Structural Dynamics Education at the University of Cincinnati
Randall J. Allemang, PhD
Structural Dynamics Research Laboratory (UC-SDRL)
Department of Mechanical, Industrial and Nuclear Engineering
University of Cincinnati, Cincinnati, OH 45221-0072
Email: Randall.Allemang@UC.EDU
Nomenclature
BOK
CHEM
CO-OP
ENFD
INDS
MATH
MECH
MINE
MTEN
PHYS
Breadth of Knowledge Course (Humanity/Social Science)
Chemistry
Cooperative Education
Engineering Fundamentals
Industrial Engineering
Mathematics
Mechanical Engineering
Mechanical, Industrial and Nuclear Engineering
Materials Engineering
Physics
Abstract
Modal analysis and structural dynamics has been a research area of interest at the University of Cincinnati since
the early 1960s. Much of the focus of this research activity has been concerned with the experimental approach
to understanding the vibration, acoustic and general structural dynamics of structures and systems. During this
time, the research and technology that has been developed has found its way into the educational program at
both the undergraduate and graduate levels, including both required and elective coursework. The impact of
research as well as the changing emphasis of structural dynamics technology has clearly impacted the
educational process, including what material is taught as well as how the material is taught at all levels.
This paper reviews some of the lessons learned during this ongoing development over nearly fifty years. Insights
on what material students easily assimilate versus the material that is harder to understand, the use of theoretical
and practical data projects and the integration of experimental methods into the coursework are all reviewed.
1. Introduction
The University of Cincinnati has a rich history of educating students in the field of structural dynamics over the last
40+ years. This history developed based upon a number of individuals who pioneered technology and application
in experimental and analytical structural dynamics beginning with research on machine tool dynamics in the
1960s. As the technology matured and moved into other applications areas (automobiles, air and space
structures, disc drives and other consumer electronics, etc.), the concepts, methods and material that were
developed for the research program gradually moved into the graduate and then the undergraduate education
programs. Graduates of both the undergraduate and graduate programs in mechanical engineering (ME) from
the University of Cincinnati over this time period have had a great influence on the application of structural
dynamics technology in industry. Figure 1 shows a diagram of structural dynamics companies that evolved
substantively from the University, either directly as spin-off companies or indirectly as spin-offs from the initial
spin-off companies. In addition to this impact, graduates from the University actively impacted many other
companies in the use of structural dynamics technology as research, test and project engineers. Today, the
influence of what is taught in structural dynamics, and how it is taught, at the Senior undergraduate level and the
MS and PhD graduate level now reaches down into the coursework that is taught at the Freshmen level.
Figure 1: UC Impact on Structural Dynamics Technology Companies
This paper is a brief overview of the structure of the Mechanical Engineering Program at the undergraduate and
graduate levels. While no direct discussion of the research activity will be included, the courses are taught so that
students are prepared to go to industry to utilize, or work in conjunction with, structural dynamics technology or so
that students are prepared to go on to work on research programs at other Universities, industry or government
labs extending the state of the art of structural dynamics technology. With this in mind, the curriculum will be
discussed from the viewpoint of required undergraduate coursework, transition coursework that is elective and
may be taken by Seniors or first year graduate students (also referred to as dual level coursework) and graduate
coursework that is only available to MS and PhD students. Some specific examples of course structure and
course material will be noted; further details are available by contacting the author.
2. Curricula Background
The University of Cincinnati, College of Engineering has a long history of mandatory, co-operative education at
the undergraduate level. The co-op program began in 1906 and is considered the oldest co-op education
program in the world. The co-op program means that, beginning after the fourth academic quarter, students
alternate quarters between the University and a work assignment with a company that participates in this
program. While the University is involved in finding and screening companies that participate in the co-op
program, the student interviews and chooses his/her company assignment based upon job description, location
financial considerations, etc. just as in a normal job search process. The University requires that a student
participate in as many work quarters as are available to the student, from the time they begin until the first quarter
of their Senior year. This allows most students six work assignments unless they choose to be involved in special
programs or need to attend classes an extra quarter or two to make up for missed classes due to illness or poor
performance.
The requirement of co-op work assignments means that the normal program at UC is a five year program in order
to get twelve quarters (four years) of engineering education and six quarters of work experience. While some
view this as a disadvantage, the focus of the students who co-op is much better since they are gaining experience
in an engineering field. From a financial point of view, the work assignments amount to a form of scholarship
valued at approximately $40,000 over the five year period ($15/hour x 40 hour/week x 11 weeks/quarter x 6
quarters). Even with this advantage, the College of Engineering is now adding a new program (ACCEND,
ACCelerated Engineering Degree) for students who enter the College with advanced standing from high school to
allow these students to pursue a MS degree and a BS degree in five years by reducing the number of required
co-op work assignments to five and adding courses along the way. For complete information concerning the coop program in Engineering at UC, please visit the UC Co-Op Engineering Web Site
(www.eng.uc.edu/prospectivestudents/coop/).
