Session R4F Methods to Formulate and to Solve Problems in Mechanical Engineering

Transcription

1 Methods to Formulate and to Sole Problems in Mechanical Engineering Eusebio Jiménez López, Luis Rees Áila, Francisco Jaier Ochoa Estrella 3, Francisco Galindo Gutiérrez 4, Jaier Ruiz Galán 5, and Esteban Soto Islas 6 Abstract - During their learning process, engineering students must sole man technical problems. The students hae difficulties either formulating the problems, or deeloping them. It is necessar to elaborate schemes and general methods, to help students to formulate and to generate the problematics that the find during their studies. In this paper, we present four methods to formulate and to sole engineering problems, the are: Snthetic or Simplified, Analtic or Modeling, 3 Research Method, and 4 Snthetic-Analtic or Combined. The methods are applied in the solution of a Dnamics problem in which the concepts of impulse and linear momentum are used. The methods can be applied to an area of the engineering knowledge. Finall the adantages and disadantages of the four proposed methods are described. Inde Terms - Solution Methods, Dnamics, Engineering Education. INTRODUCTION The new challenges that the modern world imposes to humanit force to present and future engineers to get the best tools. The products demanded b the actual market are more comple than those demanded fift ears ago, that s wh the new engineer must dominate the best tools, so the can gie better solutions. Modern engineers must hae a sstematic thinking and it must be supported b their tools, which are: A Theoretical knowledge B Mathematical tools C A collection of methods D Computational tools E Practical knowledge (eperience During the formation period of engineering students, the must formulate and sole man problems generall aided b some technological resources (calculators, computers, etc. Widel speaking, problem soling has two general goals: the first is to deelop logical abilities in the student, and the second is to get understanding b practice. To understand and manipulate knowledge the student must be sensibilized of the importance of the theories and phsical and mathematical laws, he must know that the will be the reference points needed to locate the solution of a gien problem. The more problems the student soles, the more its understanding will improe, because of this, its er important to state methods to help the teacher and the student to share and assimilate knowledge. Traditional books on Mechanical Engineering [, ] and new researches [3] suggest methods to formulate and to sole problems; howeer, these methods need to be complemented and enriched. This paper suggests four different methods to formulate and to sole problems with the mere objectie of being an alternatie to the learning process of the aerage student. Adantages and disadantages of each method are ealuated and some case studies are gien. ON THE IMPORTANCE OF PROBLEM UNDERSTANDING Engineering students sole man problems during their academic formation, to do that, the theor of a specific topic is eplained and then the are thrown to sole problems. In this part, our main goal is to gie some useful recommendations for the student to understand a problem before tring to sole it, understanding the problem is er important because the selection of a solution method is based on this preious understanding. Consider the following steps [4]: A Read carefull the problem tet. B Identif from the problem s tet the unknowns. C Identif from the problem s tet the known data. D Analze the figures looking for known or unknown data. E Identif if the unknown is: a scalar a function 3 a ector 4 a matri. F Appl the step E to the know data. G If the problem tet is understood we can tr to sole the problem now, but before this, it s desirable to appl the net model to it: Gien X, find Y, where X represents known data and Y represents unknown data. H Document eerthing possible. I Once the problem tet is understood, identif the main formulas or laws from which the solution is deried. J Identif the secondar rules or formulas, if necessar. K Use the following model (if possible before tring to sole the problem: Gien X, find Y, such that Z is satisfied, here Z are the main formulas of point I. Eusebio Jiménez López, Uniersidad La Salle Noroeste, Cd. Obregón, Sonora, Méico, ejimenez@ulsa-noroeste.edu.m Luis Rees Áila, Instituto Meicano del Trasporte, Pedro Escobedo, Querétaro, Méico, lrees@imt.m 3 Francisco Jaier Ochoa Estrella, Instituto Tecnológico Superior de Cajeme, Cd. Obregón, Sonora, Méico, fochoa@itesca.edu.m 4 Francisco Galindo Gutiérrez, Impulsora de Desarrollo Dinámico S.A. de C.V., Cd. Obregón, Sonora, Méico, galindogtz@msn.com 5 Jaier Ruiz Galán, Uniersidad La Salle Noroeste, Cd. Obregón, Sonora, Méico, jaier.ruiz.galan@gmail.com 6 Esteban Soto Islas, Uniersidad La Salle Noroeste, Cd. Obregón, Sonora, Méico, estebansi@gmail.com San Juan, PR Jul 3 8, 006 R4F-6

