Session R4F
Methods to Formulate and to Solve Problems in
Mechanical Engineering
Eusebio Jiménez López1, Luis Reyes Ávila2, Francisco Javier Ochoa Estrella3, Francisco Galindo Gutiérrez4, Javier
Ruiz Galán5, and Esteban Soto Islas6
Abstract - During their learning process, engineering
students must solve many technical problems. The
students have difficulties either formulating the problems,
or developing them. It is necessary to elaborate schemes
and general methods, to help students to formulate and to
generate the problematics that they find during their
studies. In this paper, we present four methods to
formulate and to solve engineering problems, they are: 1)
Synthetic or Simplified, 2) Analytic or Modeling, 3)
Research Method, and 4) Synthetic-Analytic or Combined.
The methods are applied in the solution of a Dynamics
problem in which the concepts of impulse and linear
momentum are used. The methods can be applied to any
area of the engineering knowledge. Finally the advantages
and disadvantages of the four proposed methods are
described.
Index Terms - Solution Methods, Dynamics, Engineering
Education.
INTRODUCTION
The new challenges that the modern world imposes to
humanity force to present and future engineers to get the best
tools. The products demanded by the actual market are more
complex than those demanded fifty years ago, that’s why the
new engineer must dominate the best tools, so they can give
better solutions. Modern engineers must have a systematic
thinking and it must be supported by their tools, which are:
A) Theoretical knowledge
B) Mathematical tools
C) A collection of methods
D) Computational tools
E) Practical knowledge (experience)
During the formation period of engineering students, they
must formulate and solve many problems generally aided by
some technological resources (calculators, computers, etc…).
Widely speaking, problem solving has two general goals: the
first is to develop logical abilities in the student, and the
second is to get understanding by practice. To understand and
manipulate knowledge the student must be sensibilized of the
importance of the theories and physical and mathematical
laws, he must know that they will be the reference points
needed to locate the solution of a given problem. The more
problems the student solves, the more its understanding will
improve, because of this, its very important to state methods to
help the teacher and the student to share and assimilate
knowledge. Traditional books on Mechanical Engineering [1,
2] and new researches [3] suggest methods to formulate and to
solve problems; however, these methods need to be
complemented and enriched. This paper suggests four
different methods to formulate and to solve problems with the
mere objective of being an alternative to the learning process
of the average student. Advantages and disadvantages of each
method are evaluated and some case studies are given.
ON THE IMPORTANCE OF PROBLEM UNDERSTANDING
Engineering students solve many problems during their
academic formation, to do that, the theory of a specific topic is
explained and then they are thrown to solve problems. In this
part, our main goal is to give some useful recommendations
for the student to understand a problem before trying to solve
it, understanding the problem is very important because the
selection of a solution method is based on this previous
understanding. Consider the following steps [4]:
A) Read carefully the problem text.
B) Identify from the problem’s text the unknowns.
C) Identify from the problem’s text the known data.
D) Analyze the figures looking for known or unknown data.
E) Identify if the unknown is: 1) a scalar 2) a function 3) a
vector 4) a matrix.
F) Apply the step E) to the know data.
G) If the problem text is understood we can try to solve the
problem now, but before this, it’s desirable to apply the
next model to it: “Given X, find Y”, where X represents
known data and Y represents unknown data.
H) Document everything possible.
I) Once the problem text is understood, identify the main
formulas or laws from which the solution is derived.
J) Identify the secondary rules or formulas, if necessary.
K) Use the following model (if possible) before trying to
solve the problem: “Given X, find Y, such that Z is
satisfied”, here Z are the main formulas of point I).