2.1 Undergraduate BSME
The undergraduate BSME curriculum at UC in substantively similar to most ME programs at any university that is
accredited by the Accreditation Board for Engineering and Technology (ABET). ABET reviews all engineering
programs in the US at least every six years in order to maintain somewhat common standards among US
engineering programs. However, each ME Program is allowed to structure their coursework as they deem
appropriate which means courses will have different names and material may be spread across various courses
in different ways. In the end, if a program does not meet criteria set by ABET, the program is subject to loss of
accreditation is adjustments are not made. Figure 2 is an overview of how the material is divided between
different courses and different years of the BSME Program at UC. Naturally, this Figure gives a good curriculum
outline but is very concise and may not be completely understandable unless the course outline for each course is
consulted to see what material is covered in each course.
While it is clear that most courses at an undergraduate level provide the background for some aspects of the
structural dynamics area, a few select courses have been targeted specifically in order to prepare the students for
structural dynamics material. One good example of this is the Computer Programming course taught in the first
quarter of the Freshman Year to all BSME students. This computer language course has historically utilized one
of the traditional programming languages (Fortran, Basic, Pascal, C) in order to teach programming language
concepts. The language of choice has changed over the last 40 years based upon the most appropriate
programming language for technical, engineering programming as time has passed and computer languages
have evolved. Today, many students will have a overview of programming in high school that may include some
or all of these languages as well as a web scripting language like HTML or JAVA when they enter the University.
The current programming language that is being used for this course is Matlab® supplied by The Mathworks, Inc..
While some have tried to argue that this is not a true programming language, it contains all of the normal
programming constructs and satisfies the goal of teaching a computer language to engineers. Furthermore,
Matlab® contains significant capability in numerical analysis, plotting and data presentation, and links to other
relevant engineering software including data acquisition. Knowledge of Matlab® provides the students with a tool
that can be utilized in many successive classes. The focus and history of Matlab® in applied linear algebra
makes it an extremely useful tool for an engineering student.
Mechanical Engineering Curriculum (Class of 2010)
Fall
(17)
CHEM101
Chemistry
Freshman Year
Winter
(17)
ENFD112
Comp
Language
Second Year
Sum/Fall
Win/Spr
(17)
(17)
CHEM102
Chemistry
ENGL101
Eng Comp
MATH251
Calculus I
Spring
(17)
MECH210
Thermo I
ENGL102
Eng Comp
MATH252
Calculus II
MATH253
Calculus III
MATH254
Calculus IV
Third Year
Sum/Fall
Win/Spr
(17)
(17)
Fourth Year
Sum/Fall
Win/Spr
(17)
(19)
Fall
(17)
MECH311
Fluid Mech
MECH312
Thermo II
MECH413
Heat Xfer
MECH414
Moment
MINE585
Clinic I
MINE586
Clinic II
MINE587
Clinic III
MECH320
Kin/Dyn
MECH321
Mach Anal
& Design
MECH422
Comp Des
MECH423
Sys Design
MINE
Elective
MINE
Elective
MINE
Elective
MECH480
Vibs I
MECH481
Controls
Guided
Elective
Guided
Elective
Guided
Elective
MATH273
Diff Eqn
MATH256
Calc Lab
MATH257
Calc Lab
MATH258
Calc lab
PHYS201
Physics I
PHYS202
Physics II
PHYS302
Physics III
ENFD375
Strength of
Materials
PHYS211
Physics
Lab
PHYS212
Physics
Lab
PHYS213
Physics
Lab
MINE340
Stats I
ENFD250
Graphics
ENFD101
Mechanics
I
ENFD102
Mechanics
II
ENFD103
Mechanics
III
PD120
Intro to
coop
Spring
(17)
Half each quarter
ENFD371
Circuits
COOP120
Coop Eval
MECH571
S/M Lab
MECH370
Meas &
Instru
MECH572
HT/F Lab
MINE
Elective
BOK
BOK
MECH342
Num Anal
INDS354
Mfrg Proc
MINE451
Eng Econ
MTEN201
Intro to
Metallurgy
MECH100
Intro to ME
Senior Year (48)
Winter
(17)
BOK
PD502
PD II
BOK
BOK
BOK
BOK
BOK
COOP120
Coop Eval
COOP120
Coop Eval
COOP120
Coop Eval
COOP120
Coop Eval
COOP120
Coop Eval
Figure 2: Basic Undergraduate Curricula for BSME Students at UC
This brings up the issue of the use of software in the BSME curriculum. Software is ubiquitous in an engineering
curriculum and must be carefully utilized to teach principles rather than expertise in running a black-box software
program. An even bigger problem is that there seems to be several individual software programs that could be
utilized in every course which would mean that students would be spending all of their time learning new software
in each course and little time concentrating on the fundamentals. Finally, there is the recurring problem of
software bugs in any software that often yield the software ineffective and frustrating for the students, and faculty,
to use. For this reason, the focus is to find software that integrates a number of concepts and, thus, can be used
by several courses as a platform and/or tool to demonstrate principles. Likewise, software that is open
architecture that allows bugs to be easily evaluated and fixed is of great interest as well. Software tools are like
any tool in that the best tools are those that can do many things and solve many problems. For that reason, the
software chosen for the Graphics class in the Freshman Year is a 3-D modeling program, Solid Edge® from UGS,
which integrates with other more advanced, 3-D modeling programs like NX®, also from UGS and also has strong
integration into finite element modeling programs, kinematic modeling and simulation programs like ADAMS® and
manufacturing programs that can control rapid prototyping or CNC machining equipment. Software programs that
are implemented in Matlab® are very interesting in that they build on the student’s knowledge of theat
programming environment and are generally somewhat open in architecture, at least from the data structure.