2 PROBLEM SOLVING METHODS In order to get the solution of a gien problem, mechanical engineering students use different methods taken from tet books and teachers or inented b themseles. Different students hae different abilities; it is possible to differentiate three main problem soling abilities in teachers and students: Students mainl analtic Students mainl snthetic Students with a balance between snthetic and analtic abilities. Each tpe of abilit is obserable b looking the wa that some student soles a gien problem. This part discusses four useful methods to formulate problems; these methods must be applied onl after following thoroughl the steps A to K. I. Simplified method The snthetic or simplified method is a procedure that summarizes the analsis [4]; this means that it reduces the number of steps necessar to reach the final result b replacing the preiousl known data and the data obtained during the solution process in the general formulas, until the are greatl simplified due to algebraic operations. The written eplanation of the process is minimal. This method is widel used b the students with snthetic abilities. II. The modeling method The modeling method or analtic method, consist in deeloping the problem with the purpose of generating a general model of the problem and then appl it to the particular case corresponding to the current problem. This method could be carried out eplaining each step of the process or without doing it. The deelopment could be done deductiel or inductiel. The model must integrate, if possible, eer data (without numerical substitutions related with the problem or, in some cases, the model must be simplified b substituting the ariables that cannot be handled in ranges, such as phsical constants. This method is used generall b students with analtic abilities. III. The research method The research method is a procedure which allows to sole a problem and to describe it erball with a logical and eplicit discourse. Eer data (figures, tables, formulas and data in general associated with the problem must be related eplicitl with each other. This procedure allows to construct models sstematicall. Indeed, this method could be applied in a deductie or in an inductie wa. The research method could be deeloped b describing each step or without doing it. Howeer this method requires a good abilit to describe things in an eplicit and logical manner. IV. The combined method (Snthetic-Analtic FIGURE SCHEMA OF THE SYNTHETIC-ANALYTIC METHOD This is another method that helps us to sole problems; this method is named Combined or Snthetic-Analtic [4]. The combined method is deeloped in a graphical schema composed b four parts: Problem statement (P Laws or snthetic rules (P 3 Problem deelopment (P3 4 Formulas or analtic rules (P4 The graphical schema is showed in Figure. The problem statement (P could be described identicall as it is written in the books or using the model: Gien X, find Y such that Z is satisfied. The second part (P (Figure describes the snthetic rules, is to sa, those formulas that represent the laws that goern the phenomena, this is particularl important in phsics, but if the problem is of mathematics the snthetic rules are the aioms. The snthetic rules could be applied directl without substitutions or ariable changes, these are laws that hae been deeloped from the theor eplanation. The third part (P3 in Figure is the problem deelopment which could be described step b step or schematized, this part concentrates the main flow of application of the snthetic and analtic rules, this means that it represents the relations between a collection of laws of the phenomena and the analtic formulas of mathematics, here are represented onl the results of each step of the modeling process and we do not sole for an ariable, at the end the problem solution is presented. Furthermore, the analtic rules (P4 are indeed the mathematical laws used to model the problem and the re used to compose or decompose the sstem of relations and propositions of the problem statement. This part is diided into two parts: analtic rules and the deelopment of the analsis. In the analtic rules we onl represent the general rules of mathematics, the substitutions and results are done in the discourse. San Juan, PR Jul 3 8, 006 R4F-7