1
Eusebio Jiménez López, Universidad La Salle Noroeste, Cd. Obregón, Sonora, México, ejimenez@ulsa-noroeste.edu.mx
Luis Reyes Ávila, Instituto Mexicano del Trasporte, Pedro Escobedo, Querétaro, México, lreyes@imt.mx
Francisco Javier Ochoa Estrella, Instituto Tecnológico Superior de Cajeme, Cd. Obregón, Sonora, México, fochoa@itesca.edu.mx
4
Francisco Galindo Gutiérrez, Impulsora de Desarrollo Dinámico S.A. de C.V., Cd. Obregón, Sonora, México, galindogtz@msn.com
5
Javier Ruiz Galán, Universidad La Salle Noroeste, Cd. Obregón, Sonora, México, javier.ruiz.galan@gmail.com
6
Esteban Soto Islas, Universidad La Salle Noroeste, Cd. Obregón, Sonora, México, estebansi@gmail.com
2
3
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9th International Conference on Engineering Education
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Session R4F
PROBLEM SOLVING METHODS
IV. The combined method (Synthetic-Analytic)
In order to get the solution of a given problem, mechanical
engineering students use different methods taken from text
books and teachers or invented by themselves. Different
students have different abilities; it is possible to differentiate
three main problem solving abilities in teachers and students:
• Students mainly analytic
• Students mainly synthetic
• Students with a balance between synthetic and analytic
abilities.
Each type of ability is observable by looking the way that
some student solves a given problem. This part discusses four
useful methods to formulate problems; these methods must be
applied only after following thoroughly the steps A) to K).
I. Simplified method
The synthetic or simplified method is a procedure that
summarizes the analysis [4]; this means that it reduces the
number of steps necessary to reach the final result by
replacing the previously known data and the data obtained
during the solution process in the general formulas, until they
are greatly simplified due to algebraic operations. The written
explanation of the process is minimal. This method is widely
used by the students with synthetic abilities.
II. The modeling method
The modeling method or analytic method, consist in
developing the problem with the purpose of generating a
general model of the problem and then apply it to the
particular case corresponding to the current problem. This
method could be carried out explaining each step of the
process or without doing it. The development could be done
deductively or inductively. The model must integrate, if
possible, every data (without numerical substitutions) related
with the problem or, in some cases, the model must be
simplified by substituting the variables that cannot be handled
in ranges, such as physical constants. This method is used
generally by students with analytic abilities.
III. The research method
The research method is a procedure which allows to solve a
problem and to describe it verbally with a logical and explicit
discourse. Every data (figures, tables, formulas and data in
general) associated with the problem must be related explicitly
with each other. This procedure allows to construct models
systematically. Indeed, this method could be applied in a
deductive or in an inductive way. The research method could
be developed by describing each step or without doing it.
However this method requires a good ability to describe things
in an explicit and logical manner.
FIGURE 1
SCHEMA OF THE SYNTHETIC-ANALYTIC METHOD
This is another method that helps us to solve problems; this
method is named “Combined” or “Synthetic-Analytic” [4].
The combined method is developed in a graphical schema
composed by four parts:
1) Problem statement (P1)
2) Laws or synthetic rules (P2)
3) Problem development (P3)
4) Formulas or analytic rules (P4)
The graphical schema is showed in Figure 1. The problem
statement (P1) could be described identically as it is written in
the books or using the model: “Given X, find Y such that Z is
satisfied”. The second part (P2) (Figure 1) describes the
synthetic rules, is to say, those formulas that represent the
laws that govern the phenomena, this is particularly important
in physics, but if the problem is of mathematics the synthetic
rules are the axioms.
The synthetic rules could be applied directly without
substitutions or variable changes, these are laws that have
been developed from the theory explanation. The third part
(P3 in Figure 1) is the problem development which could be
described step by step or schematized, this part concentrates
the main flow of application of the synthetic and analytic
rules, this means that it represents the relations between a
collection of laws of the phenomena and the analytic formulas
of mathematics, here are represented only the results of each
step of the modeling process and we do not solve for any
variable, at the end the problem solution is presented.
Furthermore, the analytic rules (P4) are indeed the
mathematical laws used to model the problem and they’re
used to compose or decompose the system of relations and
propositions of the problem statement. This part is divided
into two parts: analytic rules and the development of the
analysis. In the analytic rules we only represent the general
rules of mathematics, the substitutions and results are done in
the discourse.
