alumno a tener una mirada sobre la relación entre las disciplinas que componen su formación profesional.
Para lograr que el proceso de enseñanza sea eficaz, es importante diseñar actividades curriculares basadas en el

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teoría-práctica-aplicaciones y lograr espacios de aprendizaje que habiliten el uso de información numérica y simbólica durante el desarrollo de los temas y reforzar la conceptualización mediante diseños gráficos cuya complejidad requiere de habilidades y capacidad de análisis.
Se pretende que la interacción entre las distintas áreas del conocimiento permita potenciar la capacidad de los estudiantes en la adquisición de conceptos para su consiguiente utilización en los modelos requeridos para la resolución de problemas ingenieriles.

3 METODOLOGÍA

La problemática de fondo de como impartir la enseñanza reside en crear las condiciones para que los esquemas de conocimiento que construye el alumno evolucionen en un sentido determinado. La cuestión clave no reside en si el aprendizaje debe conceder prioridad a los contenidos o a los procesos, sino en asegurarse de que sean significativos y funcionales.
El alumno necesita disponer de conocimientos previos

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conceptos teóricos desarrollados lo que redunda en la conceptualización del tema.

4 SISTEMA DE ESTUDIO: FLUJO DE FLUÍDOS

Las actividades que se describen están orientadas hacia los alumnos de la asignatura Cálculo Avanzado de la carrera Ingeniería Mecánica, y toman como premisa los conocimientos básicos del tema Funciones analíticas de Variable compleja.
Los alumnos desarrollan la experiencia en el Laboratorio Informático y Multidisciplinar de Ciencias Básicas y utilizan como herramienta computacional el software WOLFRAM MATHEMATICA, en su versión 8.0. MATHEMATICA es uno de los programas de matemática más completos para realizar cálculo simbólico, numérico y gráfico; con una cantidad considerable de funciones integradas, y con aplicaciones a distintas disciplinas.
La propuesta que se presenta se puede realizar gracias a las posibilidades tecnológicas de representación y cálculo con las que se cuentan en la actualidad, ya que en un pasado no muy lejano, muchos casos relativos a la Variable compleja eran

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plano, son esencialmente las mismas en todo plano paralelo. Lo que permite confinar el estudio del fluido a un plano simple, el cual se desarrolla en el plano complejo z. Las figuras construidas en este plano se interpretan como secciones transversales de cilindros infinitos perpendiculares al plano.
El flujo es estacionario o uniforme: La velocidad del fluido en un punto depende de la posición y no del tiempo.
Las componentes de la velocidad derivan de una función potencial.
El fluido es incompresible: El fluido tiene densidad constante.
El fluido es no viscoso: El fluido no tiene viscosidad o no tiene fricción interna. Si no hay viscosidad, las fuerzas de presión sobre la superficie son perpendiculares a dicha superficie.
Con las consideraciones realizadas se desarrolla la propuesta dando el significado físico a las funciones utilizadas y su derivada, conectando conceptos e investigando comportamientos [3-5].

4.2 2ª Sesión: Modelización del sistema en estudio

La resolución del modelo matemático que responde al sistema

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de velocidad

De las ecuaciones (1) y (2) se pudo expresar la velocidad del fluido con la ecuación (3)

Derivando el potencial complejo a partir de la ecuación (1) se obtuvo la ecuación (4)

Una condición necesaria para que F sea una función analítica en una región R, es que en R, y cumplan las condiciones expresadas por las ecuaciones de Cauchy-Riemann:

Si las derivadas parciales en la ecuación (5) son continuas en R, entonces las ecuaciones de Cauchy-Riemann son condiciones suficientes para que F sea analítica en R.

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Para poder calcular el módulo de la velocidad del fluido |v| se utiliza la ecuación (10)

Además, si F es analítica, entonces sus componentes y son funciones armónicas en R². Una función se dice armónica en R² si se satisface la ecuación de Laplace. En las ecuaciones (11) y (12) se enuncian las expresiones de la Ecuación de Laplace que deben satisfacer y .

4.3 3ª Sesión: Análisis del potencial de velocidad complejo

Se suministra como dato el Potencial de velocidad complejo F, cuya ley es:

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realizar las representaciones gráficas en el plano y en el espacio asociadas a F, con el propósito de interpretar sus características.
Todas las actividades se desarrollan en el Laboratorio Informático y Multidisciplinar de Ciencias Básicas con la asistencia del cuerpo docente, y se trabaja con las computadoras que se tienen disponibles, y además algunos alumnos, por su comodidad, traen sus computadoras personales.
En el desarrollo de la clase, todos los alumnos trabajan con el programa MATHEMATICA, ya que con dicho programa se realizan los trabajos prácticos de Cálculo Avanzado, y también se han realizado los trabajos prácticos de Laboratorio en primero y segundo año en las asignaturas del área Matemática y en Fundamentos de Informática.
En primera instancia los alumnos con el dato de la ley del Potencial complejo F discriminan la parte real y la parte imaginaria de dicha función, expresión que se enuncia en la ecuación (14) y por medio de ella pueden realizar la gráfica de la Fig. 1.

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Fig.1 Líneas de corriente y curvas equipotenciales de F.

Se citan a continuación el resto de las conclusiones a las que arribaron los alumnos respecto a su trabajo con el Potencial complejo F:
Las líneas equipotenciales están marcadas con línea de puntos y son ortogonales a las líneas de corriente, que están dibujadas con curvas de trazo continuo. Además para este caso observaron que todas las curvas equipotenciales particulares asociadas a la ecuación (13), son asintóticas a las rectas

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los puntos del plano complejo z de coordenadas: y , mientras que para F el único polo simple que posee se localiza en un punto el plano complejo de coordenadas: .
Además concluyeron que el centro del obstáculo de sección plana circular coincide con las coordenadas del polo en F. Entonces si no estuviera el obstáculo, el fluido desaparecería. Dicho punto que presenta esta característica se denomina sumidero de F.

Fig.2 Ceros y polos de F.

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Fig.4 Curvas de nivel para identificar el polo de F.

Los alumnos indagan y verifican además, que hay una condición matemática que relaciona directamente la existencia de un polo con la velocidad del fluido. Dicha condición se enuncia en la ecuación (17), y se cumple cuando existe un polo simple para sistemas físicos cuyos Potenciales de flujo son del tipo como el que fue propuesto en la ecuación (13).
En esta instancia, se puede afirmar que los alumnos no sólo aplican los conceptos inherentes al tema de la Variable Compleja al estudio de la trayectoria de los fluidos, sino que

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Fig.5 Ceros de F.

Las curvas de nivel de la gráfica de la Fig. 6, muestran que el polo y los ceros están en la zona más opaca de la figura. El polo es el punto de intersección entre los dos lóbulos oscuros, y los puntos medios de cada uno de los mismos ubican a los ceros de F.

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Fig.7 Puntos estacionarios de F.

5 CONCLUSIONES Y TRABAJOS FUTUROS

Es interesante destacar como el uso de funciones de variable compleja ha permitido el estudio de sistemas físicos hipotéticos dándole sentido a los aprendizajes y permitiendo que el proceso de enseñanza involucre varias disciplinas favoreciendo la comprensión y fomentando el interés por la Matemática y sus desarrollos.
La importancia de trabajar con diversas fuentes documentales, problemas, guía de estudio, libro de texto, bibliografía asociada,

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REFERENCIAS

1. Tinnirello, A.; Gago, E.; Dádamo, M.: Integración multidisciplinar: Matemática computacional -Mecánica de fluidos. Trabajos III CAIM -2012. Tercer Congreso Argentino de Ingeniería Mecánica. pp. 430-441. (2012)

2. Tinnirello, A.; Gago, E.; Dádamo, M.: Designing Interdisciplinary Interactive Work: Basic Sciences in Engineering Education. The International Journal of Interdisciplinary Social Sciences; Publisher Site: http://www.SocialSciences-Journal.com; Vol. 5, Nº 3, pp. 331-334. (2010)

3. Bird, R; Stewart, W.; Ligthfoot, E.: Distribuciones de velocidad con más de una variable independiente: Fenómenos de Transporte; Editorial Limusa Wiley; pp. 141-150 (2012)

4. Mott, R.: La naturaleza de los fluidos y el estudio de su mecánica: Mecánica de Fluidos; Editorial Prentice Hall; pp. 10-51 (2006)

5. Cengel, J.; Cimbala, J.: Mecánica de Fluidos: Fundamentos y aplicaciones; Editorial McGraw-Hill. pp. 136-158. (2006).

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DESIGN AND SIMULATION OF MECHANICAL EQUIPMENT BY DESIGN TOOLS AND MULTIPHYSICS PLATFORMS Descargar pdf

Alicia Tinnirello^1,2, Eduardo Gago¹, Mariano Valentini<sup>1

1</sup> Universidad Tecnológica Nacional Facultad Regional Rosario (ARGENTINA)
² Universidad Nacional de Rosario Facultad de Arquitectura, Planeamiento y Diseño (ARGENTINA)

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1 INTRODUCTION

The new challenges in Engineering Education, require a

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working first with a design software CAD (Computer Aided Design) and then with a simulation software multiphysics that applies the technology CFD (Computer Fluid Dynamics) COMSOL.

2 METHODOLOGY

It is designed two prototypes of condensers whose difference is the incorporation of a plate of shock to one of them, which prevents fluid press directly on the tubes. By analyzing and comparing both prototypes using the graphics that generates the program succeeds in establishing the advantages in the variation of the design.
The analysis is done by considering the following stages: design of the geometry with a CAD software, the import of the geometry designed to a CFD simulation software, definition of the parameters for the entry of the fluid: velocity, pressure, temperature, type of fluid moist air, it then sets out the physics of the problem, in this case not isothermal flow, selection and generation of one type of mesh in the geometry, resolution of the model with the survey module of the software and

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2.2 Imported Geometry

Fig.2 Geometry Recognition in COMSOL

The CAD Import Module contains a package to import geometry in the COMSOL Multiphysics software from SolidWorks interfaces.
Once the design phase in CAD geometry submitted to multiphysics analysis, it imports to COMSOL (Fig. 2) by applying the "geometry" command. In addition, multiphysics platform modules have the advantage of using an option to correct any

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2.4 Equations for flow non-isothermal

The specification of a physical or well the coupling of several physical, turns out to be an important advantage afforded by the technology that applies. Each of the physical which includes the program, are bounded by a system of equations that allow you to perform the analysis in the model. You can also modify the equations which are defined in the program, either manually or with the help of a specific module.
The analysis of the relationship of the mechanisms of transport is achieved from a system of equations formed by the Navier-Stokes equations and the continuity equation for isothermal flow not steady-state.

Where: : Density, : Velocity flow. : Dynamic viscosity. : Identity matrix. : Temperature. : Constant.

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through the tubes directly with the velocity of the same income (Fig. 3).
Prototype 2: In this case the fluid enters the condenser first hits the plate and then passes through the tubes. Thus avoiding the direct impact of the drops on the tubes (Fig. 4).

Fig.3. Prototype design 1. No strike plate.

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considering the incompressible flow, and moreover ignoring the load losses. The tube temperature is 285 K (12 C) and the inlet temperature of the steam is 373 K (100 C). The mixture of moist air pipeline emerges from the bottom, at a pressure of 0 Pa (relative pressure) and the water is discharged in a tank of a reservoir at atmospheric pressure.
The impingement plate design is proposed to prevent the mechanical erosion of the tubes, analyzes the pressure on them and the outlet temperatures in both prototypes.
The walls and the intermediate plate of the equipment are considered adiabatic as and is not the subject of this analysis the condensation of the fluid, therefore, is taken at the fluid without latent heat and not be regarded as the percentage of moisture in the fluid and these parameters objects in a later study, and more deeper.

