An Attempt to Strengthen “Outcome k” in a Mechanical Vibrations Course
David Che
Department of Engineering and Computer Science
Geneva College
Beaver Falls, PA 15010
Abstract
One of the challenges in teaching undergraduate mechanical vibrations curriculum is in the
necessity of extensive use of advanced math in deriving and solving differential equations (DFQ).
How to engage students in this seemingly tedious theoretical development is paramount for
effective teaching. There is no way to get around this theoretical development – actually the
instructor not only has to engage the students with the DFQ aspect of math, but he/she also has to
teach some fundamentals in dynamic modeling of mechanical systems in order to come up with
these DFQ equations. How to keep this derivational work interesting to the students is a
perennial challenge.
Another challenge is the need to prepare our students in using ever changing engineering
software and tools that are being used more widely in the industry. Commonly termed as
“outcome k”, ABET accreditation pressure just adds to such needs.
Having taught this course for seven years in a roll, the author had gone through several textbook
changes and tried various techniques and pedagogies with varying levels of success. In this paper,
a variety of tools in enhancing student learning are shared, which include the use of animation,
simulation and virtual lab assignments. Examples in each category are given. For animation and
simulation, Mathematica and Matlab/Simulink are used. For virtual lab development, an open
access web-based resource has been adapted for use in the undergraduate level mechanical
vibration curriculum. All these tools are easy to use and easy to implement in coursework. Some
outcome assessment data are shared.
Proceedings of the 2015 ASEE Gulf-Southwest Annual Conference
Organized by The University of Texas at San Antonio
Copyright © 2015, American Society for Engineering Education
Introduction
MEE 410 (Mechanical Vibrations) is a senior or junior-level engineering elective course for
engineering majors at Geneva College. It has an ABET accredited general BSE program with
concentrations in mechanical, civil, environmental, computer, chemical, electrical and
interdisciplinary engineering. Due to the EGR 214 (Dynamics) pre-requisite, which is only an
elective for concentrations other than mechanical engineering, almost all who took MEE 410
were mechanical engineering students over the past seven years (2009-2015).
The course was redesigned in spring of 2009 with increased emphasis on using modern
engineering tools such as MATLAB and Simulink in solving engineering vibration problems. This
change was mainly motivated by the instructor’s desire that students be taught the most advanced
engineering tools currently being used in the industry. Later on it was realized that this effort
also helped address an ABET assessment requirement for outcome k (ability to use the
techniques, skills and modern engineering tools necessary for engineering practice)1. And
teaching students these software and tools is a nice change of pace from the typical theoretical
derivations inherent in teaching this course. It helps engage students in classrooms and help them
see the applications more clearly.
One of the first challenges was finding an appropriate textbook that would support this pedagogy.
Over a dozen mechanical vibrations textbooks were evaluated. Table 1 shows the results of this
evaluation according to each text’s use of engineering software and other computer tools in
support of the mechanical vibrations curriculum. It can be seen that Matlab is the consensus tool
adopted by the majority of the authors. However, only one text used Simulink extensively in the
curriculum. Many textbooks were developed using Matlab, but the books themselves did not
include or only included very little coverage on how to use Matlab to solving vibration problems.
Despite of its many shortcomings (for example, as the first edition, it has many typos and errors),
we decided to use William J. Palm III’s text2 due to its excellent coverage of Matlab and
Simulink applications. It was the first mechanical vibrations textbook that teaches Simulink, and
it is still the only vibration textbook that teaches Simulink to this day, according to our
knowledge. Palm’s text has a strong "system dynamics" and "feedback control" flavor because of
the author's background (he authored several textbooks such as System Dynamics, Control
Systems Engineering, and Modeling, Analysis, and Control of Dynamic Systems). Over the years
we had tried some other textbooks, such as Benson Tongue's Principles of Vibration3 and Daniel
Inman's Engineering Vibration (3rd edition)4, and used Palm's book as a supplement on teaching
students how to use Simulink.