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Arrows are used in order to visualize the model, these
arrows link the blocks with the discourse, they point to the
correct flow of understanding of the problem, because of this
is very important to know the parts of the combined method
and the directions of the arrows, the tip of the arrow takes
from the synthetic rules to the problem development so the
student could understand how the rule is applied to the
problem development, sometimes the tip of the arrow goes
from the problem development to the synthetic rules meaning
that some new synthetic rule has been stated based on the
results. If the tip of the arrow goes from the problem
development to the analytic rules then we can assume that we
cannot apply the synthetic rule directly and therefore the
synthetic rule must be decomposed using mathematical
analysis.
When the arrow’s tip comes to the problem development
from the analytic rules, then we can suppose that the result of
the analysis could be applied directly in a synthetic 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 any particular or general problem.
F = 200 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: “Given m = 100 kg, v1
= 0 m/s, t1 = 0 s, t2 = 10s, F = 200N y θ = 45˚, find v2
(m/s)”.
H) Documentation and recommendations: In the problem
description it's clear that the box "C" is in rest at the
beginning, this implies that v1 = 0 m/s, and that there are
no frictional forces. Finally we should point out that the
force is constant and it doesn't depends on time, is to say,
F(t) = F. In Figure 4 we present the free-body diagram of
the problem.
F
Y
CASE STUDIES
Now we present the application of the four methods described
above. We’ll develop a common problem of dynamics about
momentum and linear impulse. Consider the following
problem statement:
The box “C” shown in Figure 2 has a mass of 100 kg and
is originally in rest over a horizontal non-frictional surface. If
we apply a force F = 200 N on the box during 10 seconds with
an angle of 45 degrees, determine the final velocity of the box
during the considered time period
m = 100 kg
Box in rest (v1 = 0 y t1 = 0)
t2 = 10 s
W
θ
X
+
=
v2= (vx)2
N
FIGURE 4
FREE-BODY DIAGRAM OF THE PROBLEM
I)
The most general mathematical expression that models
the problem is:
t2
mv1 + ∫ F (t )dt = mv 2
t1
J)
FIGURE 2
PROBLEM DESCRIPTION
We must point out that due to the method we’re
explaining here the equations aren’t numbered.
Some secondary rules are:
The projection of the force in the 'x' axis is: Fx = FCosθ.
The projection of the force in the 'y' axis is: Fy = FSenθ.
W = mg, here W represents the box's weight and g = 9.81
m/s is the gravitational acceleration.
K) Final problem formulation is: "Given m, v1, t1, t2, F, and
t2
θ, find v2, such that mv1 + F (t )dt = mv 2 is satisfied".
∫
t1
I. Application of the understanding method
II. Simplified method
A) Read carefully the problem description.
B) The problem is to determine the final velocity (v2) after 10
seconds.
C) Known data is: mass of the box m = 100 kg., initial
velocity v1 = 0 m/s, the surface’s friction is non-existent,
the impressed force is 200 N at 45º (Figure 3).
D) Make a free-body 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 (velocities are commonly taken as vectors, here
v2 is the magnitude of the vector).
Suppose that the problem is formulated and understood
following the steps mentioned above, now we apply the
simplified method to develop the problem:
1) The general formula is:
t2
mv1 + ∫ F (t )dt = mv2
t1
2) But since v1 is equal to zero, then:
t2
mv1 + ∫ F dt = mv 2
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9th International Conference on Engineering Education
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Session R4F
3) Determining velocity by analyzing the ‘x’ axis,
IV. Research method
t2
∫ F dt = m(v
x
)
x 2
t1
4) Expanding and substituting,
t2
∫ F cos θ
dt = 100(v x ) 2
In this section we’ll solve the same problem using the
Research Method. According with the formulation of the step
K) of the understanding method, the mathematical expression
that models the problem is that of the principle of the
conservation of impulse and linear momentum, which is:
t2
t1
F cos θ t
10
0
mv1 + Σ ∫ F (t )dt = mv2
= 100 ( v x ) 2
400 cos 45 (10) = 100(v x ) 2
5) As result we obtain,
( v x ) 2 = 14.1m / s
t1
By analyzing the free-body diagram (Figure 4), it’s clear
that,
t2
m(v x )1 + Σ ∫ Fx (t )dt = m(v x ) 2
III. Modeling method
t1
t2
The development of the modeling method is as follows,
1) Write the general equation that models the problem:
m(v y )1 + Σ ∫ Fy (t ) dt = m(v y ) 2
t1
t2
mv1 + ∫ F (t )dt = mv 2
t1
2) The coordinates involved in the problem are (x,y),
therefore,
t2
m(v x )1 + ∫ Fx (t )dt = m(v x ) 2
t1
In the same way, we could see that it’s easy to solve for
v2 using the equation that applies the above mentioned
principle to the ‘x’ axis, because due to the known data the
equation becomes a single variable equation.