This allows students to implement their own solutions by working with the data directly.
The primary courses that support the structural dynamics area in the required course structure of the
undergraduate BSME program at UC are highlighted in Figure 2 in yellow and green. The courses highlighted in
yellow are supporting material that are classic in any ME Program and appear in most ME undergraduate
curricula in the US. The courses highlighted in green are courses that have been substantively altered to
emphasize structural dynamics material and experimental methods that experience suggests are important to
practicing engineers in this technical area. Most of this material is developed in addition to use of classic
textbooks. This material is available to students electronically in Adobe PDF format that the students can
download at no cost from the official UC Blackboard Web Site (www.blackboard.uc.edu) or from the UC-SDRL
Web Site (www.sdrl.uc.edu).
From the courses listed in Figure 2, a number of the courses specifically utilize problems from experimental
structural dynamics technology areas to highlight course material. For example, MECH370 (Measurements and
Instrumentation) and MECH571 (Structures/Motion Lab) are both laboratory courses where students are
introduced to and utilize sensors and data acquisition. Calibration methods are introduced in MECH 370 and
emphasized in MECH 571 as are time/frequency relationships together with the FFT algorithm. In parallel with
the first lab class, MECH 370, an applied math class, MECH342 (Numerical Analysis), reinforces least squares
methods and eigenvalue and singular value decomposition/solution methods. This applied math class also
utilizes Matlab® as the platform for investigating these linear algebra methods and once again reinforces the
programming concepts introduced in the Freshman year. A number of other classes in the Structures-Motion
Stem of the curriculum, as well as in the Heat Transfer-Fluids Stem of the curriculum, reinforce Matlab® as a tool
to execute the ongoing day-to-day coursework. The analog circuits course, ENFD371, which follows the math
course, MATH273 (Differential Equations), introduces RC and RLC filters from an theoretical point of view which
allows AC and DC coupling to be explored in the following lab courses MECH370 and MECH571 from an
experimental point of view.
Probably the most significant sequence of courses for experimental structural dynamics, is the MECH480
(Mechanical Vibrations I) and MECH481 (Controls) sequence which is followed by Senior or Dual Level Electives
in the Senior Year. These courses introduce typical material (one and two degree of freedom (DOF) lumped
mass systems in MECH480 and open and closed loop systems in MECH481) but emphasize the experimental
nature of the input-output relationships in both cases. Significant effort is made to provide students with physical
examples of the complex valued math relationships present in these two courses and of the linkage between the
frequency variable, ω, used in vibrations and the frequency variable, s, used in controls. This is done in two
ways: 1) by using the classic Fourier and Laplace Transform approach and 2) by using direct substitution, always
assuming a damped solution as the general case in vibrations following the general approach of most controls
textbooks and by using Euler’s Identity for sine and cosine to move from real to complex valued notation. Also,
considerable effort is made to acquaint the student with what measurements will look like, particularly the
frequency response function, for simple one and two degree of freedom systems. Elementary modal parameter
estimation relationships and methods are introduced in this course and then utilized in the Senior level lab course
MECH 571. Further comments and guidelines on how this sequence of courses is taught will be addressed in a
later section (Section 3).