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Table 1 Survey of Textbooks that Support the Use of Modern Engineering Software and
Tools
Textbook listed by
author(s)
Mathematica
Matlab
Some examples
Extensive use – many examples
throughout the book
Recommended Matlab in the preface
but no examples or coverage in the
text
Some examples and toolboxes
William Palm III2
Benson H. Tongue3
Daniel J. Inman4
H. Benaroya & M.
Nagurka5
S. Graham Kelly6
Recommended but no examples or
coverage in the text
Mentioned in the preface but no
coverage in the text
W. T. Thomson & M.
D. Dahleh7
Some examples
B. Balachandran & E.
B. Magrab8
Some coverage provided by
publisher’s website
T. Schmitz & K. S.
Smith9
Alok Sinha10
Singiresu S. Rao11
Some examples
Simulink
MathCad
Extensive use –
many examples
Some
examples
Some examples
Some examples
The revised course description says:
MEE 410 Mechanical Vibration (3 credits) Introduction to the modeling, analysis and design
of mechanical vibrating systems. Steady state and transient analysis of systems with a single
or multiple degrees of freedom. Free, harmonic and forced responses of such systems.
Extensive use of modern engineering computational tools such as MATLAB and Simulink.
Spring semester. Prerequisites: MAT 261 (Calculus III), MAT 405 (Differential Equations),
EGR 214 (Dynamics).
For course outcomes, the syllabus stated that after successfully completing this course, the
student will:
1. Be able to use the MATLAB ode functions and/or Simulink to solve ordinary differential
equations.
2. Be able to obtain the harmonic response of systems having a single degree of freedom.
3. Be able to obtain the transfer function from the equation of motion.
4. Be able to use the plots or the frequency transfer function to determine the steady-state output
amplitude and phase that result from a sinusoidal input.
5. Be able to determine the resonance frequency, peak response, and bandwidth.
6. Be able to use MATLAB and/or Simulink to analyze harmonic response.
7. Be able to apply the Fourier series method to obtain the response of a linear system to a
periodic forcing function and apply the amplitude spectrum plot.
8. Be able to use MATLAB and/or Simulink to obtain the response of a linear system to a variety
of forcing functions, such as step, impulse and other arbitrary input functions.
9. Be able to use MATLAB and/or Simulink to obtain the response of a nonlinear system to a
variety of forcing functions, both linear and nonlinear.
Proceedings of the 2015 ASEE Gulf-Southwest Annual Conference
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Most of the course objectives above relate to outcome k. Detailed outcome assessment data
on using Matlab/Simulink is included in a later section of this paper.
The reason we like Simulink is due to its versatility in handling dynamic systems. Simulink
is built on top of Matlab. It has a very user-friendly Graphical User Interface (GUI). You can
build a dynamic system simply by dragging and dropping and connecting some blocks of
functions and operators. It is intuitive and fun to learn. Many students develop a love of
mechanical vibrations as a subject thanks to the pleasures they got from using Simulink.
Simulink is especially advantageous in handling nonlinear or discontinuous functions as
input to dynamic systems. It is well equipped to handle periodic and non-periodic forcing
functions such as a train of pulse (rectangular, triangular or trapezoidal). With a graphical
interface, user can easily enter special waveforms. This saves users a lot of effort developing
Fourier series of representation of these waveforms. With this tool, coverage on forcing
function transformation using Fourier series could be shortened.
Other uses of these tools include student capstone senior designs. Pakkala introduced a
vehicle dynamics model based on Simulink for design and simulation of SAE Baja vehicles
at Milwaukee School of Engineering21.
Animation of System Responses Using Mathematica
One of the challenges in teaching undergraduate mechanical vibrations curriculum is in the
necessity of extensive use of advanced math in deriving and solving differential equations (DFQ).
How to engage students in this seemingly tedious theoretical development is paramount for
effective teaching. There is no way to get around this theoretical development – actually the
instructor not only has to engage the students with the DFQ aspect of math, but he/she also has to
teach some fundamentals in dynamic modeling of mechanical systems in order to come up with
these DFQ equations. How to keep this derivational work interesting to the students is a
perennial challenge.