2.7 Meshing

Meshing Fig. 5 it is achieved with the "mesh" function of CFD simulation software. This part of the process is performed after setting the multiphysics working conditions with the need

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3 RESULTS

3.1 Prototype 1

The results obtained in the analysis of the prototype 1 are below:
In the graph in Fig. 6 are seen cutting patterns that realize the magnitude of the velocity of fluid within the unit.

Fig.6 Plans fluid velocity. (a) To 0.5 m / s. (b) To 1 m / s.

In the 3D graphics in Fig. 7 below shows how the particles of humid air that enter the condenser through the tubes hitting against them.

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Fig.8 Contours of pressure. (a) To 0.5 m / s. (b) To 1 m / s.

In the 3D graphics in Fig. shows the temperatures inside the condenser. It can be seen that are no significant changes in the temperature gradient before a considerable variation in the velocity.

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Fig.10 Plans fluid velocity. (a) To 0.5 m / s. (b) To 1 m / s.

In the 3D graphics in Fig. 11 shows the path that the particles of moist air within the condenser and the sector where the tubes are affected by the clash of the particles. In this case first the fluid collides with the plate and then passes through the tubes.

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In the 3D graphics in Fig. 12 shows the range of variation of the pressure exerted by the air-water mixture on the plate, mix colliding particles of water is observed. It is clearly seen as the fluid first hits the board and then against the tubes in the back, causing future sources of mechanical erosion that produced the consistent corrosion. The affected area is the same for both velocities of the fluid. In the 3D plot of Fig. 13 fluid temperatures are displayed within the condenser with the striking plate.

Fig.13 3D Temperature. (a) To 0.5 m / s. (b) To 1 m / s.

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generates lower temperature exchange, a higher velocity and lower density to the output.

4 CONCLUSIONS

From the results obtained is observed as influences the direct impact of the fluid flow and the velocity of the same on the tubes of the heat exchanger, to simulate the equipment by placing an intermediate plate shock wasting the mechanical erosion flow directly from the tubes by direct contact of the particles of water-air.

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3. Cengel, J., Cimbala, J. (2006). Mecánica de Fluidos: Fundamentos y aplicaciones. Edit. McGraw- Hill, 2ª Ed, pp. 471-502.

4. Comsol. Introduction to CFD (2011). Módulo, versión 4.2a.

5. Raviculé, M., Mocciaro, C., Ramajo, D., Nigro, N. (2012). Aplicaciones de fluido dinámica computacional en la industria del petróleo desarrolladas en la dirección de tecnología de YPF. Mecánica Computacional Vol. XXXI, pp. 3715-3739.

6. Panton, R. (2013). Incompressible flow. Edit. Wiley. 4th edition, pp. 220-257.

7. Frank, W. (2008). Mecánica de Fluídos. Edit. Mc Graw Hill 6ª edic, pp.601-646.

8. Patil, S., Kostic, M., Majumdar, P (2009). Computational fluid dynamics simulation of openchannel flows over bridge-decks under various flooding conditions. Proceedings of the 6th WSEAS International Conference on fluid mechanics (FLUIDS'09), pp 114-120.

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SYSTEMS ANALYSIS AND MODELLING TECHNIQUES IN PHYSICAL DOMAINS Descargar pdf

Mónica Dádamo, Lucas D’Alessandro, Alicia Tinnirello, Eduardo Gago

Computer Basic Sciences Laboratory
Universidad Tecnológica Nacional – Facultad Regional Rosario
Zeballos 1341, Rosario, Santa Fe
ARGENTINA
atinnirello@frro.utn.edu.ar, egago@frro.utn.edu.ar

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This new approach includes the concepts of causation, retroaction and ports, as well as the emergent vision of programming paradigm focused on objects in order to create bases that allow us choosing between different resolution techniques, also adding those concepts that enable the resolution of systems with more than one degree of freedom, non-linear, with multiple inputs and outputs. Linear time invariant dynamical systems are gradually developed to study the model that represents the early feedback stage in simple description of physical phenomena. This approach facilitates software engineering professionals when designing systems with controllers, where the feedback is considered a main feature [1].
The advantages of the Bond Graph method application will be demonstrated; it uses physical concepts and takes advantage of the fact that energy provides a system the dynamic of its performance, and this is a direct consequence of the energy exchange between the components of a system. BG only uses two types of variables, named conjugated variables of power: effort (e) and flow (f), both defining the interaction of power, remaining that the result of them is power.

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consider one effective way of dealing with subject of study. It is connected directly with the comprehension of the topic. In complex systems, the relation between the operational knowledge and the justification and explanatory areas is potentially more diffuse. We can notice one example in the analysis of the simple dynamic systems such us a mass-spring system (M_S) either an electrical system R-L-C: In these cases, it is easy to get the model through an Ordinary Differential Equation (EDO) raising its constitutive and structural relations Fig.1.

Fig.1 Dynamic systems and EDOs

An order n EDO can be rewritten as n EDOs order 1 system,

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- To get the EDO, from the consecutive and structural relations, it is necessary algebraic effort.
- The obtained models have not got visual similarities with the originals schedules. That is why to obtain continuous systems models it is advisable using graphical languages with Diagram Blocks (DBs). The DBs simplify the algebraic manipulation of constitutive and structural relations.
Then, we consider two systems separately, a CC engine (Fig.2a) and a pulley (Fig.2b), and their respective models are made with DBs (Fig.3d). In the event of couple mechanically the systems (Fig.2c): Is it possible to obtain DB of the new system coupling the partial DBs of the engine and pulley?

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Fig.2 Dynamic systems, a) cc engine, b) pulley, c) engine + pulley, d) modelling with DBs

In Fig.2d is shown it is not possible, since the variable (t), which links the systems, appears as input in both DBs.
In addition to establishing mathematic relations, the DBs establish causal relations between the variables. For this reason, each sub-system sees the interaction variables as ‘input –

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4 DEVELOPMENTS

In engineering, the modelling of systems energetic performance involve the breakdown of a system in elementary conceptual sub-systems, which represent basic physical processes and determinate the system performance from the energetic point of view. This basic physical process can be classified as: storage processes, transporting and distributing, dissipation, and exchange of energy.
The final product of the breakdown could be represented with a connecting graphs diagram. On this way, a priori the system for modelling is intended as a net which interconnects basic elements. These elements, storage, dissipate or transform the energy. When these interact with each other, the product is a bidirectional and non – causal flow. The components of the system interact with each other determining the dynamic of the system. In the same line of though, it is considered one fundamental fact when two systems (S1,S2) or elements interact, none of them can impose both variables. If one of them sends an effort signal (power, voltage) the other one answers with a flow signal (speed, current); in inverse, if one of them

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Table 1: Variables of effort, flow, moment and displacement in different physical domain

The physical meaning of the effort and flow variables will depend on the physical domain where the system of study be established. Apart from the mentioned variables, the method

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between the effort and flow in which there is energy loss. These elements can represent electric resistances, shock absorbers, friction efforts, head loss in pipes, etc.
Capacitive elements K: They can be modelled as flow stored elements. These elements are capable of storing energy and return it to the system without losses. A capacitor can represent springs, torsion bar, electric capacitors, hydro pneumatic accumulators, etc. Contrary to resistive elements, the capacitive ones relate the effort associated to the graph with the displacement produced. Whereas the displacement is defined as the flow integral in the time, in the capacitive elements:

Inertial or inductive elements L: They can be modeled as effort stored elements. These elements are capable of storing energy and return it to the system without losses and they relate the flow with the P variable defined as the effort integral in the time. The condition is met:

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mechanical systems that share the same effort, electrical systems that share the same tension, etc. Union 1: It is also called serial connection. All graphs connected have the same flow and the product of the sum of the efforts is zero. These connections represent mechanical systems that share the same speed, electrical systems that share the current, etc.

Fig.6 Representation in BG method of flow elements: a) of effort b) of flow.

Source elements: They provide power to the system. This power can be provided through a flow or an effort, in such a way there will be two kinds of sources: known effort (excitation force in mechanical systems, voltage generators in electrical systems, etc.) and known flow sources (electrical current sources, hydraulic pump as flow rate source, etc.). Fig.6 shows the flow representation in BG.
Transforming elements: They transmit power variables with a

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Fig.8 Modeling of systems with BG notation. a) M_S b) RCL combined c) pulley

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Fig.10 BG modelling of CC engine + pulley.

Looking at the shaded area in the BG of Fig.10 and assuming the non-casual nature of BG, it is possible to couple directly both systems joining the two links which have 𝜏(𝑡) as a variable, in this way to get the complete model of the system. It is shown in Fig.11.

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Fig.12 Causality between two elements.

One advantage of BG is there are a lot of procedures to get the causality assignation, for this reason this task is reduced to a sequential and simple action, as long as the ‘causal dependence’ of each one of the basic elements below be respected. These elements are classified by the causal assignation point of view, as follow: Fixed Causality: They are the flow source ones (causality flow) and effort source (causality effort). By definition, they provide information of one kind or another.
Indistinct causality: Resistive elements.
Preferred causality: They are energy stored elements (capacitive or inductive) which can have integral or differential causality in order to the relation between the flow and the effort. Integral causality is more preferred than differential causality because of the integration is a more efficient computational procedure.
Multi ports with restricted causality: Transformers and unions. It

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5 CONCLUSIONES

This analysis focus on trying separately the resolution technique, as the case may be: mathematical, numerical or computational, so that the knowledge acquired does not depend on outdated methods or technologies. We want to highlight the portability of the model; that the same pattern of analysis may be applicate regardless of the character of the system (mechanical, electrical, hydraulic, etc.). This new approach about system analysis makes it possible to simplify the variables in complex systems with different components. It breaks with the ‘unique model of thought’ and prepares the future engineers in the management of an actual complexity, this day, unavoidable.

REFERENCES

1. A. Tinnirello, M. Dádamo, E. Gago, Integration techniques in mathematical application projects for mechanical engineering, 6th International Technology, Education and Development Conference, 2012, pp. 1844-1850.

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Simulation of Hybrid Systems: A Hybrid Bond Graph Approach, Simulation: Transactions of the Society for Modeling and Simulation International, Vol. LXXXVII, No.6, pp. 467-498, 2010.

8. V. Damic, J. Montgomery, Mechatronics by Proceedings of the European Computing Conference Bond Graphs, Springer, 2003.

9. I. Roychoudhury, G. Biswas, X. Koutsoukos, Using Factored Bond Graphs for Distributed Diagnosis of Complex Systems, Proceedings of the 9th International Conference on Bond Graph Modeling and Simulation, 2010, pp. 11- 18.