As part of this sequence of courses, a number of different Electives can be taken in the Senior year to provide
further depth for the interested student. In Figure 2, most of these courses are identified as Technical Electives or
Guided Electives. Technical Electives are generally engineering courses within the ME Program. Guided
Electives are courses within or outside the Program that have been identified as possible selections for students
and may be engineering courses or a course needed for a career path needed by a number of engineering
students (business or law courses for students interested in more of a business career, anatomy and physiology
for biomedical engineering, etc.). With respect to the structural dynamics area, common courses are listed in
Table 1. In Table 1, 500 series courses are available to Seniors only, 600 series courses are available to Seniors
and graduate students (thus referred to as Dual Level) and 700 series and above are for graduate students only.
In addition to the courses listed in this table, courses are available in civil engineering, materials engineering and
aerospace engineering that will be interesting to students studying in the structural dynamics area.
For Senior students interested in the structural dynamics area, Table 1 lists a number of courses that students
can choose from. Naturally, a two quarter, finite element course sequence is available, strictly at the Senior level,
to introduce students to this analytical methodology (a more advanced sequence in finite elements is available for
graduate students). This sequence can also be supplanted with Dual Level courses in advanced strength of
materials, elasticity and advanced dynamics. There are a number of Dual Level courses offered for entering
graduate students or Senior level students that follow the vibrations/controls sequence. These courses include a
sequence of two more quarters of vibrations, one more quarter of controls, a course in Fourier and related
transforms and a two course sequence in acoustics. These courses will be explained more fully in the following
sections.
Course
Credit
Title
MECH525
MECH526
3
3
Intro to Finite Element Methods I
Intro to Finite Element Methods II
MECH621
MINE636
MINE638
MINE639
MECH660
MECH662
MECH663
MECH664
MECH666
MECH667
MECH668
3
3
3
3
3
3
3
1
3
3
3
Modeling and Simulation
Introduction to Robotics
Robot Design
Robot Vision
Fourier Transform Techniques
Mechanical Vibrations II
Mechanical Vibrations III
Mechanical Vibrations III Lab
Acoustics I
Acoustics II
Vehicle Noise and Vibration
MECH781
MECH794
MECH795
MECH796
3
3
3
3
Nonlinear Vibrations
System Dynamics I
System Dynamics II
System Dynamics III
Table 1: Elective Courses in Structural Dynamics
2.2 Dual Level Courses – BSME and MS
The courses listed in Table 1 with 600 series numbers are Dual Level courses, available to Seniors and all
graduate students but primarily the MS students. Of these courses, the primary sequence of courses in the
structural dynamics area is the sequence of courses involving Fourier Transform Techniques (MECH660) and
Vibrations (MECH662, MECH663, MECH664). These courses serve to introduce students to both the theoretical
side of general multiple degree of freedom systems and also the practical aspects of working with these concepts
on an experimental level. The concept of measurement degrees of freedom versus theoretical degrees of
freedom is introduced so that students can understand the limitations of measured data with respect to theory.
Specialized notes and experimental exercises emphasize these issues and prepare the students for further work
in other classes or work with commercial hardware and software in industrial situations. While the general
aspects of hardware and software is discussed, students are expected to develop an appreciation for the big
picture from this coursework rather than become proficient in the details and nuances of measurements and
parameter estimation. Major characteristics of these courses will be discussed in Section 3.
2.3 Graduate Level Courses – MS and PhD
The primary sequence of courses that follow the Fourier Transform Techniques and Vibrations sequence is the
System Dynamics Analysis sequence (MECH794, MECH795, MECH796). This sequence of courses followed
fro a complete year examines the details of the numerical methods involved in making measurements utilizing
FFT methods, the details in modal parameter estimation methods, and the details in modal and impedance
modeling methods. Students, working in groups of 3 to 5, develop specialized software using Matlab® in order to
process a large matrix of time data into measurements, then to estimate modal parameters from the
measurements and finally to estimate structural modifications with modal and/or impedance modeling methods.
By programming these methods in Matlab®, the students understand fully the theory involved as well as the
practical measurement and numerical issues on which structural dynamics methods are based.
3. Course Structure/Content
In this section, specific characteristics and curriculum content will be highlighted for a subset of courses in the ME
Program that are effective in providing the students with an appreciation and comfort level for structural dynamics.
3.1 Mechanical Vibrations I (MECH480)
As the first course that emphasizes the dynamics of structures, Mechanical Vibration I has to undo the bad habits
that students develop from earlier courses that teach fundamentals but then work simplistic problems that do not
require rigorous application of the fundamentals. An example of this is a simple translating cart with a pendulum
hanging from the cart representing a 2 degree of freedom problem. Moving reference frames involving rotational
coordinates are rarely a problem that is encountered before a dynamics course that considers 2 degree of
freedom problems. Students rarely can handle this problem without relearning some fundamental mechanics and
developing a physical feel for the problem. While Newton’s Law is well understood theoretically, the application to
this problem is troublesome for students. The introduction of a two-sided free body diagram (applied and internal
forces on one side and inertia forces/moments on the other side in a modified d’Alembert solution method has
greatly assisted students in understanding what they did not understand previously.