3 Arrows are used in order to isualize the model, these arrows link the blocks with the discourse, the point to the correct flow of understanding of the problem, because of this is er important to know the parts of the combined method and the directions of the arrows, the tip of the arrow takes from the snthetic rules to the problem deelopment so the student could understand how the rule is applied to the problem deelopment, sometimes the tip of the arrow goes from the problem deelopment to the snthetic rules meaning that some new snthetic rule has been stated based on the results. If the tip of the arrow goes from the problem deelopment to the analtic rules then we can assume that we cannot appl the snthetic rule directl and therefore the snthetic rule must be decomposed using mathematical analsis. When the arrow s tip comes to the problem deelopment from the analtic rules, then we can suppose that the result of the analsis could be applied directl in a snthetic rule. Also if the arrow s direction is straight down in the same block indicates that we re following a logical step. This method could be applied to an particular or general problem. CASE STUDIES Now we present the application of the four methods described aboe. We ll deelop a common problem of dnamics about momentum and linear impulse. Consider the following problem statement: The bo C shown in Figure has a mass of 00 kg and is originall in rest oer a horizontal non-frictional surface. If we appl a force F 00 N on the bo during 0 seconds with an angle of 45 degrees, determine the final elocit of the bo during the considered time period FIGURE PROBLEM DESCRIPTION We must point out that due to the method we re eplaining here the equations aren t numbered. I. Application of the understanding method A Read carefull the problem description. B The problem is to determine the final elocit ( after 0 seconds. C Known data is: mass of the bo 00 kg., initial elocit 0 m/s, the surface s friction is non-eistent, the impressed force is 00 N at 45º (Figure 3. D Make a free-bod diagram of the problem (Figure 4, showing the relation between known and unknown data. E Determine the kind of mathematical object that could represent the unknowns. In this case all unknowns are real numbers (elocities are commonl taken as ectors, here is the magnitude of the ector. F 00 N θ 45 FIGURE 3 PROBLEM DATA F Determine the kind of mathematical object that could represent the know data, here the known data is real too. G The initial problem formulation is: Gien 00 kg, 0 m/s, t 0 s, t 0s, F 00N θ 45, find (m/s. H Documentation and recommendations: In the problem description it's clear that the bo "C" is in rest at the beginning, this implies that 0 m/s, and that there are no frictional forces. Finall we should point out that the force is constant and it doesn't depends on time, is to sa, F(t F. In Figure 4 we present the free-bod diagram of the problem. X Y ( N FIGURE 4 FREE-BODY DIAGRAM OF THE PROBLEM I The most general mathematical epression that models the problem is: m t F t + ( dt m J Some secondar rules are: The projection of the force in the '' ais is: F FCosθ. The projection of the force in the '' ais is: F FSenθ. W mg, here W represents the bo's weight and g 9.8 m/s is the graitational acceleration. K Final problem formulation is: "Gien m,, t, t, F, and θ, find, such that m II. Simplified method t F t + ( dt m is satisfied". Suppose that the problem is formulated and understood following the steps mentioned aboe, now we appl the simplified method to deelop the problem: The general formula is: m t F t + ( dt m But since is equal to zero, then: m 00 kg Bo in rest ( 0 t 0 t 0 s F θ + t + Fdt m W San Juan, PR Jul 3 8, 006 R4F-8

4 3 Determining elocit b analzing the ais, t F dt m 4 Epanding and substituting, t ( F cosθ dt 00( F cosθ t ( 400cos 45 (0 00( 5 As result we obtain, 4.m / s III. Modeling method ( The deelopment of the modeling method is as follows, Write the general equation that models the problem: m t F t + ( dt m The coordinates inoled in the problem are (,, therefore, t ( + F ( t t ( + F ( t 3 Select one unknown to determine; in this case we ll sole for analzing the ais. 4 Soling for t ( + F ( t 5 We know F ( t F and F F cosθ so we substitute these equations to get, t ( + ( FCosθ 6 Deeloping the integral we obtain, t m t] ( ( t ( FCosθ 7 Then we determine the component of oer the ais, t FCosθ ( m 8 Substitute the data to get the particular solution for the problem, if 00 kg, t 0 s, t 0 s, F 00 N and θ 45º then, 00(0 + (00Cos45(0 00Cos45(0 ( 00 ( 4.m / s IV. Research method In this section we ll sole the same problem using the Research Method. According with the formulation of the step K of the understanding method, the mathematical epression that models the problem is that of the principle of the conseration of impulse and linear momentum, which is: t Σ F t San Juan, PR Jul 3 8, 006 R4F-9 m + ( dt m B analzing the free-bod diagram (Figure 4, it s clear that, t ( + Σ F ( t t ( + Σ F ( t In the same wa, we could see that it s eas to sole for using the equation that applies the aboe mentioned principle to the ais, because due to the known data the equation becomes a single ariable equation. Note that the projection of the force F oer the ais is F FCosθ and F(t F (F is constant, so we can rewrite the equation as follows, t ( + Σ ( FCosθ Alter deeloping the integral the epression is, Or, ( t] t ( t FCosθ Then, we hae to sole for the final elocit t FCosθ ( m Now that we hae our final epression which represent the most general case in our problem, we can use it to get the numerical alue of the final elocit oer the ais b substituting 00 kg, t 0, 0, t 0 s, F 00 N θ 45, the result is, ( 4.m / s V. The Snthetic-Analtic Method The application of the snthetic-analtic method (or combined is eemplified b the schema of the Figure 5 (in the last page. The schema is composed as follows: The problem formulation is written in the part (P as it is eplained in the step K of the problem understanding method. the snthetic rules are described in the part (P, for this case, the principle of the conseration impulse and linear momentum is the most important snthetic rule, in this part are located all the formulas to be deeloped during the analsis and the known data of the problem. The part (P3 presents the problem deelopment without showing eplicitl the analsis, it just shows the direct application of the snthetic rules and it receies formulas resulting from the analtic deelopment.