10. W. Borutzky, Bond Graph Modelling of Engineering Systems. Theory, Applications and Software Support, Springer, 2010.

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Ingeniería Matemática: Modelado y Simulación de Sistemas Complejos en
el contexto de las Tecnologías Básicas y Aplicadas

Homologado por la Secretaría de Ciencia y Tecnología bajo la identificación Nº UTI3793TC

Fecha de inicio: 01/05/2016
Fecha de finalización: 30/04/2019

IJPS 2016

ANALYSIS AND PREDICTION OF ELECTRIC FIELD INTENSITY AND POTENTIAL DISTRIBUTION ALONG INSULTOR STRINGS BY USING 3D MODELS Descargar pdf

Alicia Tinnirello, Eduardo Gago, Lucas D’Alessandro, Mariano Valentini,

Computer Basic Sciences Laboratory
Universidad Tecnológica Nacional – Facultad Regional Rosario
Zeballos 1341, Rosario, Santa Fe
ARGENTINA
atinnirello@frro.utn.edu.ar, eagago@gmail.com

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with the chains (cables, brackets, accessories), transient efforts as lightning, operation switches, contamination in the chains, wind temperature changes, additional charge by presence of ice, rain, etc.
All of these factors affect the normal performance of the lines leading to the emergence of phenomena such as corona discharge and partial discharge. These phenomena produce consequences such as loss of power, generation of light, audible noise and interference, vibration, deterioration of materials, generation of ozone, nitrogen oxides and moisture between others and, if reaches certain importance, produces corrosion in drivers because of the formed acid [1].
Currently both the University and the professional scope of electrical engineering the study of these phenomena is carried out by analytical calculations based on empirical methods. These calculations allow us to determine the potential gradient for which appears ionization on the conductor surface called surface critical gradient under certain operating conditions from which it is possible to calculate the losses and other consequences due to corona discharge or verify the levels of tension and line conditions, but is not analyzed the behavior

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transformer since 1912. Diameters of wires analyzed by Peek experimentally were from several millimeters to an inch [2]. Subsequently, a large number of researchers is interested by this phenomenon, among them L. B. Loeb who headed one of them groups of research that more have contributed to the knowledge of the corona discharge. Loeb published in 1965 the book entitled "Electrical coronas" which is still considered a very important reference work [1]. The works and research oriented the analysis of the field electric in insulators of lines electric, are these of glass tempered, porcelain or synthetic through numerical methods due to [3], who define a method to determine an electrical field of low frequency in standard contaminated insulators, whereas models axisymmetric are considered and applying finite element method.
Rasolonjanahary analyzes the three-dimensional behavior of insulators contaminated using the numeric method of border elements establishing a theoretical formulation and comparing the results with obtained in analytical and experimental way [4].
Due to the continuous advancement in computer systems, can be currently studies extremely complex for electromagnetic phenomena using experimental and numerical techniques. The

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Fig.1 Porcelain electric insulator

Strings made of insulators are used in high and medium voltage by a variables number of them dependent from voltage operating (Fig.2).

Fig.2 Insulator stringsr

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gradient is a function of the conductor diameter , the condition of its surface and the relative density of the air which in turn depends on the pressure and temperature . Peek, obtained empirically formula that is the most used for the calculation of the critical gradient in cylindrical conductors. The critical gradient "Ec", kV tip/cm, is expressed as:

Where
: disruptive critical gradient of air 29.8 kVpunta / cm.
: conductor radio, en cm.
: relative air density
: state superficial coefficient

Where P air pressure, en mm Hg, and T air temperature, in °C. The analysis and evaluation of the models was conducted using the computational tool Comsol Multiphysics. This is a package of software analysis and resolution by e.m.f for various

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Fig.3 a) Geometric design in CAD

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implemented in a computer. [8]. The main advantages that brings the use of this technology are: the electric field prediction, the design and protothype system avoiding experiments of high cost, the visualization and the animated system in the variables terms modifying the components properties and the environment.

3.3 Meshing Sequences

To solve the proposed moddels we must make a mesh geometry. The meshing of Fig. 4 is achieved with the function "mesh" CFD simulation software, this part of the process is of utmost importance and it is always done after adding and defining multifisics conditions, it is here in which defines the type of mesh necessary for the analysis by finite element.

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insulators on a line of 33kV corresponding to those used for the distribution of energy in medium voltage in Argentina.

Fig.5 Electric field distribution in porcelain insulator

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Fig.7 Distribution of electric field in insulator of ethyl propylene.

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Fig.9 Flow lines of electric field in porcelain insulator.

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analyzed, Fig. 11 and 12. One of the alternatives to reduce the electric field in this area is to add a semi-conducting screen between conductor and insulator. The Fig. 13 and 14 shows the evaluation of such alternative.

Fig.11 Level surface of Electrical field in length of arc for different materials

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Fig.13 Level surface of Electrical field in length of arc for different materials with semiconductor plate.

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Fig.15 Level surface of Electric field distribution for damaged string insulator

Fig.16 Electric field distribution for damaged string insulator

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Fig.18 Distribution of electric field for the chain of insulators with wet surface.

5 MODELING RESULTS AND DISCUSSION

Use materials of low permittivity, ethyl-propylene case with

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of the outputs of services from electrical power lines.
The presence of water on the surface of the insulator generates a reduction of the dielectric strength of the same with an electric field more intense in a damp insulator that being dry as shown in Fig. 11. On the other hand, the products of rain moisture favors the appearance of disruptive discharges through the air due to the ionization of the same reason of humidity. A layer of humidity on the insulator implies an increase in the conductivity of the same favoring the emergence of currents that outlined the outside of it.

4 CONCLUSION

The corona and partial discharges occur in non-uniform fields, in areas with large field intensities and is favored by the presence of humidity or contamination in the insulators, so these should be issues to take into account in the design of the chains according to the conditions in which it will operate the same.
These discharges in an insulating material usually begin in hollow filled with gas within the dielectric. After a partial

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3. Asenjo, E., Morales, N., Valdenegro, A., Solution of low frequency complex fields in polluted insulators by means of the Finite Element Method. Journals and Magazines IEEE Transactions on Dielectrics and Electrical Insulation, Vol. 4, No. 1, 1997, pp. 10-16.

4. J. Rasolonjanahary, Computation of Electric Fields and Potential on Polluted Insulators using a Boundary Element Method. Journals and Magazines IEEE Transactions on Magnetics. Vol. 28, No. 2, 1992, pp. 1473-1476.

5. Zhao, T., Comber, G., Calculation of Electric Field and Potential Distribution along Nonceramic Insulators Considering the Effects of Conductors and Transmission Towers. Journals and Magazines IEEE Transactions on Power Delivery, Vol. 15, No. 1, 2000, pp. 313-318.

6. K Kiousis, A. Moronis, E. Fylladitakis, Finite Element Analysis Method for Detection of the Corona Discharge Inception Voltage in a Wire-Cylinder Arrangement, Recent Advances in Finite Differences and Applied & Computational Mathematics, WSEAS Press, 2013.

7. K. Kantouna G. Fotis et al., Analysis of a Cylinder-Wire-Cylinder Electrode Configuration during Corona Discharge, Latest Advances in

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IJCE 2016

DIAGNOSIS OF ROTOR FAILURES CURRENT POWER INDUCTION MOTORS BY SPECTRAL ANALYSIS METHODS Descargar pdf

Alicia Tinnirello, Eduardo Gago, Lucas D’Alessandro, PaolaA Szekieta

Computer Basic Sciences Laboratory
Universidad Tecnológica Nacional – Facultad Regional Rosario
Zeballos 1341, Rosario, Santa Fe
ARGENTINA
atinnirello@frro.utn.edu.ar, eagago@gmail.com

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1 INTRODUCTION

Nowadays, as many of the tasks that a modern industry must carry out are performed by induction motors, induction motors have become the core of most common industrial processes.
In industrialized countries between 40% and 50% of all energy produced is used for these machines [1]. As the market for these engines has grown to have control and continuous and efficient

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and the implementation of these maintenance methods. Failure statistics report that the engine components that tend to fail more often are: Stator (38%), Rotor (10%), Bearings (40%), Others (12%), [3] Broken bars in the rotor represent 10% of the failures of a motor. Depending on its severity, failures can range from malfunction to an engine stop/halt, resulting in large financial losses for businesses due to unexpected changes in strikes production. There are various techniques and methodologies for detecting engine failure as for example, the impedance negative sequences, the analysis of the frequency spectrum, the vector Perk, the Hilbert transform, the analysis of the electrical signature, circuital engine analysis, vibration analysis among others [4]. Among the variables used to check the condition of a component or equipment are vibration, temperature, voltage, current, and oil level and insulation resistance.
This paper presents the analysis and implementation of an alternative method for the detection and diagnosis of fault rupture bars in the squirrel cage induction motors using spectral analysis of the feed stream known as MCSA (Motor Current Signature Analysis). The work aims to create a system of

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promptly in the task of signal analysis and subsequent diagnosis, leaving aside the proper signal acquisition as shown in the Fig. 1.

Fig.1 Global scheme for the use of MCSA

It is based on a study and implementation using LabVIEW software platform method based in applying the Fourier Transform for spectral analysis of the feed stream called induction motors MCSA.

2.1 MCSA foundation

For analysis of the effect of the broken bars on the supply current (stator current) of an induction motor we will use the

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Suppose the rotor rotating under load at a certain speed n in r.p.m. The speed difference between the field and the rotor is the relative rotational speed with which the field lines intersect the rotor conductors and under this velocity difference are induced in the rotor winding e.m.f. and currents of a frequency f₂ expressing in equation (1).

The rotor polyphase currents in turn create a rotary field at a speed , in relation to the rotor in question and in the same direction as the stator field following the inductive sequence which it comes from. It is shown that relative to the stator, the rotor rotates at the speed field as:

That is, it rotates at the synchronous speed, regardless of the rotor’s itself. Fields, stator and rotor remain stationary for one another and could be combined into a single rotating field that ultimately is left as resulting in the machine. It is important to

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delay, which turns sliding speed, this is Sn1 respect to the rotor [5]. With the presence of this field a stationary observer in the stator windings a rotating field will observe a resultant speed n_r defined:

It clearing of (5) the motor speed n in replacing (8) is obtained:

Multiplying both sides of (9) by the number of pairs of poles 𝑝 and considering the expression (1) is obtained:

As the rotating magnetic field frequency f_r short the stator windings, is induced in them an e.m.f. and hence a current with the same frequency of the rotating field, called f_nr, since it corresponds to the frequency of the current through the stator windings as well:

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sidebands around the fundamental frequency in an induction motor without failure is normal, but the amplitude of these sidebands are intensified with the broken bars and precisely is the amplitude ratio between the components critical current and these sidebands those considered for diagnosis.
Table I shows the criteria to diagnose broken bars in induction motors using MCSA [6].

Table 1 Fault diagnosis for MCSA

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sufficient for the existence of equation (14), which is a corrected form of the equation of Dirichlet, and posed as f must:

- be absolutely integrable in , or is convergent.
- have a number finite of maximum and minimum and be continuous by sections. In addition, equation (14), can be written:

To the expression defined in (15), it is called Fourier transform of f and provides a representation in the domain of the frequency of a function not periodic f[7]. FThas been approached from the format, closer to their use in methods and computational algorithms discrete signal; whose formulation is presented in equation (16).

Where n=0,1,2...(N-1) and 𝑁 is the number of samples of the window to be analyzed,Tis the sampling period (inverse of

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2.3 Frequency analysis implemented with LabVIEW

LabVIEW is a graphical programming software created by National Instruments and it has integrated functions for data acquisition, instrument control, measurement analysis and data presentation. The main feature of this software is its ease of use, since it has an extensive library of powerful tools to create relatively complex applications without writing code lines of text. Programs developed with LabVIEW are called Virtual Instruments (VIs) which can be used in turn as building blocks for more complex programs. Each VI consists of two parts: the front panel, which is the interface used to interact with the user when the program is running and the block diagram, which is the program itself and where the icons are placed that interconnect with each other and perform the functions established [10].
Power Expectrum.vi the FFT Power Measurements Waveform tool corresponding to the library Signal Processing shown in Fig. 2 was used to implement the FFT algorithm.

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Fig.3 Front panel

Through the control call signal to be analyzed it can choose the signal source to study, with the options: Simulated corresponding to a signal generated by the same system through appropriate controls, Stored in disco corresponding to a signal that has been taken offline and stored in a file, and finally Acquired from, or Acquired from abroad which refers to

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Fig.5 Options control signal to be analyzed

Since our interest is focused on signal analysis, rather than technical data acquisition and considering choosing LabVIEW as a deployment platform, the acquisition and signal conditioning, our tests were carried out with signs stored on disk. Note that the system is able to process signals from the outside once it has the corresponding procuring plate.

3 ANALYSIS AND DISCUSSION

The vibrations in a machinery are directly related with the useful

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Fig. 6 and Fig. 7 shows the results of testing a damaged motor rotor

Fig.6 Time response of the motor current damaged

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Fig.8 Motor test report damaged

Failures should be detected when they still do not significantly affect the machine so as to perform in this way the corresponding preventive maintenance. Otherwise, the fault would affect significantly the operation of the engine at the moment of its identification, resorting to the model of

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Fig.10 Frequency spectrum of the current healthy motor

Fig. 11 corresponds to the report of the frequency response in the range of interest and diagnosis of the engine as the view expressed in table I.