Note that the projection of the force F over the ‘x’ axis is
Fx = FCosθ and F(t) = F (F is constant), so we can rewrite the
equation as follows,
t2
t2
m(v x )1 + Σ ∫ ( FCosθ ) dt = m(v x ) 2
m(v y )1 + ∫ F y (t )dt = m(v y ) 2
t1
t1
3) Select one unknown to determine; in this case we’ll solve
for v2 analyzing the ‘x’ axis.
4) Solving for v2
t2
m(v x )1 + ∫ Fx (t )dt = m(v x ) 2
t1
5) We know F (t ) = F and Fx = F cosθ so we substitute
these equations to get,
Alter developing the integral the expression is,
m(v x )1 + ( FCosθ )t ]tt12 = m(v x ) 2
Or,
m(v x )1 + ( FCosθ )t 2 − FCosθ t1 = m(v x ) 2
Then, we have to solve for the final velocity
(v x ) 2 =
m(v x )1 + ( FCosθ )t 2 − FCosθ t1
t2
m(v x )1 + ∫ ( FCosθ ) dt = m(v x ) 2
t1
6) Developing the integral we obtain,
m(v x )1 + ( FCos θ )t ]tt12 = m(v x ) 2
m(v x )1 + ( FCos θ )t 2 − ( FCos θ )t1 = m(v x ) 2
7) Then we determine the component of v2 over the ‘x’ axis,
m(v x )1 + ( FCosθ )t 2 − FCosθ t1
(v x ) 2 =
m
8) Substitute the data to get the particular solution for the
problem, if m = 100 kg, t1 = 0 s, t2 = 10 s, F = 200 N and
θ = 45º then,
100(0) + (200Cos 45)(10) − 200Cos 45(0)
(v x ) 2 =
100
(v x ) 2 = 14.1m / s
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m
Now that we have our final expression which represent
the most general case in our problem, we can use it to get the
numerical value of the final velocity over the ‘x’ axis by
substituting m = 100 kg, t1 = 0, v1 = 0, t2 = 10 s, F = 200 N y
θ = 45˚, the result is,
(v x ) 2 = 14.1m / s
V. The Synthetic-Analytic Method
The application of the synthetic-analytic method (or
combined) is exemplified by the schema of the Figure 5 (in the
last page). The schema is composed as follows:
The problem formulation is written in the part (P1) as it is
explained in the step K) of the problem understanding method.
the synthetic rules are described in the part (P2), for this case,
the principle of the conservation impulse and linear
momentum is the most important synthetic rule, in this part are
located all the formulas to be developed during the analysis
and the known data of the problem. The part (P3) presents the
problem development without showing explicitly the analysis,
it just shows the direct application of the synthetic rules and it
receives formulas resulting from the analytic development.
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The analytic development is explicitly done in the part
(P4), in this part the mathematical operations are explicited.
The arrows indicate the process’ sequence and the explicit
relation between the figures and formulas.
V. Integrating the Four Methods
It is possible to use the four methods jointly while solving a
problem taking the next considerations:
A) It is mandatory to explain de theory behind the problem.
B) Select the simplest case o someone very simple.