5 The analtic deelopment is eplicitl done in the part (P4, in this part the mathematical operations are eplicited. The arrows indicate the process sequence and the eplicit relation between the figures and formulas. V. Integrating the Four Methods It is possible to use the four methods jointl while soling a problem taking the net considerations: A It is mandator to eplain de theor behind the problem. B Select the simplest case o someone er simple. C Follow the steps of the understanding method. D Deelop the problem with the snthetic method. E Tr to structure the problem as it resulted in the step 4 using the modeling method in order to get a model of the problem. F Rewrite the problem enumerating the formulas and the figures, then use the research method, this means, to eplain the problem with a eplicitl logical discourse and rigorousl relating the formulas and the figures. G Schematize the results of the past steps using the Snthetic-Analtic method and eplain the problem using a logical discourse supported b graphics. It's clear the information described in this process is sometimes repetitie, but it's time worth because b this wa the student can see the problems from different points of iew and he can differ the seeral forms that a problem can take, due to this it's important to take if not the simplest, a simple case, after this process, the teacher can freel state more difficult problems to students and the will be able to sole them using the method that best fits their likes and skills. ADVANTAGES AND DISADVANTAGES The adantages that could be obtained from the simplified method are: It s possible to deelop man particular problems and it is eas to handle and dominate, the discourse is simplified, man steps are aoided in the analsis, simple calculators could be used, in general the most part of the students are used to the simplified method and it s useful to sole eams because of the saed time. Disadantages are: Limited abilit to build models, doesn t justifies the use of sophisticated calculators, the problem s discourse is limited, the method it s almost useless when we want to restud the content and it limits the understanding and sometimes is eas to forget the principles. The adantages of the modeling method are: It s possible to deelop general models, it could be used to sole particular cases b restricting the ariables range, step b step analsis, it s eas to stud the content and sophisticated calculators are justified. Modeling method s disadantages are: it s time consuming, simple calculators doesn t help too much, the more the analsis the more the error s probabilit due to abstract deelopment (without ariable substitution. The adantages of the research method are the following: logical clarit in the problem deelopment, useful to generate models, restud is facilitated, snthetic-analtic deelopment in the discourse and eplicit logical eposition of the problems. The disadantages of the research method are summarized in the following points: a clear master of logic, snthesis and analsis is required, it s time consuming, and it s difficult for aerage students because it needs good grammatical abilities. The following are the adantages of the combined method: knowledge could be understood in general and particular terms, a schematized and logical deelopment of the problems, it eposes the snthetic-analtic abilities of the students and it sstematizes the analsis procedures. Some disadantages are: it s time consuming, requires a deep understanding of the principles and logic, the schematized logical deelopment is sometimes difficult and repetitie. CONCLUSIONS The main conclusions deried from this article are snthesized in the net points: It s necessar to deelop procedures to help students to understand, to analze and to formulate problems. In this sense the steps of the understanding and formulation method could be useful to reach such goals. Once that the problems are formulated, it s necessar to sole them, in this paper four methods are proposed hoping to ease the problem soling procedure to an kind of student (snthetic, analtic or snthetic-analtic students The recommendations proposed for the understanding and formulation of problems and the four methods could be applied to an field of engineering, to achiee it the person who applies the methods must be able to adapt the specific knowledge of the field to the methods. For the use of the methods it is recommendable, at least at first, to appl them to simple case studies and after that to continue with more comple cases. It is possible to use the four methods jointl in order to sole one problem and to fortif the understanding of the theor behind it. ACKNOWLEDGMENT Thanks must go to those institutions that brought their support to make this work happen: Uniersidad La Salle Noroeste (ULSA, Instituto Tecnológico Superior de Cajeme (ITESCA and Uniersidad Tecnológica del Sur de Sonora (UTS who jointl form La Red Regional Alfa. REFERENCES [] Beer, F., P., Johnston, E., R., DeWolf, J., T., Mechanics of Materials, Third Edition, McGraw-Hill, New York, NY, 00. [] Hibbeler, R., C., Mechanics of Materials, Fifth Edition, Prentice Hall, Upper Saddle Rier, NJ, 003. [3] Joseph, J., Rencis, Hartle, T., Grandin, Jr., Educating Students to Question, Test and Verif Problem Solutions, Proceedings of the 004 American Societ for Engineering Education Annual Conference & Eposition. [4] Jiménez, E., Ochoa, F., Martínez, M., Ruiz, J., Métodos para el planteamiento solución de problemas: aplicaciones a problemas de impulso momentum lineal, Folleto interno de diulgación #, Red Alfa, ISBN: , Méico. San Juan, PR Jul 3 8, 006 R4F-30

6 FIGURE 5 FULL DEVELOPMENT OF COMBINED METHOD San Juan, PR Jul 3 8, 006 R4F-3