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It is recommended that the fault diagnosis is made with the engine close to the rated load.
The amplitude of the side harmonics is proportional to the degree of fault of cage motor, so the difference in dB between the magnitudes of fundamental and harmonics decrease as more broken rotor bars exist in the cage.
Broken rotor bars on the motor creates two lateral bands on current’s frequency spectrum, located on doubles frequencies of the frequency slide (2sf₁) among fundamental frecuency f₁.
Analyzing voltage signals in addition to the current, the method can diagnose failures both electrical, and mechanical among which are: broken rotor bars breaking rings rotor shorts the stator coils, rotor eccentricity, bearing failures, among other. The possibility along with the implementation of the signal acquisition stage is presented on the horizon as a continuation of this work in the future.
MCSA requires a frequency analysis, and having digitalized and filtered signals, it must calculate a FDT ( Fourier discrete transform), with the purpose to optimize the computational process, it uses the FFT (Fourier fast transform) which is an algorithm than allow to calculate the FDT faster.

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the engine periodically in order to determine the evolution of the motor.
This technique can be fairly simply, or complicated, depending on the system available for data collection and evaluation. MCSA technology can be used in conjunction with other technologies, such as motor circuit analysis, in order to provide a complete overview of motor system health.
Using LabVIEW technology allows the implementation to be user friendly and facilitate the collection, processing, data storage and the ability to generate reports and statistics in a simple way.

REFERENCES

1. W. Thomson, M. Fenger, Current Signature Analysis to detect induction motor faults, IEEE Industry Applications Magazine, Vol.7, No.4, 2001, pp. 26-34.

2. D. Raheja, J. Llinas, R. Nagi, Data fusion: Data mining based on architecture for condition based maintenance, International Journal of Production Research, Vol.44, No.14, 2006, pp. 2869-2887.

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2002.

9. H.Kwakernaak, R. Sivan, Modern Signals and Systems, Prentice Hall, 2000.

10. V. Lajara, S. Pelegrí, LabVIEW: Entorno gráfico de programación, Alfaomega, 2006.

11. W. Thomson, R. Gilmore, Motor Current Signature Analysis to Detect Faults in Induction Motor Drives-Fundamentals, Data Interpretation, and Industrial Case Histories, Proceedings of 32nd Turbomachinery Symposium, 2003, pp. 145-156.

12. N. Mehala, R. Dahiya, Motor Current Signature Analysis and its Applications in Induction Motor Fault Diagnosis, International Journal of Systems applications, Engineering & Development, Vol.2, No.1, 2007, pp. 29-35.

13. M. Castelli, J. Fossati, M. Andrade, New methodology to faults detection in induction motors via MCSA, Transmission and Distribution Conference and Exposition: Latin America, 2008 IEEE/PES, August 2008,pp. 1-6

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EDUCATIONAL TECHNOLOGY INTERVENTION FOR THE DEVELOPMENT OF ADVANCED CALCULUS APPLICATIONS Descargar pdf

Alicia Tinnirello, Eduardo Gago, Mónica Dádamo

Universidad Tecnológica Nacional (ARGENTINA)

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It is noteworthy that students are encouraged when they leave aside traditional forms of paradigmtype problem solving, and when they are imposed to a significant and reflexive learning dynamic that involves engineering situations which ignore idealism and are related to the ones to be carried out in practice. Sometimes it is observed that the students do not follow abstract and general processes, or find it difficult to adapt to them. This is why the trend in contemporary education needs the implementation of a system that places the student in centred processes that identify him as an active and reflective learning subject.
Project based learning has proved to be a suitable method to demonstrate the need of mathematics in professional engineering. Students are confronted, complementary to their regular courses, with problems that are of a multidisciplinary nature and demand a certain degree of mathematical proficiency.
The curricula of the Mechanical Engineering programs at our university include Advance Calculus at the third year of the engineering studies; the authors’ experience is that students increase their interest and their appreciation for the contents

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is linked in direct way with the understanding of the topic. When complex systems are studied, the relationship between the operational knowledge and the explanatory and justification areas is potentially more diffuse. An example of this is observed in the study of the SLI, in which through its calculation the learning of the following themes is jointly introduced: Laplace transform, Fourier transform, transfer matrix, convolution, and so on.
One of the not desired outcomes of this conjunction of elements owned to SLI is considered to the respective transfer function (FT) as obtained by means of the Laplace transform, when the input is the unit impulse.
If we focus on the properties inherent to the linear system, highlighting that the FT is characterized by the linearity of the differential operator, that describes the model, the constant coefficients described a SLI system, so that it is selected the "side" that corresponds to the system, separating it from the input function.
With a SLI of the obtained solutions, an analytical approach by means of differential equations can link with the ones obtained from an systemic approach, with simulation systems, by means

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ones, but also those of recent development with different techniques of analysis and mathematical models involved, a systemic approach is presented, - key component in the professional software for modeling and simulation-, providing new strategies in the study, through the use of virtual platforms for modelling and simulation of dynamic systems.
In the undergraduate degree in engineering, feedback is treated in mathematical studies of SLI systems, and usually associated with programs that implement it for the treatment of complex systems, with little or no mention of the systemic approach.
In this presentation the concept of feedback shown efficient and compatible management of mechanics Newtonian. Although the form of relating it shows a rupture with the thoroughness traditional of the analytical approach, is consistent with the paradigm of the complexity, where the reasoning analog, added to it effectiveness verifiable, is considered valid.
In a simple way, we mention the systemic approach in early stages, which is of scarce academic treatment currently and of intensive use in the engineering professional career.

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aspect.
This possibility not only gives place to the analysis in different conditions but also allows observing instant to instant the transition between states, which contributes considerably in the understanding of topics associated to dynamic issues. This methodology has acquired special relevance since it is essential part of any control system.

4 PRELIMINARY CONCEPTS

In order to build a meaningful learning, both the basics of Laplace's transform and its application to the resolution of systems using differential equations as an introduction to the algebra of blocks and rudimentary knowledge of programming in the graphic language, are needed (in this work, LabVIEW).
It requires also a family management of the Fourier transform and the analysis in the time and frequency domain. These concepts will give the student a satisfactory view of the current state of the science and technology development linked to the continuous SLI, with the exception of recent systemic studies, exceeding the objectives of advanced basic training.

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An 'order n EDO' can be rewritten as 'n EDOs order 1 system', named States Equations. On this way, it is defined for the equation (1) the system is

Where are called state variables. In circuit RCL, defining , we get the system:

Where is the state variable

Even though the continuous systems are represented through differential equations models, it is very difficult to obtain the

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quadratic [2]. The students work analyzing the different possibilities for the response, using the software LabVIEW, of simple and practically intuitive way, is develops the structure graphic that models the system oriented to the analysis of the nature of these roots [3].

Fig.2 Underdamped response

These diagrams allow us to analyze all the states that adopt the

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implies there are no oscillations.

Fig.3 Critically damped system

The relationship of the oscillation of the transient response can observe as the complex conjugate poles are moving towards higher values of imaginary component and less than real component, to the extent of becoming totally oscillatory for pure imaginary poles.

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The Fig. 4 shows complex conjugate poles with little damping response with more oscillations, in this case the real negative part of poles is small. The system is called critically stable and the response has sustained oscillations.
The sliding controls allow to modify the system parameters and to observe at the same time its evolution in the graphic, facilitating the teaching and complementing the explanation of themes as: systems mathematical modeling, limitations of operating in the domain of time with differential equations, giving rise to a new domain using the Laplace transform and other related topics.
The application allows to move the poles and to locate the place of roots showing both, the locations of closed loop poles and the natural frequency, damping coefficient and response to the step of the system. Alternatively, we can specify the real and imaginary components directly in the legend below the graph, the advantages that this graphs provides for teaching are evident, showing the relationship between the parameters displayed and its influence in the temporary response.

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Fig.6 System with wide range of phase and gain, oscillatory and slow response

By the observation of the effect of different types of system’s drivers in the analysis and errors calculation in stationary state, and the response of the step and ramp input at the same time, it is easy to display as such changes affect the frequency response by means of bode diagrams and the corresponding gain and phase margins [6].This is very useful as it allows the student to demonstrate graphically how the effects of a modification in a controlled system manifest themselves differently depending on the domain in which they are being analyzed, whether it’s the time response and/or the frequency

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Fig.8 System unstable

From Fig. 6, 7 and 8, we can observe that the margin of profit is positive and the system is stable if the magnitude, to the crossing of phase of-180 °, is negative (in db). I.e., the axis 0 db, will be negative and an unstable system. The phase margin is positive and the system is stable if the phase is greater than 180 ° at the junction of 0 dB gain. I.e., the margin of phase is measured above the shaft-180 °. If the margin of phase is measured down the shaft-180 °, the margin of phase is negative, and the system is unstable.

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ACKNOWLEDGEMENTS

The authors would like to express their recognition to the student Lucas D’Alessandro for his performance during the development of his project work.

REFERENCES

1. K. Astrom, R. Murray, “Feedback Systems: An Introduction for Scientists and Engineers”, Princeton University Press, pp. 201-226, 2009.

2. A. Batatunde, A. Ogunnaike, W. Harmon Ray, “Process Dynamics, Modeling, and Control” Oxford University Press, pp. 430-459, 1994.

3. V. Lajara, S. Pelegrí, “LabVIEW: Entorno grafico de programación”, Edit. Alfaomega, Madrid, pp. 175-185, 2006.

4. G. James, et al, “Matemáticas avanzadas para ingeniería”, Edit. Prentice HALL, México, pp. 178-202, 2002.

5. K. Ogata, “Ingeniería de control moderna”, Edit. Pearson, Madrid,

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VIRTUAL INSTRUMENTS INTEGRATING MATHEMATICAL MODELING FOR ENGINEERING EDUCATION Descargar pdf

Alicia Tinnirello, Eduardo Gago, Lucas D’Alessandro, Mónica Dádamo

Universidad Tecnológica Nacional (ARGENTINA)

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creative and original proposals to solve problems; ability to learn; unlearn and relearn; learn how to make decisions and be independent; apply the methods of abstract thought; being able to identify problems and find solutions [1].
In addition, the advent of computer media, so attractive and friendly to students, not only because of its wide possibilities of work, but the familiarity with which students operate with them, have impacted heavily on education. All these issues translate into challenges posed by the need to maintain horizontal and vertical connections with the subjects of the curriculum. Students demonstrate a lack of motivation being experienced in different subjects of the Basic Cycle, either because their find its content disjointed and abstract or, they cannot relate it with their chosen studies [2].
Simulation and modeling of problematic situations help to conceptualize and visualize calculation topics involved in the development of subjects of the upper cycle of their studies or in their professional life, and facilitate learning modes.
The didactic proposals presented allow explaining subjects in Higher Mathematics through means of research and application of analytical, qualitative, graphical and numerical methods, in

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from the learner and a real commitment from the teacher in what refers to the directionality, coordination and pedagogical help.
In such sense, an integrated will be developed: theoretical-practical and theoretical-technological, in an attempt to have a different experience based on: dialogue, convergence of criteria and active participation of students.
To follow the evolution of their mathematical thinking is necessary to awaken the interest of student in applications and induce them to form notions and to discover by themselves the mathematical relations, trying not to impose a single way of thinking, taking advantage of the originality of their creative spectrum and new points of view.
As long as the student is allowed to relate each piece of knowledge to incorporate with the previous knowledge already acquired, learning will be more meaningful and functional. Relevant and properly sequenced questions, so as to guide the student’s line of thought through reasoning that allow them to reach certain conclusions -convergent thinking-, will make its exposure more dynamic and participatory [1] [2].
The teacher will act as a learning facilitator, guiding with

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with technological resources.