Another area that students have problems with is understanding when to apply approximations (small angle
theory) and linearizations. In previous coursework involving rigid body kinematics and dynamics, rigorous
attention to dynamics is required but most textbooks in dynamics and vibrations immediately apply linearization
methods so that students can work the problems with linear second order differential equation solutions more
easily. This is confusing to students because it seems like a different methodology is at play. Ultimately, this
linearization is required but students lose sight of the fundamentals if this is done at the start of a problem rather
as the last step before solution. No insight as to nonlinear behavior can be discussed if an immediate small angle
assumption is made when setting up the problem and structuring the free body diagram. This classes that the
exact solution, in terms of differential equation, must first be found before any linearization is applied.
Finally, students tend to view undamped and damped problems separately since textbooks present the material
that way. Historically (meaning before computers), this academic presentation was required since solution via
complex math was too difficult for more than a one degree of freedom system. This has not been the case for
over 25 years but textbooks still follow this time honored tradition. Today, utilizing Matlab® as a tool, general
damped problems can be introduced with the same Laplace frequency notation (s) that is used in controls
coursework. First of all, this integrates the dynamics and controls coursework and presents the general solution
for the damped problem as a starting point. Undamped cases are then presented as a special case of the
general case, as they should be. This also emphasizes general, second order differential equation solution,
Fourier and Laplace transforms as well as Euler identity relations. Students working in structural dynamics need
to be comfortable with this material.
In order to do this, additional class notes need to be utilized in parallel with any classical vibrations text. In the
ME Program at UC, a number of texts have been used over the years but the text, Mechanical Vibrations with
Applications [2] is now used. This text is out of print and the copyright has been returned to the author, Professor
Emeritus I.E. Morse, Jr., who now allows the text to be copied for educational use without cost. This text is made
available to the students electronically at no charge. The notes that are used to supplant the material are also
available to the students electronically [5]. The course is actually taught from this material and the textbook is
utilized as a source of problems and as a reference. Extensive use of Matlab® is included in this course for
solving homework problems; sample Matlab® scripts are made available to students electronically to solve the
general type of problem assigned but students must alter the script to work the specific problems assigned.
3.2 Structures/Motion Lab (MECH571)
In order to optimize both student and faculty time in experimental lab coursework, the laboratory work that might
be spread across many individual courses has been placed in two Senior level required courses, one in the
Structures/Motion Stem and one in the Heat Transfer/Fluids Stem of the ME Program, for all students in the ME
Program. This means that some laboratory work is delayed from the initial contact with the course material but
enough course credit can be given in this format to reward both students and faculty for the time commitment of a
lab course and lab reports. The structure of lab reports can also be formalized and students can benefit from
repeating the lab report format over several lab exercises.
Structures/Motion Lab highlights material from the structural dynamics area of the curriculum through several lab
exercises. First of all, sensor calibration as well as measurement and instrumentation methods are emphasized
in an individual lab as well as in each lab during the 10 week quarter. The hardware and software in this lab is
chosen to be similar to what students may find when they begin working in industry. In some cases, the students
already have experience with the equipment from their co-operative work experience. While different hardware
and software has been utilized over the years, VXI Technology data acquisition hardware, PCB Piezotronic
sensors and m+p International software is being used currently. In all cases, the hardware and software is used
to get data into Matlab® and all data processing and presentation is done in Matlab®.
Two specific laboratory exercises emphasize material from previous coursework in the structural dynamics area.
First of all, Figure 3 shows the set-up for a beam vibration experiment. This exercise requires that the students
utilize theory in a pre-lab exercise to estimate the frequency of the first six deformable modes of vibration for a
free-free beam. This requires the students to recognize that there are modes of vibration in both bending and
torsion in two primary axes of the beam. Once in lab, the students must decide on an measurement grid and
measure the appropriate frequency response functions (FRFs) to be able to identify the frequency and mode
shape for these six modes. The FRF data is moved to Matlab® for presentation and analysis rather than using a
commercial or even the UC-SDRL modal analysis software, X-Modal II.
Figure 3: Beam Vibration Lab
Figure 4 shows the set-up for a rotating unbalance lab. In this lab exercise, the students utilize vibration theory to
balance a multiple plane unbalance problem in two planes using a trial weight method. The students are
required to show that the rotating shaft is essentially rigid and therefore can be balanced by this method. The
students use the practical lessons learned from the beam vibration lab to document the frequency of the first
bending mode of the rotating shaft, in order to show that this frequency is above the frequency (speed) of
operation and the speed at which balance will be calculated.