Typically the teaching of mechanical vibrations starts with introducing students to the simple
harmonic motion, as represented by a sinusoidal function of the form
𝑥(𝑡) = 𝐴𝑠𝑖𝑛(𝑤𝑡 + ∅)
The instructor needs to explain what each of the parameter represents in the oscillation behavior
of the system. But it is kind of hard to visually demonstrate the effects or contributions of each
individual parameter with mere words and equations. Previous work in this area included
extensive GUI programming in either Matlab or LABView12, 13. Aung20 used an example from
Wolfram to demonstrate the simulation of vibration of a string. However, it is a canned GUI and
is not readily transferrable to other situations and applications. The author found a useful tool in
Mathematica that could be easily adapted for a variety of applications. It is the “Manipulate”
Proceedings of the 2015 ASEE Gulf-Southwest Annual Conference
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function in Mathematica14. A simple command below would generate a window as shown in
Figure 1.
Manipulate[Plot[A*Sin[wn*t + phi], {t, 0, 10}], {A, 0, 10}, {wn, 0, 5}, {phi, 0, 100}]
Figure 1 Mathematica’s Manipulate function as used in the demonstration of a sinusoidal
function
In the above window, students can use mouse to drag the bars and change in real time the
amplitude, natural frequency, and phase angle. As these parameters are being changed, students
can observe its impact on the vibration signal. It is powerful! Previously boring material like this
becomes so fun to learn that it literally becomes “playing!” for the students. And once students
are introduced to the use of this function, later on in the semester they can easily adapt it to
observe more complex damped oscillations.
Figure 2 Under-damped oscillation demonstrations
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The following command could be used to show the typical under-damped solution 𝑥(𝑡) =
𝐴𝑒 −𝑡/𝜏 sin(𝑤𝑛 ∗ 𝑡 + ∅) in Mathematica:
Manipulate[Plot[A*Exp[-t/tau]*Sin[wn*t + phi], {t, 0, 10}], {A, 0, 10}, {wn, 0, 5}, {phi, 0, 100}, {tau, 1, 20}]
Figure 2 shows the under-damped response. Figure 3 shows a near-critically damped response.
All four parameters (amplitude, natural frequency, phase angle, and time constant) could be
changed by students using the slider bars on top.
Figure 3 Near-critically-damped oscillation demonstrations
Virtual Labs
As echoed by Jouaneh and Palm23, “most mechanical engineering curricula include courses in
system dynamics, controls, mechatronics and vibrations, but at most schools, these courses do
not have a laboratory component. Even at schools that have such a component, laboratory access
is often limited, and thus there is a need to increase students’ laboratory experience.”23
Virtual and remote laboratories are gaining popularity in engineering education due to its
flexibility on staff, time and space. According to Chen, et al22, virtual laboratories are based on
software such as LabVIEW, Matlab/Simulink, Java Applet, Flash or other software to simulate
the lab environment. University of Colorado Boulder developed a series of online interactive
teaching tools to help with teaching science and math subjects15. Termed PhET Interactive
Simulations, it is a non-profit open educational resource (OER) project founded in 2002 by Nobel
Laureate Carl Wieman16. PhET began with Wieman’s vision to improve the way science is
taught and learned. Their stated mission is "to advance science and math literacy and education
worldwide through free interactive simulations."
Proceedings of the 2015 ASEE Gulf-Southwest Annual Conference
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We developed two virtual mechanical vibration labs from two simulation tools provided by
PhET. These are spring and mass lab (Figure 4) and pendulum lab (Figure 5). Normally these
two labs are assigned during the first two weeks of the semester and students can pretty much
work on these assignments with minimum instructions. It helps students visualize and get a “feel”
for the mechanical vibration phenomenon at hand.
Figure 4 Spring and Mass Virtual Lab17
Figure 5 Pendulum Virtual Lab18
For the spring and mass lab, students can hang masses from springs and adjust the spring
stiffness and damping. Students can even slow time to help with measurements. Students can
also transport the lab to different planets. It includes a chart showing the kinetic, potential, and
Proceedings of the 2015 ASEE Gulf-Southwest Annual Conference
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thermal energy for each spring. Unique questions were developed for this lab – due to page
limits in this paper, these questions are not included here. If anyone is interested in using them,
please contact the author and he would be glad to email you the questions (the author will start
teaching at Anderson University in Anderson, Indiana, in the fall of 2015, so make sure you
google and find his new email address at Anderson). Tables 2 and 3 give a summary of what,
how and when some of the basic educational outcomes were introduced or reinforced by these
lab activities. Not surprisingly, there has been some staggering of coverage, which is good to
stimulate students’ thinking and reinforcement the basic concepts required of a vibrations course.