C) Follow the steps of the understanding method.
D) Develop the problem with the synthetic method.
E) Try 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
explain the problem with a explicitly logical discourse
and rigorously relating the formulas and the figures.
G) Schematize the results of the past steps using the
Synthetic-Analytic method and explain the problem using
a logical discourse supported by graphics.
It's clear the information described in this process is
sometimes repetitive, but it's time worthy because by this way
the student can see the problems from different points of view
and he can differ the several 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 freely state more
difficult problems to students and they will be able to solve
them using the method that best fits their likes and skills.
ADVANTAGES AND DISADVANTAGES
The advantages that could be obtained from the simplified
method are: It’s possible to develop many particular problems
and it is easy to handle and dominate, the discourse is
simplified, many steps are avoided in the analysis, simple
calculators could be used, in general the most part of the
students are used to the simplified method and it’s useful to
solve exams because of the saved time. Disadvantages are:
Limited ability 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 restudy the
content and it limits the understanding and sometimes is easy
to forget the principles.
The advantages of the modeling method are: It’s possible
to develop general models, it could be used to solve particular
cases by restricting the variables’ range, step by step analysis,
it’s easy to study the content and sophisticated calculators are
justified. Modeling method’s disadvantages are: it’s time
consuming, simple calculators doesn’t help too much, “the
more the analysis the more the error’s probability” due to
abstract development (without variable substitution).
The advantages of the research method are the following:
logical clarity in the problem development, useful to generate
models, restudy is facilitated, synthetic-analytic development
in the discourse and explicit logical exposition of the
problems. The disadvantages of the research method are
summarized in the following points: a clear mastery of logic,
synthesis and analysis is required, it’s time consuming, and
it’s difficult for average students because it needs good
grammatical abilities.
The following are the advantages of the combined
method: knowledge could be understood in general and
particular terms, a schematized and logical development of the
problems, it exposes the synthetic-analytic abilities of the
students and it systematizes the analysis procedures. Some
disadvantages are: it’s time consuming, requires a deep
understanding of the principles and logic, the schematized
logical development is sometimes difficult and repetitive.
CONCLUSIONS
The main conclusions derived from this article are synthesized
in the next points:
• It’s necessary to develop procedures to help students to
understand, to analyze 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 necessary to
solve them, in this paper four methods are proposed
hoping to ease the problem solving procedure to any kind
of student (synthetic, analytic or synthetic-analytic
students)
• The recommendations proposed for the understanding and
formulation of problems and the four methods could be
applied to any field of engineering, to achieve 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 apply them to simple case studies and after that to
continue with more complex cases.
• It is possible to use the four methods jointly in order to
solve one problem and to fortify the understanding of the
theory behind it.
ACKNOWLEDGMENT
Thanks must go to those institutions that brought their support
to make this work happen: Universidad La Salle Noroeste
(ULSA), Instituto Tecnológico Superior de Cajeme (ITESCA)
and Universidad Tecnológica del Sur de Sonora (UTS) who
jointly form La Red Regional Alfa.
REFERENCES
[1]
Beer, F., P., Johnston, E., R., DeWolf, J., T., “Mechanics of Materials”,
Third Edition, McGraw-Hill, New York, NY, 2001.
[2]
Hibbeler, R., C., “Mechanics of Materials”, Fifth Edition, Prentice Hall,
Upper Saddle River, NJ, 2003.
[3]
Joseph, J., Rencis, Hartley, T., Grandin, Jr., “Educating Students to
Question, Test and Verify Problem Solutions”, Proceedings of the 2004
American Society for Engineering Education Annual Conference &
Exposition.
[4]
Jiménez, E., Ochoa, F., Martínez, M., Ruiz, J., “Métodos para el
planteamiento y solución de problemas: aplicaciones a problemas de
impulso y momentum lineal”, Folleto interno de divulgación #1, Red
Alfa, ISBN: 968-5844-14-3, México.
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FIGURE 5
FULL DEVELOPMENT OF COMBINED METHOD
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9th International Conference on Engineering Education
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