Figure 4: Rotating Unbalance Lab
Matlab® is extensively used included in this course for processing an presenting data in every lab exercise.
Several labs involve least squares estimation of parameters from experimental data and emphasis in each lab is
on comparing theoretical predictions to experimental results. This also reinforces the need for students to
understand the applicability of theory and derived equations to the practical situation found in applications
(applying the wrong equation) as well as checking for units consistently versus common-sense physical results
(order of magnitude problems).
Other labs utilize concepts from Mechanical Vibrations I and Controls as the basis for the lab exercise. For
example, one lab uses a torsional pendulum to estimate the mass moment of inertia of an object which is based
upon single degree of freedom theory from Mechanical Vibrations I. Another example is the modeling and
simulation of an electro-mechanical system (shaker or speaker) which is a coupled MKC and RLC system which
can be modeled with common controls methods.
3.3 Fourier Transform Techniques (MECH660)
Fourier Transform Techniques is the first, Dual Level course available to Seniors or entry level MS students that
emphasizes the numerical issues associated with utilizing Fourier transform in measurement and data acquisition
situations. Extensive use of synthesized data and well as measured data sets are utilized as the basis for
processing data, understanding errors such as leakage and understanding data processing methods such as
windowing, averaging, etc. as well as data presentation methods utilizing linear and log formats. Formal
processing of the data into power spectra, frequency response functions and coherence is performed in Matlab®
and once again, sample scripts are often made available to initiate the students work on a particular problem.
3.4 Mechanical Vibrations II (MECH662)
Mechanical Vibrations II is the sequential course that follows Mechanical Vibrations I. While all students must
take Mechanical Vibrations I and Structures/Motion Lab, this is the first elective course that the students can take
in the vibrations area. This course follows the same philosophy as Mechanical Vibrations I and presents the
classical material in the same way. Ironically, the undergraduate students who have taken Mechanical Vibrations
I do very well in this course while the graduate students struggle for the same reason that the students in
Mechanical Vibrations I struggle. The first half of this course is very conventional, covering multiple degree of
freedom material from both Newtonian (d’Alembert) and Lagrangian (energy) methods. The emphasis in the
second half of this material is to relate the lumped parameter models using theoretical degree of freedom
concepts to continuous structures that are represented by input and output measurements or what is referred to
as measured degrees of freedom. A complete set of notes has been developed to demonstrate this relationship
in terms of measured functions and relationship to theoretical (continuous) models that relate measured FRFs to
lumped parameter [M], [C], and [K] matrices [3]. Once again, Matlab® is used extensively to solve traditional
lumped parameter problems, to formulate synthesized measurements in different display formats, and to solve
practical problems with a dynamic absorber from both a lumped parameter and measured FRF point of view.
3.5 Mechanical Vibrations III (MECH663) and Mechanical Vibrations III Lab (MECH664)
The combination of Mechanical Vibration III and Mechanical Vibration III Lab constitutes an experimental vibration
course. The first half of the course reviews measurement and instrumentation material, modal parameter
estimation material, as well as application material that has been gathered into a set of course notes [4] that are
available to the students electronically. Some of this material is review for some students but it generally has not
been presented previously in a consistent fashion. For this material, laboratory time is utilized to demonstrate the
material in practical measurements and familiarize the students with data acquisition as well as to begin to
introduce the students to a formal modal analysis program. This course uses the X-Software programs (XAcquisition and X-Modal II [6,7]) developed by the UC-SDRL for most of the lab exercises. This software is built
on top of Matlab® and is open architecture so that the students can see not only the results but also the way the
data is being processed at any step in the process. Emphasis in this course is placed upon understanding the
concepts involved in data acquisition and measurement estimation and the concepts in modal parameter
estimation rather than on understanding the details of how these methods are implemented. At the conceptual
level, the emphasis is placed upon why certain methods should be used and why the methods behave the way
they do based upon theory or practice. This is what dictates successful and appropriate use of methods in
application to structural dynamics problems.
This course is marked by three major lab exercises in which all students participate. The first exercise involves
FRF data, with additive random noise, that is synthesized from a 20 degree of freedom model of a bladed disc.
This system is analyzed with modern modal parameter estimation methods to familiarize the student with time
and frequency domain methods and their sensitivities. Since the error in the FRF data (random) matches the
error minimization form of these parameter estimation methods (least squares methods), the students develop
confidence in the parameter estimation concept. This model of the bladed disc contains a number of modes with
close frequencies, nearly repeated roots as well as multiple roots due to the close symmetry of the structural
model.
The second exercise utilizes a measurement set-up that is provided to the students for an apparently simple
structure of a simply supported circular plate. A circular plate made of aluminum, shown in Figure 6, is excited via
two exciters using a multiple input, multiple output using response taken at 30 locations using accelerometers.