Table 2 Coverage of Basic Vibration Concepts and Principles by the Spring and Mass Lab
Questions
Basic concepts/experimental techniques
Definition of stiffness of springs (or spring rate);
measurement thereof
Damping/internal friction
Factors contributing to oscillation frequency
Free vs. forced vibration - definition
Oscillation period and its measurement
Conservation of energy; energy conversion
Gravitational constant and its effect
Initial condition and its effects
Response parameters (amplitude, oscillation
frequency, phase angle)
Equilibrium position and its significance
Appeared in these questions
1; 3; 4; 8; 10
2; 5; 7
2; 6; 8; 9; 11; 12
2
6; 8; 11; 12
7;
8;
9; 10; 12;
9; 12
10; 12
Table 3 Coverage of Basic Vibration Concepts and Principles by the Pendulum Lab
Questions
Basic concepts/experimental techniques
Nonlinear vs. linear solutions (exact but need
numerical solution vs. small angle approximation
but closed-form solution)
Damping/internal friction
Factors contributing to oscillation frequency
Velocity and acceleration
Oscillation period and its measurement
Conservation of energy; energy conversion
Gravitational constant and its effect
Initial condition and its effects
Response parameters (amplitude, oscillation
frequency, phase angle)
Free vs. forced vibration
Appeared in these questions
2; 3
7; 9
4; 6
8; 11
1; 2; 3; 4; 6; 10
7; 11
5; 10; 11
1; 10; 11
11
9; 11
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For the pendulum virtual lab18, one of the key concepts we try to instill in students’ mind is the
non-linear response of the system. Due to small angle approximation, we can simplify the
solution of the differential equation and obtain a closed-form solution. But there are errors in it.
How large are the errors? We ask students to measure the oscillation periods at different initial
conditions (release angles). Later on in the semester, once we get into teaching students using
Matlab ode solvers to solve for the nonlinear equations, we again ask students to plot the percent
difference (error) between the linear solution and nonlinear solution.
Assessment
The first time we implemented these two virtual labs and the use of Mathematica Manipulate
function was in spring semester of 2012. There were 16 students in the class. At the end of the
semester, an anonymous survey was conducted and 14 responses were collected. The results are
shown in Table 4.
Table 4 Student Responses to the Survey
Strongly agree
Agree
Disagree
Strongly disagree
The use of “Manipulate” function in Mathematica helped me understand the
contribution of various parameters in a vibration model
3
10
1
0
For the virtual labs, we only have some indirect evidence, since the survey question was on
students’ reaction to the “practicum” sessions we offered in class, which included not only the
virtual labs but also some problem solving activities. A majority of the students responded
positively to the practicum sessions. There were, however, two students who felt the practicum
sessions took valuable class time and they’d prefer it be assigned as homework. Since then we
have been assigning the virtual labs as homework. From a year to year comparison of student
IDEA evaluations, there has been a significant increase in scores in the areas of stimulating
student interest and encouraging student involvement.
In spring semester of 2013, roughly 33.8% of the overall homework grade and 64% of the final
exam grade were based on a student’s performance in using Matlab/Simulink. The students were
given two lectures (1 hour and 20 minutes is the typical length of each lecture) of training on
using fundamentals of MATLAB at the beginning of the semester. Continued training and
application of the software were conducted throughout the semester. Usually the instructor used
the first 40-50 minutes of class time to teach on theory and concepts, and then devoted the
remainder of the class time to hands-on use of the software to solve problems. The classroom
was a fully equipped computer lab and each student had a computer to use. Most of the time
students worked individually under the supervision of the instructor, with some discussions
among themselves. The instructor also encouraged them to help each other out as much as
Proceedings of the 2015 ASEE Gulf-Southwest Annual Conference
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possible in and outside of class. Sometimes students would come to ask software related
questions during office hours and the instructor would go with the student to a computer lab and
help them debug their codes or teach them how to use certain features of the software. Table 5
shows the rubric that was used to assist in the Matlab/Simulink teaching effectiveness
assessment process and the assessment results for various outcomes19. There were a total of nine
students in spring semester of 2013, two are seniors and seven are juniors. Except for one student,
who was quite proficient in using the software throughout the course, all other eight students
needed some assistance and occasionally applied the tool inappropriately.