3.1 Case No. 1: Signal response of 2^nd order system

In this case, the methodology of presenting the mathematical content is based on the search of models that simulate the situation that we require to formulate or technical situation in mathematical terms. This is why a simplified situation is presented by translating the situation in mathematical language, creating the model for classroom work.
Considering an RLC circuit formed by a resistor 160 ohms, a capacitor 10-4 F and an inductance of 1 H serially connected to a voltage source 20 V. Before closing the switch at time t = 0, both the charge and the resulting current is zero. For determining the charge on the capacitor and the resulting current in the circuit students propose guiding questions to investigate the system [3]:
− What is the system solution?
− What does each term in the solution represent?
− What are the characteristics of the system?
− What is the electrical and power curves burden the system

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The maximum load on the system is , while the current flowing through the circuit after a short time is zero Students analyzed the system behavior using the LabVIEW platform, as shown in Fig. 2.

Fig.2 Circuit behavior report

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Students will analyze a system to detect failures due to the bars that break in the squirrel cage induction motor, and specifically, they will perform a task that involves analyzing the signal with the frequency of the electrical currents in the engine to later to make a diagnosis leaving aside the acquisition of the signal itself.
Students will study and implement using LabVIEW programming platform application based on the Fourier Transform method for spectral analysis of the supply current of induction motors with MCSA technique [4].
We request students to research the theoretical guidelines needed to analyze the operation of two induction motors, to later determine which engine have the squirrel cage bars in good condition and which motor has the damaged one. For this purpose the students fill in a report that consists of two parts.

3.2.2 Theoretical knowlegde

For the analysis of the effect of the broken bars on the supply current (stator current) of an induction motor students have the general equation linking frequency (f), with the speed of the rotating field (n) and the number of pair’s pole of the machine

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speed difference between the field and the rotor (n₁-n) is the relative rotational speed with which the field lines intersect the rotor conductors, and under these speeds differences are induced in the rotor winding f.e.m. and currents of the frequency f₂, as expressed in the equation (3):

The rotor polyphaser currents in turn create a rotary field speed (n₁-n), with respect to the rotor, in question and in the same direction as the stator field following the inductive sequence which they proceed from. It is shown that relative to the stator the rotor rotates at the speed field as:

That is to say, at the synchronous speed, regardless of the rotor itself. Fields, stator and rotor remain stationary from one another and could be combined into a single rotating field that ultimately it is left as resulting in the machine. It is important to note that in any healthy motor whatever the speed of rotation of

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From equation (7) and by replacing in (10) is obtained:

Multiplying both sides of (11) by the number of pole pair’s p and considering the expression (3) It is obtained:

As the rotating magnetic field frequency short the stator windings, It is induced in them an f.e.m. and hence a current with the same frequency f_r of the rotating field, called f_1r, since it corresponds to the frequency of the current that circulates through the stator windings:

This implies that under conditions of asymmetry, as a consequence of the rupture of the bars in an induction motor,

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Table 1 Fault diagnosis MCSA

Fourier transform (FT) is a mathematical tool used to convert a signal from the time domain to a signal in the frequency domain, in order to observe the behavior of a function at a specific time. The FT is studied from the format of a discrete signal, closer to its use in methods and computational algorithms; whose formulation is presented in the equation (15). [1]

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4 OPERATION ANALYSIS

For the implementation of the algorithm FFT, Students use the LabVIEW tool: FFT Power Expectrum.vi Waveform Measurements, corresponding to the Signal Processing library. In Fig. 3, the front panel of the system is observed, and in Fig. 4 the main program block diagram provided in the report done by students [8].

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Fig.4 Block Graph

Since the control called “signal to be analyzed” can choose the signal source to study, with the options; “simulated” corresponding to a signal generated by the same system through appropriate controls, “signal stored on disk” this being a signal that has been taken offline and stored in a file, or finally

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The Fig. 5 shows the graphic work resulting from a motor with damaged rotor

Fig.5 Current in a damaged engine: a) temporary response, b) frequency spectrum.

Fig. 5 students indicate that faults detection in motors is facilitated by increasing the engine load, since the higher the load, the lower the speed, the greater the slip, resulting in greater distance between the harmonic frequencies fundamental and those located in the sidebands, and therefore easier to distinguish failures. Therefore they recommend diagnosis of faults is performed with the engine load close to

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Furthermore, students conclude that the amplitudes of the harmonic frequencies of the sidebands is proportional to the degree of breakdown of the motor cage, therefore the difference in dB between the magnitudes of the fundamental frequency and the harmonics will decrease as more broken bars exist in the rotor cage [9].
Fig. 7 shows the graph obtained by the students when working with a motor with rotor without fail.

Fig.7 A healthy motor current: a) temporary response, b) frequency spectrum.

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possible only to evaluate the system with simulated signals or facilitated by files, limiting the amount of field tests that can be performed [10].

6 CONCLUSIONS

The methodology implemented to develop Higher Mathematics lessons with a scheme of multidisciplinary teaching is motivating and it is closer to the why of doing of professional engineering, since knowing how to perform is a basic skill in the work life and should be encouraged during the learning process.
By applying virtual platforms, an interconnection between computational mathematics and basic and applied technologies develops as a means to facilitate the incorporation of knowledge and to improve the efficiency of conventional education. It is noted that the learning process is facilitated through the development of an appropriate language, where problematic and complex engineering situations are proposed and solved where the investigative task is central, with a respect of learning rhythms, and promoting collaborative learning and

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2. A. Tinnirello, E. Gago, M. Dádamo, “Designing Interdisciplinary Interactive Work: Basic Sciences in Engineering Education”. The International Journal of Interdisciplinary Social Sciences, vol. 5, no. 3, pp.331-334, 2010.

3. G. James, et al, “Matemáticas avanzadas para ingeniería”, Edit. Prentice HALL, México, pp. 398-426, 2002

4. R. Puche Panadero, M. Pineda Sanchez, et al ”Improved Resolution of the MCSA Method Via Hilbert Transform, Enabling the Diagnosis of Rotor Asymmetries at Very Low Slip”, IEEE Transactions on Energy Conversion, vol. 24, no. 1, pp. 52-59, 2009.

5. W. Thomson, M. Fenger, “Current Signature Analysis to detect induction motor faults”, IEEE Industry Aplications Magazine, vol. 7, no. 4, pp. 26-34, 2001.

6. J. Bobadilla, P. Gomez, J. Bernal, “La Transformada de Fourier: Una visión pedagógica”. Dialnet, vol. 1, no. 10, pp. 41-74, 2001.

7. P. O’Neil, “Matemáticas avanzadas para ingeniería”, Edit. Thomson, pp. 135-147, 2008.

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INTEGRATING MATHEMATICS TECHNOLOGY WITH MECHANICAL ENGINEERING CURRICULUM Descargar pdf

Alicia Tinnirello, Eduardo Gago

Universidad Tecnológica Nacional (ARGENTINA)

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1 INTRODUCTION

New technologies are based on complex algorithms that manage to produce, through the use of appropriate technologies, works with unexpected complexity 20 years ago. These algorithms have a strong mathematical basis and allow conceptualize other working methods capable of solving the problems in the practice of engineering.
The study of this new mathematic required deepen and broadening the field of knowledge. To achieve the analysis and utilization of complex design systems and generate experimental non-linear models, the contributions of the new math will be critical, both in the stage of formation of the engineer and his professional life [1].
The engineer must use during the exercise of their professional activity, primarily the reasoning, in order to understand

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integrate knowledge in order of increasing complexity. An inclusive material aims to create, throughout the career, a multidisciplinary space synthesis allowing student to understand the characteristics of the practice of engineering, on the basis of the resolution of open problems of the Mechanical Engineering.
The way in which the horizontal and vertical articulation of the content takes place is given, mainly, through two areas: One 1st and 2nd level and the other 3rd, 4th and 5th level. Inclusive materials are: Engineering Mechanics I, Engineering Mechanics II, Engineering Mechanics III, Elements of Machines and Final Project. These subjects include the principal horizontal connection of engineering relationships with disciplines and sciences that are issued every year and vertical from one year to another [2].
The current curriculum is organized into four interrelated blocks: Basic Science, Basic Technologies, Applied Technologies and Complementary Materials, being aimed at the training of professionals for two levels of hierarchy according to the context of Argentina.
The first application includes tasks for use and operation of

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3 THE EMERGENCY OF TECHNOLOGY: MATHEMATICS EDUCATION

Different works describe the various innovative experiences implemented in the classrooms of Advanced Calculus, the career of mechanical engineering, where mediation between the content of the subject and responses expected from the student group, was given through the gradual incorporation, in the dictation or workshops classes, different technological applications (Mathematica, LabVIEW, Comsol) and the use of virtual learning platforms.
Last years, have been implemented methodological strategies with different characteristics, the activities and results achieved have being integrated as contributions made to the training in engineering.
Educational interventions were developed by using technological applications available to explore concepts of modeling, estimation of parameters, numerical simulation and case analysis, pursuing better understanding of concepts and procedures.
Besides, the results achieved by the students were emphasis

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are the reduction of the curricula of engineering careers that go from 6 to 5 years of nominal duration; the commitment with the need to achieve a strong student training in basic cycle of engineering careers; to homogenize the knowledge; to make changes in the subjects content in basic cycle; and to introduce the computational tools in teaching, to the realization of practical work of all subjects in the area of Mathematics [5]. Use of technology made it possible to make more explicit the role of modes of representation. In particular, the way in which the complementarity between graphic, numerical and symbolic representation, produce best comprehension using technology and help develop coordination processes. Thus, the descriptions of the construction process followed by students have allowed relating aspects of the particularization to the reflective abstraction derived from the modes of representation in construction knowledge. These early educational experiences continued with intervention and interaction activities with students, inside and outside the laboratory, through virtual platforms, always favoring the approach and requests. The use of ICTs in general, relied on the publication of information in sites accessible over the Internet, so that students

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uncertainty, adaptation, non-linearity.
To analyze the profile of the entrants to the career and to take into account the performance in the University seminar, we arrived at the conclusion that the students were admitted to the University with low levels in certain competitions such as ability to abstraction and analytical thinking, among others. A diagnostic study was developed to measure levels of mathematical knowledge gained in previous years perceived by Advanced Calculus Engineering students. The Mathematical knowledge search consisted of five dimensions: Integration, Differentiation, General subject, Graphics and Limit, related to subject of Mathematics curriculum [4], [5], [6].
There was a very significant difference between the students that successfully have completed the Mathematical Analysis I, Mathematical Analysis II courses during the 1st two years of the careers and those failing those courses. To solve the difficulties to understand concepts view in previous years, practical sessions were implemented at the beginning of Advance Calculus where several applications were showed through special material referred to modeling physical systems and to explain the significant of different mathematical tools in

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the signals responses producing others or any desired result, as well as build and develop skills in management methodologies and mathematical tools for signals treatment in continuous and discrete time systems, in the temporary or frequency field, deterministic or stochastic. Identified strategies are designed to develop skills in students to adopt a style of active learning that favors the ability to deal with risks and a competent behavior to deal with difficult situations, for example, requiring discover solutions, manage conflicts, give feedback and learn to delegate.
Modeling and simulation are, require or imply mathematical formalization (previous) work, but this is not always true, sometimes the mathematization can be done later as a verification or demonstration of what has been modeled or simulated. It is necessary to consider that simulation and modelling demand prior work, conceptual or theoretical, that leads to analyze algorithmic or computational problems.

7 ADVANCE CALCULUS AND APPLICATIONS

Examples of assignment work and computer projects

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knowledge of the calculation of probabilities and dynamic models and apply it to real processes.
Dynamically signals were simulated for the study of vibrations in rotating equipment, to display different analysis techniques and mathematical models involved. Patterns of different signs were analyzed to incorporate prior knowledge about the behavior of the mechanical systems operating in theoretical conditions of functioning. Simulated signals were compared with the real signals, checking the signals with the patterns when noise is added that disturb the output vibration signal.
They simulate signals with and without noise, the noise was incorporated by stochastic generators. This procedure provides a more realistic estimate signal which can deal with the ones that show the measuring equipment, allowing a more genuine comparison of the graphics information that they provide.
The results of these works are shown below:
a) Patterns of behavior of a signal of a gear with wear, sine wave and equation mixture of harmonics without and with noise, Fig. 1, Fig. 2.