The students are required to choose the measurement methods but are not responsible for the basic set-up. This
seemingly simple structure has repeated roots at most peaks in the FRF measurements but is lightly damped and
relatively easy to test. However, due to the light damping, the leakage bias error is very difficult to control and
students quickly see this difficulty as well as the impact of this bias error on the parameter estimation process.
Figure 5: Experimental set up for the lightly damped circular plate
The third lab exercise involves students working in a group of three on a realistic test structure. Figures 6-8 are
common test structures that are available to the students although other test objects are often solicited from
industry for this exercise. Students are responsible for choosing all aspects of the data acquisition and set-up as
well at the data processing and parameter estimation. These structures are much harder to test and rarely yield
data that looks like the synthesized FRF data that they have seen previously. This normally gives the students a
good experience with realistic data. Emphasis is placed upon understanding the process, and any mistakes
made, rather than on performing the experiment perfectly or mistake free.
Figure 6: Test Structure: Truck Body and Frame
Figure 7: Test Structure: Truck Frame
Figure 8: Test Structure: Fully Trimmed Vehicle
3.6 System Dynamic Analysis I, II, III (MECH794, MECH795, MECH 796)
Once students have completed the Vibrations sequence, the System Dynamic Analysis sequence follows and
mirrors the material presented in earlier courses with several differences. The first difference is that all methods
that are further studied are methods that are based upon experimental application to real structures. Data
acquisition and measurement formulation is the basis for System Dynamic Analysis I, experimental modal
parameter estimation is the basis for System Dynamic Analysis II and modal and Impedance modeling are the
basis for System Dynamic Analysis III. Emphasis in this course sequence is placed upon understanding the
details involved in data acquisition and measurement estimation, the details in modal parameter estimation and
the numerical issues of how these methods are implemented and the details in how modal and impedance
modeling are implemented. The details of all methods are studied in addition to the larger focus of understanding
why these methods are problematic in some applications.
The second major difference in the System Dynamic Analysis sequence is that students program each method in
Matlab® rather than simply study the method using commercial software. This requires the student to understand
the technical details and to integrate various methods in a common framework. With respect to modal parameter
estimation, the Unified Matrix Polynomial Approach (UMPA) [8-10] is the basis for the student’s study. In addition
to the technical details, though, the numerical issues, including algorithm development, memory requirements,
data organization and dynamic range of data and solution method applicability, are studied in order to obtain a
practical and manageable solution. These studies are normally performed on a team basis where students work
together to understand and implement the methods in groups of 5 or 6. Data sets are provided from known
analytical and experimental systems so that students have a priori ideas about the answers. The data is in both
time domain histories and FRF matrices measured from theoretical 15 or 20 DOF models or measured from
relatively simple structures like the circular plate in Figure 5 or the H-Frame structure in Figure 9. This data can
be utilized by existing software (X-Acquisition and X-Modal II) where interim structure of data matrices is available
within the Matlab® structure of these programs.
The H-Frame structure in Figure 9 is frequently used for many of these studies. While the geometry is very
simple and symmetric, this structure is very uncoupled in the three principle directions. Modes of vibration are
dominantly in only the X, Y or Z direction with very little cross coupling. This type of structure presents interesting
challenges with respect to data acquisition and modal parameter estimation as well as modal and impedance
modeling. The structures is configured to add lumped masses at any extremity and to add stiffness between any
two extremities. There is also the ability to add a sub-structure directly to the top of the H-Frame as well as to add
a sub-structure via automotive body mounts to the top of the H-Frame structure.
Figure 9: Test Structure: H-Fame Structure
The final difference is that an extensive amount of time is spent in these studies. In the Experimental Vibrations
coursework (primarily Mechanical Vibrations III and Lab) approximately 10 weeks are used to cover this material
once the background vibrations theory has been covered. In the Dynamic Analysis Sequence, approximately 30
weeks is required to cover the material in great detail. In both cases, the same set of electronic notes, together
with handout presentation slides are used to present the course material.
4. Lessons Learned
As the coursework and coursework changes that have been implemented in the BSME Program at the University
of Cincinnati with respect to structural dynamics are viewed retrospectively, a number of lessons have been
learned that can be highlighted as follows:
• Extremely solid (depth) of knowledge of a minimum number of fundamentals is more important than
breadth of knowledge, particularly if breadth sacrifices depth. An example is differential equations where
students are taught several methods for solving linear, constant coefficient second order differential
equations rather than being proficient with one method that is learned very well. If fundamentals are well
learned, additional methods can be self taught.
•
•
•
•
•
•
•
•
Focus on the general solution first and then add special cases, rather than the reverse. The poor
example of this is learning undamped systems before damped systems and learning MDOF systems
before SDOF systems. While this is not always possible, the framework of the general solution has to be
introduced initially so as to sensitize the students to the special case that they are studying.