Table 5 Rubric19 and Assessment Results (Numbers in the table are the number of
students at the level indicated in the heading)
Using Matlab
to solve
ordinary
differential
equations (ode
solver)
Using Matlab
to find roots to
polynomial
equations
Clueless –
Unable to use
tool and does
not try
(0)
Attempts to
use tool but
does so
incorrectly
and
consistently
gets incorrect
results
(1)
Proficient use
of tool with
minimal
assistance and
consistently
applies tool
appropriately
(4)
a,
b, c
8
1
3.1
a,
b, c
8
1
3.1
8
1
3.1
8
1
3.1
Using Simulink
a,
to model a
b, c
second order
system
Using Matlab
a,
to obtain
b, c
frequency
response of a
system
a. Observation of students during lab
b. Homework assignments
c. Final Exam
Occasionally
able to use
tool correctly
by guess
work (2)
Average Score
Able to use tool
but needs some
assistance and
occasionally
applies the tool
inappropriately
(3)
Means for Observation
Technique,
Skill,
Tool
Figure 6 shows the aggregated scores (in percentage) of each student for their performance that
relate to outcome k in their homework and final exam. Six of the nine performed better in the
final exam than in their homework, which indicates that the majority of them show a continuous
improvement in the use of these tools. Student #5 seems to have the solution manual based on
the instructor’s observation from his homework (probably he purchased an illegal copy from
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online), which explains why he did better in homework than the final exam. Figure 7 through 10
show the detailed homework evaluation results for individual outcomes as previously listed. In
Figure 7, student #3 did not turn in his homework.
120%
100%
80%
60%
homework
40%
final exam
20%
0%
1
2
3
4
5
6
7
8
9
Figure 6 Aggregate percent scores for each of the 9 students in portions of the homework
and final exam that are related to the use of MATLAB and Simulink software
120%
100%
80%
60%
40%
20%
0%
1
2
3
4
5
6
7
8
9
Figure 7 Percent scores for each of the nine students in a homework assignment that is
related to the use of MATLAB ode solver
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100%
95%
90%
85%
80%
75%
70%
1
2
3
4
5
6
7
8
9
Figure 8 Percent scores for each of the nine students in a homework assignment that is
related to the use of Simulink and Laplace transform to obtain forced response of a second
order dynamic system
120%
100%
80%
60%
40%
20%
0%
1
2
3
4
5
6
7
8
9
Figure 9 Percent scores for each of the nine students in a homework assignment that is
related to the use of Matlab/Simulink to find bandwidth of a dynamic system and plot
amplitude spectrum
120%
100%
80%
60%
40%
20%
0%
1
2
3
4
5
6
7
8
9
Figure 10 Percent scores for each of the nine students in a homework assignment that is
related to the use of Matlab/Simulink to solve a dynamic system with either a nonlinear
model or a nonlinear input
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Overall, it can be concluded that the entire class achieved a satisfactory outcome in mastering the
use of these software tools. It can also be shown that students’ critical thinking and problem
solving skills are enhanced with the use of these tools.
Summary
Using animation, simulation and virtual labs is an economical way of enhancing student learning
in classes where traditionally a lot of “dry” theoretical derivation and equation solving are
involved. It has no limitation in terms of lab space and lab hours. As long as students have access
to a computer, they can do the virtual labs on their own time and at their own pace. The lab
questions help them dig deep into the course materials. The staggered coverage of fundamental
concepts and principles help reinforce what was covered in class. As long as the school has
Mathematica software, the use of “Manipulate” function proves to be easy to implement in class
demonstration of complex mathematical or physical relationships. Students need not know how
to use Mathematica beforehand. The learning curve is virtually non-existent, but the benefit is
significant in helping students to visualize what they are learning in theory. Another part of the
simulation activities we had engaged the students with was the extensive use of Matlab and
Simulink later in the semester. As evidenced by the assessment data, this enhanced students’
learning. As part of the engineering curriculum renewal effort at Geneva College this year, all
mechanical engineering students will be required to take a Matlab-based programming course to
substitute the traditional C++ based programming course. This will further enhance the effort to
strengthen outcome k in the mechanical engineering curriculum.