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Figure 3. Gear signal withFigure 4. Spectral analysis
noise with noise

b) Axis of rotation slide: Equation of sine wave with changes in frequency amplitude

being , phase angle and A_j the j-th harmonic amplitude [3].

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7.2 Failures that occur in an induction motor: Simulation of an electrical signal

The objective is the signal analysis rather than the techniques of data acquisition and LabVIEW is used as a platform for simulation. Through the front panel students watch the simulation perform by the analysis of the simulated signal coming from the engine, illustrated in Fig.7:

Figure 7. Front panel

Since the student interest is focused on signal analysis, rather

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speed, greater the slip, resulting in more distance between the fundamental harmonic frequencies and those located in the sidebands, and therefore easier to distinguish failures. Therefore was concluded that diagnosis of faults is performed with the engine load close to the nominal [10], [11].

Figure 9. Damaged engine: temporary response and frequency spectrum.

In Fig. 10, students observe current peaks at frequencies of 60Hz, and 63,33Hz 56,67Hz, corresponding to the fundamental frequency and the sidebands matches with those expressed in

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equipment development in a laboratory class as an example task for mechanical engineering.
It was very interesting the treatment in the class of this engineering application as the air heaters are an interesting option to increase the performance of the cycle responsible for generating steam. The project consists in the air heater design, where the fluid enters from the top and comes in contact in descending order with the gas pipes of heat transmission, which are placed in the form of a labyrinth, the design allows the contact of the air with pipes across its surface, while the air outlet is located at the bottom.
Students work with the Comsol Multiphysics simulation platform that represents through their interactions what happens with the fluid when it's exposed to different conditions of work.
Specifically, technology used is CFD (Computational Fluid Dynamics) of COMSOL. This module is designed for the solution and simulation of fluid flow problems ranging from a single stage laminar flow to a turbulent flow and the resolution of problems of heat transfer, either by conduction, convection or radiation [14]. This module includes numerical methods and

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Figura 12. Temperature variation T0 = 60°C

It was also discussed, the variation of the speed of the fluid, Fig. 13, and the increase with the increase of temperature. The simulation design allows perform a real analysis of the pressures, variation of density, internal energy, etc. These are other features that can lead to an optimization of fluid flow analysis [15], [16].

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8 CONCLUSIONS

The teaching activity based on these types of integration discipline allows the concepts, theoretical frameworks, procedures and other elements with which they have to work with teachers and students are organized around units more global, conceptual and methodological structures shared by several disciplines. The academic activities of integration discipline contribute to the consolidation of certain values in teachers and students: flexibility, trust, patience, intuition, divergent thinking, sensitivity toward other people, acceptance of risks, learns to move around in the diversity, accept, new roles, among others. The work in Advance Calculus with technology has created a new relationship between the teachers of basic and applied technologies and research groups, which promotes the interaction, aimed at achieving the multidisciplinarity in the process of teaching, the application of new technologies, the transfer to the classroom of research for the best treatment of various subjects; and work in conjunction with the purpose of improving the quality of teaching, and thus to achieve the curriculum objectives. Besides this form of work

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4. V. Macchiarola, “Currículum basado en competencias. sentidos y críticas”, Revista Argentina de Enseñanza de la Ingeniería, Vol. 8,Issue 14, pp. 39-46,2007.

5. G. Bischof, E. Bratschitsch, A. Casey, Domago Rubesa. (2007). “Facilitating Engineering Mathematics Education by Multidisciplinary Projects”, Journal of American Society for Engineering Education. 2008

6. B. Prepelita-Raileanu, “Social Software Technologies and Solutions for Higher Education”, Proceedings of the 8th WSEAS International Conference on Education and educational Technology, 2009.

7. A. Tinnirello, “Stochastic Models in Engineering Quality Problems”, Journal WSEAS TRANSACTIONS on SIGNAL PROCESSING, Issue 2, Vol. 2, 2006.

8. A. Tinnirello, S. De Federico, “Modelos Espectrales para el Análisis de Fallas en Sistemas Mecánicos”, Congreso Argentino de Ingeniería Mecánica, 2008.

9. V. Lajara, S. Pelegrí, “LabVIEW: Entorno grafico de programación”, Madrid, Edit. Alfaomega, pp. 175-185, 2006.

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16. K. Ogata, “Ingeniería de control moderna”, Madrid, Edit. Pearson, pp. 178-202, 2010.

17. R. TORRES, J. GRAU. “Introducción a la mecánica de fluidos y transferencia de calor con COMSOL Multiphysics”, España, Edit. Addlink Software Científico S.L., 2007.

18. Y. Lin, “Grey Systems. Theory and Application”. Edit. Springer-Verlag, Berlin, pp. 475-479, 2011.

19. Thanagasundram, S. and Soares Schlindwein, F., “Autoregressive Order Selection for Rotating Machinery”, International Journal of Acoustics and Vibration, Vol. 11, N° 3, 2006.

20. W. Wang, “Autoregressive model-based diagnostics for gears and bearings”, British Non- Destructived Testing and Condition Monitoring, Vol. 50, Issue 8, pp. 414-418, 2008.

21. Y. Zhan, V. Makis, A. Jardine, “Adaptive model for vibration monitoring of rotating machinery subject to random deterioration”, Journal of Quality in Maintenance Engineering, Vol. 9, Issue 4, pp. 351-375, 2003. Descargar pdf

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INTRODUCING DISCRETE DYNAMIC SYSTEMS IN ALGEBRA TEACHING PROCESS Descargar pdf

Paola Szekieta, Alicia Tinnirello, Eduardo Gago

Computer and Basic Sciences Laboratory
Universidad Tecnológica Nacional – Facultad Regional Rosario
Zeballos 1341, Rosario, Santa Fe
ARGENTINA
paolasz@gmail.com, amtinni@gmail.com, eagago@gmail.com

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order to show the existing relations with symbolic calculus and the applications which these have with system resolution and modeling.
These applications are wide, ranging from microscopic simulations of Physics and Biology to macroscopic simulations of social and geological processes (Our translation) [1].
CAs are among the simplest mathematical representations of dynamical system that consist of more than a few – typically nonlinearly – interacting parts [2].
As such CAs are extremely useful idealizations of the dynamical behavior of many real systems, including physical fluids, molecular dynamical systems, natural ecologies, military command and control networks , economy fire spreading, epidemiology and many others [3] [4]. Because of their underlying simplicity CAs are also powerful conceptual engines with which to study general pattern formation [2].
CAs consists of a regular array of identically programmed units called cells or sites that interact with their neighbors’ subjects to a finite set of rules prescribed by local transitions. All sites make a regular lattice and they evolve in discrete time steps as each site assumes a new value based on the values of some local

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its nearest neighbor sites, called neighbors. Two kinds of neighborhoods are commonly defined for a rectangular lattice: A Von Neumann neighborhood consists of the site and the four nearest neighbors, situated above, below, right and left as shown in Fig.1 below.

Fig.1 Von Neumann neighborhood

A Moore neighborhood consists of the site and the eight nearest neighbors as shown in Fig.2 below.

Fig.2 Moore neighborhood

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The nearest neighbor left of a site on the left border is the site in the same row on the right border. In the same way, the neighbors on the right of the cells on the right border are analyzed. The nearest neighbor above site on the top border is the site in the same column on the bottom border. In the same way, the neighbors on the bottom order are analyzed.

2.3 Evolution Rule

Another basic component worth mentioning is the Evolution Rule which defines the state of each cell according to the immediate previous state of the neighborhood. This evolution is determined by a mathematical function which captures the

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This automaton is a game of cero players, which implies that its evolution is determined by its initial set-up and there is no need of any further data entry. The game unfolds over a bidimensional grid as the game board. Each position on the board is called cell and it has 8 neighbor cells which are the nearest to each of them, including the diagonal ones (Moore neighborhood). The cells have two states, living or dead, which are represented by the numbers 1 and 0 respectively. The number and arrangement of living cells on the board evolve along the discrete time units. All cell states are taken into account to calculate their state in the following time. All cells are updated simultaneously. The transitions depend on the number of neighbor cell which are alive. A dead cell with exactly 3 living neighbors will be born in the next turn. If a living cell has 2 or 3 neighbor living cells, the following turn it will still be living. In any other case, it will die or remain dead due to loneliness or overpopulation.
The game set out will continue until 2 identical consecutive states are obtained, or rather, until a certain number of predetermined transitions are reached.
To start the modeling of this game, an initial cell array over the

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It is possible to determine the number of living neighbors in each of the initial positions, counting the numbers of living neighbors which are around each cell:

Comparing the state of each cell of the game board in time t and the number of living neighbors, the following state in time t+1 can be set up.
Fig.4 below shows the transition from initial time t= 0 to t = 1. To visualize some examples, if we consider de state of de second element on first board row, it has 3 living neighbors so this cell will be born at t = 1 but first cell on third row of the board will be dead at next turn due to overpopulation.

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Fig.6 Evolution from t = 3 to t = 4

4 LABORATORY PROJECT: DEFINING AND DEVELOPING A COMPUTATIONAL SIMULATION MODEL

In the context of meaningful learning, the students' activities must be oriented in a school system based on research and development of appropriate strategies for connecting and integrating the computational mathematics and the basic technologies and applied in Engineering to promote the multidisciplinary approach to the curriculum content corresponding to the plans of study, aiming to train professionals capable of solving complex models with the use of technologies.
The existence of simulation tools transformed the programming

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is made up by the 8 neighbors around the position which is often identified with a cardinal point according to the position of the central cell: north, northeast, east, southeast, south, southwest, west and northwest. Fig.7 below shows this Moore neighborhood.

Fig.7 Moore neighborhood

It is possible to obtain a matrix that shows a particular neighbor by performing elementary operations on S_t. For example, to obtain a matrix N_t whose elements n_ij represent north neighbor of each site s_ij in S_t at a certain time 𝑡, it is necessary to move down every row on S_t by properly interchanging them. Ec. (1) shows N₀ that is north neighbors of< each site s_ij at time 𝑡 = 0.

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Fig.8 Number of living neighbors’ matrix at 𝑡 = 0

Fig.8 above shows how to obtain number of living neighbors matrix at time 𝑡 = 0.
Analyzing values of homologous elements on S_t and V_t, the next state into which s_ij will evolve can be obtained.
The evolution rule is function to the state of a cell s_ij and the number of living neighbor which it has.

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Fig.9 Game consecutive states

These rules are simple enough for anyone to understand, yet the Game of Life leads to an endless number of different patterns, and to significant complexity [11]. It is interesting to observe different these patterns or life forms. Students investigated these patterns and modified initial set-up to watch diverse evolution processes of the game. Laboratory work classes are an integral part of any educational program and

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Fig.12 Spaceship pattern

5 Engineering application: Pollutant diffusion in a liquid form

CA application is the pollutant diffusion in a liquid form, for example the water flows which are presented in the projects of environmental remedial. In fact, this tool can be useful to guide

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Fig.14 CA Simulated Pollutant Diffusion

Fig.14 above shows the evolution of the pollutant diffusion throughout time until all section under analysis is covered and a constant concentration is reached:
Students are required to interpret the software graphic output of pollutant diffusion and to compare it with other software outputs which can result from changing the initial configurations of the pollutant concentrations. This mathematical model is applied to understand the distribution of pollutants by formulating a 2D diffusion.

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REFERENCES

1 G. Merino, Use of a Cellular Automaton to Create a Diffusion Model of Pollutants in a Soil-Water System. Journal of Mathematics: Theory and Applications, Vol.18, Nro.1, 2011, pp. 63-76.