Lab and problem exercises need to be graduated in difficulty so that goals are clear and each exercise
builds upon the previous exercises. Introducing too many unknowns makes the learning lesson more
difficult for the student to grasp. Time has to be spent with students on this material and students need to
know why the material is assigned and what learning goals are involved.
Material in textbooks often mirrors material in many other textbooks, current and previous, in a given
technical area and may not reflect current students and the way they learn in today’s electronic world.
Frequently, classical texts can still be utilized if additional material in terms of course notes and exercises
are developed to be extend the basic material.
Use of software must be carefully weighed with respect to learning curve time commitment (investment)
by students and faculty versus return in the number of courses that the software can impact. Software
that requires excessive learning time but impacts only one or two courses should be avoided if possible.
Students learn from simple experiments and problems rather than overly complicated experiments. The
simple beam vibration experiment where students perform most of the data processing, analysis and
presentation is more effective than students plugging coded FRF data into a commercial modal analysis
software for the first contact with the technology. Once the flow path through the process is understood,
the added complexity of sophisticated software or more complicated experiments can be more easily
introduced.
Faculty need to be careful to balance the amount of current (research) technology that is brought into
fundamental classes. In order to excite students about where the fundamental material will eventually
lead, the use of latest research concepts seems very attractive. The complexity of this material, and the
amount of time it takes to put it in proper perspective often outweighs the any advantage gained.
Most students learn best through repeated contact with material that involves several learning styles;
hearing the material, reading the material, working theoretical and experimental problems concerning the
material, etc.
Students learn best when they can see how the material will be used to solve engineering problems. In
the case of more elementary coursework, students need to know what material from previous courses is
required for success in the present course, what the five major learning goals are for the present course,
and what future courses will require these learning goals from the present course.
These are not particularly unique lessons but can be applied to almost any area of study. Certainly, these
lessons are not all of those learned over the years of adjusting the ME Program to provide more effective
education in the structural dynamics area. Nevertheless, these lessons provide a framework for making decisions
when course material is considered for addition or subtraction to a specific course.
5. Summary/Conclusions
Much of the focus of the research activity in structural dynamics over the last 40 years has been concerned with
the experimental approach to understanding the general structural dynamics of structures and systems. During
this time, the research and technology that has been developed has found its way into the educational program at
both the undergraduate and graduate levels, including both required and elective coursework. The impact of
research as well as the changing emphasis of structural dynamics technology and the use of experimental
methods has clearly impacted the educational process, including what material is taught as well as how the
material is taught at all levels. The effective use of computers, data acquisition hardware, and software and the
restructuring of coursework to reflect and utilize this new technology is very important to effective engineering
education in the structural dynamics area.
References
[1]
[2]
http://www.sdrl.uc.edu.
Tse, F.S., Morse, I.E. and Hinkle, R.T., Mechanical Vibrations: Theory and Applications, Second
Edition, Originally Allyn and Bacon, Inc. (1963, 1978), Used by Permission from current copyright holder,
I.E. Morse, Jr., 2001, 449 pp.
[3]
Allemang, R.J., et al, Vibrations: Analytical and Experimental Modal Analysis, UC-SDRL-CN-20-263662, 160 pp. (http://www.sdrl.uc.edu/ucme662/ucme662.html)
[4] Allemang, R.J., et al, Vibrations : Experimental Modal Analysis, UC-SDRL-CN-20-263-663/664, 1999,
250 pp. (http://www.sdrl.uc.edu/ucme663/ucme663.html)
[5] Phillips, A.W., Mechanical Vibrations I, 2001, 78 pp.
(http://www.sdrl.uc.edu/ucme662/Vibs1_review_notes.pdf)
[6] http://www.sdrl.uc.edu/X-ACQuisition.html.
[7] http://www.sdrl.uc.edu/X-ModalII.html.
[8] Allemang, R.J., Phillips, A.W., "The Unified Matrix Polynomial Approach to Understanding Modal Parameter
Estimation: An Update", Proceedings, International Conference on Noise and Vibration Engineering,
Katholieke Universiteit Leuven, Belgium, 36 pp., 2004.
[9] Allemang, R.J., Brown, D.L., "A Unified Matrix Polynomial Approach to Modal Identification", Journal of
Sound and Vibration, Volume 211, Number 3, pp. 301-322, April 1998.
[10] Allemang, R.J., Brown, D.L., "A Unified Matrix Polynomial Approach to Modal Identification", Proceedings,
Indo-US Symposium on Emerging Trends in Vibration and Noise Engineering, IIT Dehli, India, Allied
Publishers Limited, pp. 379-390, March 1996.