References
1.
ABET, Criteria for Accrediting Engineering Programs, page 3,
http://www.abet.org/uploadedFiles/Accreditation/Accreditation_Process/Accreditation_Documents/Current
/eac-criteria-2012-2013.pdf, accessed on 3/7/2014
2. William J. Palm III, Mechanical Vibration, John Wiley & Sons, Inc., 2006
nd
3. Benson H. Tongue, Principles of Vibration, 2 edition, Oxford University Press, 2002
4. Daniel Inman, Engineering Vibration, 3rd edition, Prentice Hall, 2008
5. Haym Benaroya & Mark L. Nagurka, Mechanical Vibration – Analysis, Uncertainties, and Control, 3rd
edition, CRC Press, 2010
6. S. Graham Kelly, Mechanical Vibrations – Theory and Applications, Cengage Learning, 2012
7. William T. Thomson & Marie D. Dahleh, Theory of Vibration with Applications, 5th edition, Prentice Hall,
1998
8. B. Balachandran & E. B. Magrab, Vibrations, 2nd edition, Cengage Learning, 2009
9. T. Schmitz & K. S. Smith, Mechanical Vibrations – Modeling and Measurement, Springer, 2011
10. Alok Sinha, Vibration of Mechanical Systems, Cambridge University Press, 2010
11. Singiresu S. Rao, Mechanical Vibrations, 5th edition, Pearson and Prentice Hall, 2011
Proceedings of the 2015 ASEE Gulf-Southwest Annual Conference
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12. Thomas Nordenholz, Animation as the Final Step in the Dynamics Experience, ASEE Annual Conference,
2006
13. Peter Avitabile, Jeffrey Hodgkins, Development of Visualization Tools for Response of 1 st and 2nd Order
Dynamic Systems, ASEE Annual Conference, 2006
14. Wolfram Mathematica 9 Documentation Center, Manipulate,
http://reference.wolfram.com/mathematica/ref/Manipulate.html, accessed on 3/7/2014
15. http://phet.colorado.edu, accessed on 3/7/2014
16. http://en.wikipedia.org/wiki/PhET_Interactive_Simulations, accessed on 3/7/2014
17. http://phet.colorado.edu/en/simulation/mass-spring-lab, accessed on 3/7/2014
18. http://phet.colorado.edu/sims/pendulum-lab/pendulum-lab_en.html, accessed on 3/7/2014
19. Robe Liljestrand, Dave Clark, Proposal for Assessing Student Outcome k, Geneva College Department of
Engineering Internal Communication, 2013
20. Kendrick T. Aung, Teaching Advanced Engineering Mathematics to Graduate Students: Lessons Learned,
Proceedings of the ASEE Annual Conference, 2011
21. John E. Pakkala, A Vehicle Dynamics Design and Simulation Tool for Capstone Projects, Proceedings of
the ASEE Annual Conference, 2011
22. X. Chen, G. Song and Y. Zhang, Virtual and Remote Laboratory Development: A Review, Proceedings of
Earth and Space 2010, pp. 3843-3852, Honolulu, HI, March, 2010
23. Musa Jouaneh, William Palm, System Dynamics and Control Take-home Experiments, Proceedings of the
ASEE Annual Conference, 2010
DAVID CHE
Dr. Che currently serves as Associate Professor of Mechanical Engineering at Geneva College in Beaver Falls, PA.
He also serves as Director of the Pinkerton Center for Technology Development at the college. His research interests
are manufacturing engineering, quality control, and precision engineering. Starting in fall of 2015, Dr. Che will be
teaching at Anderson University in Anderson, Indiana.
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