2. A. Ilachinski, Cellular Automata: A Discrete Universe, World Scientific, 2001, pp. 175-185.

3. J. Quartieri, N. Mastorakis, G. Iannone, C. Guarnacci, A Cellular Automata Model for Fire Spreading Prediction, Proceedings of 3rd WSEAS International Conference on Urban Planning and Transportation, 2010, pp. 173- 179.

4. M. Dascalu, G. Stefan, A. Zafiu, A. Plavitu, Applications of Multilevel Cellular Automata in Epidemiology, Proceedings of the 13th WSEAS international conference on Automatic control, modelling & simulation, 2011, pp. 439-444.

5. R. Gaylor, P. Wellin, Computer Simulations with Mathematica: explorations in complex physical and biological systems. Springer- Verlag, 1995.

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Scientific American, Vol.251, 1984.

Descargar pdf

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COMPUTATIONAL MATHEMATICS IN ALGEBRA TEACHING PROCESS Descargar pdf

Paola A. Szekieta, Alicia M. Tinnirello, and Eduardo A. Gago

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one discrete variable by teaching the fundamental basics of cellular automata (CA) in order to show the existing relations with symbolic calculus and the applications which these have with system resolution and modeling.
These applications are wide, ranging from microscopic simulations of Physics and Biology to macroscopic simulations of social and geological processes (Our translation) [1].
CAs are among the simplest mathematical representations of dynamical system that consist of more than a few – typically nonlinearly – interacting parts [2].
As such CAs are extremely useful idealizations of the dynamical behavior of many real systems, including physical fluids, molecular dynamical systems, natural ecologies, military command and control networks , economy fire spreading, epidemiology and many others [3] [4]. Because of their underlying simplicity CAs are also powerful conceptual engines with which to study general pattern formation [2].
CAs consists of a regular array of identically programmed units called cells or sites that interact with their neighbors’ subjects to a finite set of rules prescribed by local transitions. All sites make a regular lattice and they evolve in discrete time steps as each

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2 BACKGROUND ON CELLULAR AUTOMATA

From a theoretical point of view, some main concepts play an important role in CAs models:

A. The physical environment

This defines the universe on which the CA is computed. This underlying structure consists of a discrete lattice of cells with a rectangular, hexagonal, or other topology. Typically, these cells are all equal in size; the lattice itself can be finite or infinite in size, and its dimensionality can be 1 (a linear string of cells called an elementary cellular automaton), 2 (a grid), or even higher dimensional. In most cases, a common—but often neglected—assumption, is that the CAs lattice is embedded in a Euclidean space [3].

B. The cells’ states

Each cell can be in a certain state, where typically an integer represents the number of distinct states a cell can be in, e.g., a binary state. Note that a cell’s state is not restricted to such an

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Fig.1 Von Neumann neighborhood

A Moore neighborhood consists of the site and the eight nearest neighbors as shown in Fig.2 below.

Fig.2 Moore neighborhood

D. Lattice Boundaries. Periodic Boundaries

The nearest neighbors of sites along the sites of a lattice are determined differently for various boundary conditions. The way these conditions are defined will impact directly on the automata behavior. The periodic boundaries which are used in

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The nearest neighbor left of a site on the left border is the site in the same row on the right border. In the same way, the neighbors on the right of the cells on the right border are analyzed. The nearest neighbor above site on the top border is the site in the same column on the bottom border. In the same way, the neighbors on the bottom order are analyzed.

E. Evolution Rule

Another basic component worth mentioning is the Evolution Rule which defines the state of each cell according to the immediate previous state of the neighborhood. This evolution is determined by a mathematical function which captures the influence of the neighborhood over the target cell.

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of any further data entry. The game unfolds over a bidimensional grid as the game board. Each position on the board is called cell and it has 8 neighbor cells which are the nearest to each of them, including the diagonal ones (Moore neighborhood). The cells have two states, living or dead, which are represented by the numbers 1 and 0 respectively. The number and arrangement of living cells on the board evolve along the discrete time units. All cell states are taken into account to calculate their state in the following time. All cells are updated simultaneously. The transitions depend on the number of neighbor cell which are alive. A dead cell with exactly 3 living neighbors will be born in the next turn. If a living cell has 2 or 3 neighbor living cells, the following turn it will still be living. In any other case, it will die or remain dead due to loneliness or overpopulation.
The game set out will continue until 2 identical consecutive states are obtained, or rather, until a certain number of predetermined transitions are reached. To start the modeling of this game, an initial cell array over the board is set at the time
t = 0, represented by the following grid:

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It is possible to determine the number of living neighbors in each of the initial positions, counting the numbers of living neighbors which are around each cell:

Comparing the state of each cell of the game board in time tand the number of living neighbors, the following state in time t + 1 can be set up.
Fig. 4 below shows the transition from initial time t = 0 to t = 1. To visualize some examples, if we consider de state of de second element on first board row, it has 3 living neighbors so this cell will be born at t = 1 but first cell on third row of the board will be dead at next turn due to overpopulation.

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At the time instance t = 3, a board with all dead cells is obtained as shown in Fig. 6 below. The following turn, time t = 4, the same result will be obtained, so the game will finish by obtaining two consecutive similar states.

Fig.6 Evolution from t = 3 to t = 4.

4 LABORATORY PROJECT: DEFINING AND DEVELOPING A COMPUTATIONAL SIMULATION MODEL

In the context of meaningful learning, the students' activities must be oriented in a school system based on research and development of appropriate strategies for connecting and integrating the computational mathematics and the basic

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and each of its entries is the state of a particular cell at a given time t.
The following step is to define the function which returns the number of neighbors alive of each cell of the board. To model this automaton, the Moore neighborhood is considered which is made up by the 8 neighbors around the position which is often identified with a cardinal point according to the position of the central cell: north, northeast, east, southeast, south, southwest, west and northwest. Fig. 7 below shows this Moore neighborhood.

Fig.7 Moore neighborhood

It is possible to obtain a matrix that shows a particular neighbor by performing elementary operations on S_t.

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Similarly, it is possible to obtain matrices that show a particular neighbor for each position in the state space and therefore to know the number of living neighbors of ijs adding these 8 matrices of neighbor positions.

Fig.8 Number of living neighbors’ matrix at t = 0

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Fig.9 Game consecutive states

large. For simple binary cells, with 8 neighboring cells there are 8+1 cells that influence a given cell (previous state of a cell can influence it’s next state), which leads to 2⁵¹² possible binary

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Fig.10 Glider pattern

propagating pattern known as glider, shown in Fig. 10, it consist of five living cells and reproduces itself in a diagonally displaced position once every four iteration.
Software allows visualizing an animation of any pattern evolution, shown in Fig 11. Glider gives the appearance of walking across the screen.

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Fig.12 Spaceship pattern

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Live cells survive only if surrounded by two or three cells and a new cell is born only if surrounded by exactly three living cells. Initial configurations consisting of either single or neighboring live cells immediately yield the null state.
Survival in Game of Life requires a minimum of three living cells. Fig.15 shows the fate of one three-live initial state; its evolution is a period-2 state. Structures that lead into a periodic behavior are called oscillators.

Fig.15 Oscillator

In general, it’s not possible to predict whether a particular starting configuration will eventually die out or not. There is no

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Fig.17 Evolution of three gliders

Placing four oscillator patterns on the same board evolves into a stable pattern after nine iterations as shown in Fig. 18 below.

Fig.18 Evolution of four oscillator pattern

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to watch diverse evolution processes of the game. Laboratory work classes are an integral part of any educational program and their purpose is bringing the students closer to real situations of the area of studies.

5 ENGINEERING APPLICATION: POLLUTANT DIFFUSION IN A LIQUID FORM

CA application is the pollutant diffusion in a liquid form, for example the water flows which are presented in the projects of environmental remedial. In fact, this tool can be useful to guide and support the processes applied to purify contaminated water. It is not easy to obtain a numeric solution from a mathematical expression for such a phenomenon. However, there are ways to approach the problem by mathematical modeling of fluid transport: some of these methods will be studied in higher courses of engineering careers, CAs often provide a simpler tool that preserves the essence of the process by which complex natural patterns emerge [12].

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Fig. 21 above shows the evolution of the pollutant diffusion throughout time until all section under analysis is covered and a constant concentration is reached: Students are required to interpret the software graphic output of pollutant diffusion and to compare it with other software outputs which can result from changing the initial configurations of the pollutant concentrations. This mathematical model is applied to understand the distribution of pollutants by formulating a 2D diffusion.

6 CONCLUSION

The CAs have been used in different disciplines successfully. Currently, attention is raised towards the development of models which can carry out complex tasks such as cryptography, image processing and turbulence analysis, among others.
This is the reason why we consider important to introduce CA concepts and applications in engineering teaching, taking as a starting point the study of mathematical models with discrete variable systems.

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3. J. Quartieri, N. Mastorakis, G. Iannone, C. Guarnacci, “A Cellular Automata Model for Fire Spreading Prediction”, Proceedings of 3rd WSEAS International Conference on Urban Planning and Transportation, 2010, pp. 173-179.

4. M. Dascalu, G. Stefan, A. Zafiu, A. Plavitu, “Applications of Multilevel Cellular Automata in Epidemiology”, Proceedings of the 13th WSEAS international conference on Automatic control, modelling & simulation, 2011, pp. 439-444.

5. R. Gaylor, P. Wellin, “Computer Simulations with Mathematica: explorations in complex physical and biological systems”. Springer-Verlag, 1995.

6. M. Resnick, “Turtles, Termites, and Traffic Jams”, MIT Press, 1994.

7. S. Wolfram, “Cellular automata as Model of Complexity. Nature, vol. 311, no. 5985, 1984, pp. 419- 424.

8. R. Gaylor, K. Nishidate, “Modeling Nature: Cellular Automata Simulations with Mathematica”, Springer-Verlag, 1996.

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COMPUTATIONAL METHODS: THEIR ADVANTAGES ON TEACHING COMPLEX FLUID FLOW SYSTEMS Descargar pdf

A.M. Tinnirello, E.A. Gago, M.F. Romero

Universidad Tecnológica Nacional (ARGENTINA)

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1 INTRODUCTION

The incorporation of technological resources in Higher Education allows to make substantial changes in the programs of engineering careers, because they facilitate the understanding and integration of the knowledge that students have in a compartmentalized way through the application of a tool that favors training of the professional profile.
The development of numerical methods and the emergence of simulation platforms applied to teaching, consider the need to guide the methodological approach of Mathematics programs towards new forms of learning that impact a priori on the cognitive development of the student, and in the future in a solid mental structure capable of solving problems of different kinds [1].
It is not about imparting knowledge using computer resources with abstract applications, which are not useful, but to look for models applied to engineering that incentivize the implicit capacities that students possess and that channel them to seek new ways of reasoning, fostering creativity and the acquisition of critical and reflective thinking.

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2 METHODOLOGY

The applied methodology aims to create a learning space where a set of experimental activities are developed as a fundamental line of the educational process. The programmed activities do not separate theory of practice and are developed in the Basic Sciences Area Laboratory, where the students perform technological applications with topics developed in class, linking the subjects of the subject with different levels of the area, as well as with other disciplines Through the resolution of models whose complexity is conditioned only by the basic knowledge that students have.
The teaching strategies are established from different practical activities without neglecting the theoretical foundation, respecting the multidisciplinary approach [3].
In the areas of applied technology cycle and following a multidisciplinary line, training activities are proposed, with a strong focus on professional activity. The design of the activities must present clear objectives and goals that allow updating and continuously coordinating the tasks in the areas of knowledge of university studies.

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• Value the benefits of plan different assumptions in a safe work environment, without any risk and propose diverse alternatives, since multiple tests can be performed in a short period of time.
• Analyze the pros of Creeping Flow de COMSOL Multiphysics modules, in the study of program fluid fluxes in general, and the limit layer theory in particular. [4], [5].
• Evaluate the capacity that the program has when making changes in the parameters of the system and its response.
• Evaluate the results obtained once the simulations have been carried out and arrive at pertinent conclusions about the virtual system when applied in a real environment.

4 EXPERIENCE DEVELOPMENT

The Lab experience propose analyze the behavior of limit layer generated by the interaction of a fluid in contact with a solid body.
With that purpose, commission groups of three participants each was created, and the session was structured in the next way:
1- Search, analysis and selection of bibliography material.

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coexist in it. Sometimes it is useful for the boundary layer to be turbulent [5], [6].

Fig.1 Fluid layers relative velocity

In aeronautics applied to commercial aviation, they usually opt for wing profiles that generate a turbulent limit layer, since this remains adhered to the profile at greater angles of attack than the laminar limit layer, thus preventing the profile from stalling. So, it stops to generate aerodynamic sustentation in an abrupt way by the loss of limit layer [7].
The thickness of the limit layer in the area of the leading edge is small, but increases along the surface. All these characteristics vary depending on the shape of the object (lower thickness of the limit layer, the lower the aerodynamic drag present on the surface).

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Fig.3 2D and 3D fluid-sphere geometry system.

In both cases, the fluid is water. Both the thin plate and the sphere are made of iron. The Software has incorporated a list of material that considers their physical properties, and in this study the students decided to work a constant temperature.
In those graphics are observed the contour conditions imposed for the fluid-thin plate and fluid-sphere.

Fig.4 Meshing geometry.

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Where: ρ : Density, u : Velocity flow. μ : Dynamic viscosity. I : Identity matrix. T : Temperature. F : Constant.
In addition, the Navier Stokes equation takes into account the density constant in the term convective and discarded the effects of the turmoil.

5 RESULTS

5.1 Fluid-plate System

Tests are carried out considering that the fluid circulates at velocities of 0,01 m/sec and 0,10 m/sec whose effects are observed in Fig. 5 and 6, respectively.
The students analyze the graphs provided by the software and note that the flow stream lines in the boundary layer are approximately parallel to the plate and to the fluid flow current.

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Fig.6 Fluid velocity variations.

When the students analyze the pressure gradients, they observe that fluid possess a high pressure in the instant at the point of stagnation after the plate, despite this pressure gradient, the fluid does not change its velocity after leaving the plate [9].
The pressure decreases after it moves away from the plate, where the fluid force is not capable to surpass the forces by the internal friction, so this produces pressure gradients in opposite directions.
As velocity is increases, the pressure effect at stagnation point increases significantly.
The systems pressure gradients for velocities of 0,01 m/sec and 0,10 m/sec are observed in Fig. 7 and 8, respectively.

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solid material and the variations in velocity and pressure that the fluid presents.
Tests are carried out considering that the fluid circulates with velocities of 0,01 m/sec and 0,10 m/sec and whose effects are seen in Fig. 9 and 10, respectively.
In Fig. 9 and 10, students observe that the flow has the characteristics of boundary layer, where the velocity profiles of the fluid that circulates from left to right are considered with a plane of cut located in the center of symmetry of the sphere.

Fig.9 Fluid velocity variant.

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around the sphere when the velocity of the fluid circulation increases considerably, we work with a velocity of 5m/sec and as seen in Fig. 11, it is seen that the blue wake it widens to the left around the sphere.

Fig.11 Fluid velocity variant.

Through the graphical interface of the software, different cuts can be made through different planes of the geometry observing the flow patterns in the different model sections.
For greater visibility, the example is shown in Fig. 12 for a velocity of 5 m/sec.

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Fig.13 Pressure gradients.

Fig.14 Pressure gradients.

In Fig. 15 and Fig. 16, the level curves corresponding to the pressure gradients in 2D and 3D are observed for the velocity 0,01m/sec.

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Fig.16 3D pressure levels curves.

6 CONCLUSIONS

The purpose of incorporating Multiphysics simulation platforms in university education is to establish a nexus between basic sciences and applied technologies, since they produce changes in the different learning structures, promoting the development of students' abilities.
In the experience it is shown, relating to previous works, how the same situation that can be analyzed with the complex variable, in the case of the boundary layer, is insufficient for its resolution, and the simulation software provides a real response to the model.

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REFERENCES

1. K. Sumithra, A. Dharani, M. Vijayalakshmi, “Transformation in Engineering Education: An Analysis of Challenges and Learning Outcomes”, en Proc. of the 14th International Conference on Education and Educational Technology, pp. 33-36, 2015.

2. A. Tinnirello, E. Gago, L. D’Alessandro, M. Dádamo, “Virtual Instruments Integrating Mathematical Modeling for Engineering Education”, en Proc. of 9th annual International Conference of Education, Research and Innovation, vol. 1, pp. 255-264, 2016.

3. A. Tinnirello, E. Gago, M. Valentini, Design and Simulation of Mechanical Equipment by Design Tools and Multiphysics Platforms, Proceedings of 8th. International Conference of Education, Research and Innovation, pp. 789-797, 2015.

4. R. Torres, J. Grau, Introducción a la mecánica de fluidos y transferencia del calor con Comsol Multiphysics, España: Editorial Addlink Media, 2007.

5. R. Bird, W. Stewart, E. Lightfoot, Transport Phenomena, USA,

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ALGORITHMIC MATHEMATICS IN LINEAR ALGEBRA APPLICATIONS Descargar pdf

A.M. Tinnirello, E.A. Gago, P.A. Szekieta

Universidad Tecnológica Nacional (ARGENTINA)

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1 INTRODUCTION

At the first year AL course, the issues that obstacles the learning process are diverse. Between them, the inadequate basic formation with the students enters the university, so that makes essential to plan simple models that leads to the acquisition ways of thinking and reasoning, and being triggers of motivating situations.
This paper describes an experience in a course in Algebra and Analytical Geometry, in the Chemical Engineering career, where students are expected to integrate knowledge when developing the topic Linear transformations through the use of specific software. Situations are proposed by means of which an analytical procedure is induced to relate the theoretical concepts learned in the classroom and to enhance the competences that students can develop.
The vision of teaching Mathematics in engineering careers in Higher Education institutions, aims that learning is done under certain conditions, which allow students to acquire knowledge with techniques that are inherent to problems related to their future professional work [1].

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designed, which is called the theoretical-practical-technological class where the concepts of the geometry of linear transformations are applied in the plane and in the space, trying to introduce models that have an entity in the field of Engineering and that in addition, they are related to the generation of objects of a current mathematics [4].
The Laboratory class aims to train the student in specific software, with the intention that from the use of computer resources can develop their inventiveness and dexterity, and conceptualize the content. The characteristics of the theoretical-practical technological class are to establish the guidelines of an interactive work of engineering analysis that manifests itself in an activity that produces new ideas. This methodology of teaching and learning allows us to advance first with the theoretical research of the subject, creating an environment where the student assumes an active role, leaving aside the fact of being a spectator, and allowing him to build concepts through experimentation, and the elaboration of conclusions.
The teacher, in this kind of experience, take the roll of process guide and acts like a facilitator of learning, until reaching the full

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mathematical functions in real situations, which allows to validate the student’s own skills for the basic competences development.
The interconnection between the concepts of LA, the real cases in the engineering field, and the anticipation of knowledge that will later become more complex in the subjects that deal with basic and applied technologies provide students with versatility when considering models each more sophisticated times in the specific subjects of his career [5], [6].
It is about creating a work environment where the student can be reflective, critical, make decisions and also define the mistakes made in the process. This situation incites the need for the student to know the object of study and create the internal conditions for the assimilation, in an active and independent way of the new knowledge.

4 EXPERIENCE DEVELOPMENT

A theoretical-practical-technological class is designed in two sessions of three hours chair each, in which students are to be led in a significant process of learning the proposed subject,

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class the generation of fractals is made from related transformations, which involve the concepts of linear transformations.

4.1 First session: Engineering application

4.1.1 Case 1: Center of gravity and transversal sector of a pipe.

A pipe has a transversal rectangular section as indicates on Fig. 1, and it is hanged from the ceiling in oblique position according an α angle with respect to the perpendicular that forms the ceiling with the wall. With the objective to analyze the air flux that will pass in the said enclosure, it is asked to the students to modify the pipe position and the transversal section, in a way that increases the amount of flux pass, and the center of gravity is positioned in the origin of coordinates.

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According to the request, and after investigating the theoretical knowledge of the subject, the students suggest to use first the transformation of rotation expressed through Eq. (1), which allows to rotate the position of the pipe in order to keep the sides of it parallel to the ceiling, walls and floor.

Fig.3 Pipe Rotation Sequence.

The students realize tests to determine the α angle value which places the pipe in the desired position, after try with various rotation angles, they find when α = 45º the vertical pipe position is achieved. Fig. 3 shows a sequence of some tests realized from the students until find the desired position [7].

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to double [8].

Fig.5 Double-sized pipe.

Consequently, it is indicated that if you know the cross sections of two pipes, one hexagonal and another square, located in different regions of the plane, you are asked to analyze which linear transformations you can propose to reduce them cross-sectional area and its center of gravity is the origin of coordinates. The sequence in graphic form of the results obtained by the students is shown in Fig. 6 and 7.

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students propose a simple graphing geometry that represents one of the blades of a fan as shown in Fig. 8 [9].

Fig.8 Fan Blade.

The students apply the transformation whose produces reflections respect to the coordinate axis. The eqs. (4), (5) y (6) corresponds to the reflection respect from y axis, reflection to x axis, and reflection in 3rd quadrant, respectively [7], [8].

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Fig.10 Fan Blades.

4.2 Second Session: Fractals Objects Generation

In principle, it is intended that students generate the fractal object called Sierpinski Triangle (ST), analyzing the related transformations that will be used for this process. In this context, they begin the experience starting from an isosceles triangle of vertex, and, and through the application of a system of iterated functions (SFI) they will generate the desired geometric structure [10].
The laws of related transformations applied are:

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From that point, and as consequence to keep applying SFI, the students achieve the first five graphics, observed in Fig. 12.

Fig.12 First ST iterations.

The iteration tending to infinity applying the SIF allows the students to generate the fractal called ST, in Fig. 13 the result of the experience is shown.

Fig.13 Finite ST representation.

As a result of this activity, and through the application of a new

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Fig.14 First application of SFI to the AR.

Fig.15 Generation of AR.

After generate the ST and the AR, the students analyzes similar structures but in space. The applied SFI in this case is {f₇ , f₈ }[8].

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Fig.16 3D fractal from a tetrahedron.

Fig.17 3D Fractal from a cube.

5 CONCLUSIONS

The use of linear transformations applied to the study of idealized models applied to engineering, along with the generation of fractal objects and the support of computer resources used appropriately led to a contextualized analysis that achieved the objective of understanding the subject,

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generating basic skills in the area of Mathematics.
This experience is a link between the disciplines of the University Basic Cycle with the subjects of basic and applied technologies, and allows to provide students with an autonomous character to perform the required tasks and learn new knowledge, as well as creating a host of activities to address problematic situations related to engineering tasks.

REFERENCES

1. R. Posada Álvarez, “Formación Superior Basada en Competencias, Interdisciplinariedad y Trabajo Autónomo del Estudiante”, Revista Iberoamericana de Educación, 2011.

2. G. Bischof, E. Bratschitsch, A. Casey, D. Rubesa, “Facilitating Engineering Mathematics Education by Multidisciplinary Projects”, Journal of American Society for Engineering Education, 2007.

3. M. Rosen, “Engineering Education: Future Trends and Advances”, Proceedings of the 6th. WSEAS International Conference on Engineering Education, pp.44-52, 2009.

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Madrid, Publisher Thomson, 2001.

10. Z. Zhu, E. Dong, “Simulation of Sierpinski-type fractals and their geometric constructions in Matlab environment” .Proceedings of WSEAS Transactions on Mathematics, 2013.

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