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PRACTICAL EXPERIENCES IN VIBRATION
Lecture Notes
By
Douglas E. Adams, Ph.D.Assistant Professor of Mechanical Engineering
School of Mechanical EngineeringPurdue University
West Lafayette, IN 47907-1077, USA
© Douglas E. Adams, 2002
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PREFACE
Introductory vibrations should be fun. These notes are designed to show students just
how applicable and exciting the subject of mechanical vibrations can be to engineers. It
should become clear right away in the notes that the discussions will be motivated
primarily by observationsand not by theories. Instead of starting with the fundamental
mechanics, deriving the differential equations that describe a certain system, finding a
solution to the equations, and then trying to figure out what the solution means in terms
of resonant frequencies and the like, every discussion in these notes will start off by first
observing the phenomenon of interest in some application of interest (i.e., discussions
will start with an idea of what the solution may look like and then work backwards to
find the equations from which the solution came). For instance, if a baseball bat is the
mechanical system under investigation, then the discussion will begin with a mental
experiment by ‘swinging’ the bat and observing what happens when the baseball is
struck. Questions that could be asked to better understand the behavior of the bat
include: ‘What do our hands feel like when we hit a ball and why?’; ‘What happens
when the ball hits the bat at the top versus at the bottom?’; ‘How does the bat behave
differently when it is aluminum as opposed to wood, or hollow as opposed to solid?’; and
‘What do a baseball bat, tennis racquet and golf club have in common in terms of their
vibrations?’. By observing interesting things about the way the bat behaves, and then
working backwards to discover the equations that govern how the bat responds to the
ball, the discussions in these notes will mimic the approach that many of the great minds
in science and engineering, like Newton, Rayleigh and Euler, took to lay the foundation
that engineers now use to investigate the behavior of mechanical systems in statics,
dynamics and strength of materials. Of course, design engineers will also eventually find
it useful to work in the opposite direction by beginning with equations of motion and then
analyzing those equations to discover interesting things about how the mechanical system
behaves differently if certain parameters like mass or stiffness are changed.
The notes strive to be as interesting as possible and the course, “Practical
Experiencies in Vibration”, for which these notes were written aims to do two primary
things: expose students to the phenomena, classical problems, and analytical and
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experimental techniques in mechanical vibrations that most mechanical design, analysis
and test engineers are confronted with througout their careers; and encourage
undergraduate students to pursue higher level analytical and experimental courses like
ME 563 and ME 597A for a more detailed and rigorous treatment of structural dynamics.
In other words, the goal of these notes is to pull students into the area of structural
dynamics by exciting them about engineering applications in which vibrations are crucial
to system performance. The overall theme of these notes is that when students think
vibrations are fun and practical, they automatically develop a taste for the associated
math and engineering science, which must eventually be introduced to make theoretical
and experimental techniques in vibrations useful to engineers. Students should keep
these points in mind as they read the notes, which should be read for the purpose of
understanding the concepts first and the details second.
There are many excellent textbooks written on the subject of mechanical vibrations;
however, students almost always uniformly dislike these textbooks to some degree.
Students often feel that the textbook is too dry, too theoretical, too academic, etc. or does
not sync up with the lectures that are prepared by the instructor. Some of these
complaints may be well founded when, for instance, the students have actually taken the
time to read the books, whereas other complaints might be the result of students who
have not given these books a fair read. Regardless of the reason for these complaints, it
is clear that if instructors cannot get students to read the textbook, then lectures will not
go as far in teaching the students vibrations and both the students and faculty lose
something in the process.
These notes were written to respond to these complaints and cannot possibly capture
all of the technical content and various presentation styles in numerous high quality
textbooks on mechanical vibrations that are available. Students should refer to textbooks
by Tse, Morse and Hinkle (1978), Meirovitch (1986), Rao (1995), Thomson (1998) and
Allemang (1999) among others in order to receive a full treatment of the analytical nature
of mechanical vibrations by teachers and researchers who have much to offer in the way
of pratical experience and to Ewins (1994) and Allemang (1999) for a thorough
engineering treatment of the experimental nature of structural dynamics. Student can
benefit tremendously from more than one way of thinking about a subject like vibrations.
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ACKNOWLEDGMENTS
The initial offering of this course and the accompanying roving laboratory have been
made possible through a Course Curriculum and Laboratory Improvement (CCLI) grant
to the School of Mechanical Engineering at Purdue University from the Department of
Undergraduate Education (DUE) within the National Science Foundation, grant number
DUE-0126832. The author of these notes would like to thank Dr. Ibrahim Nisanci, who
is the program monitor for this grant, for his encouragement, support, and tremendous
interest in the project. The author also thanks and acknowledges Dr. Charles Farrar, staff
member at Los Alamos National Laboratory in the Engineering Sciences and
Applications division in Los Alamos, NM; Dr. Lane Miller, Director of mechanical
systems research at Lord Corporation in Cary, NC; John Grace, Vice President of
Research and Development at ArvinMeritor in Columbus, IN; Larry Freudinger,
Measurements Lead at NASA Dryden Flight Research Center; Elias Rigas, staff engineer
with the Army Materiel Command at the Army Research Laboratory; Matthew Bedwell,
design engineer wth Caterpillar Large Engine Facility; Prof. Mete Sozen in the School of
Civil Engineering at Purdue University; and Prof. Jim Jones in the School of Mechanical
Engineering at Purdue University for providing helpful suggestions in their roles on the
industrial advisory committee for this course. Without the contributions and support of
each one of these individuals, this course and the roving laboratory would not have been
realized.
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1. INTRODUCTION
1.1. FAMOUS AND NOT-SO-FAMOUS EXAMPLES OF VIBRATION
1.1.1. Bad vibrations
Vibrations are oscillations in mechanical systems. All of the various parts that make
up automobiles vibrate. Airplanes vibrate. Bridges vibrate. Everything around us
vibrates including the air we breathe and the ground on which we walk. Sometimes
vibrations are bad and other times they are good. That is, engineers sometimes want to
suppress vibrations whereas other times engineers want to intentionally make systems
vibrate in some useful way. Perhaps the most infamous example in the engineering
community of ‘bad’ vibrations occurred during the two days preceding the catastrophic
failure of the Tacoma Narrows Bridge in Tacoma, WA in 1940 (see Figure 1.1). After a
day of large amplitude oscillations back-and-forth, the bridge material eventual gave way
due to fatigue similar to how a paper clip fails when it is opened and closed repeatedly.
Later in the course, the phenomenon that lead up to the Tacoma Narrows failure will be
analylzed in detail. For now it is interesting to note that this bridge vibrated for the same
reason that aircraft wings sometimes oscillate excessively, wine glasses humm when their
rims are rubbed in the right manner or icy cables oscillate during the winter time in strong
winds. By first examining one of these vibrating systems, it will then be possible to draw
conclusions about the entire class of vibration problems in which fluids flow over
structures. Structure-fluid interactions like this are important in engineering.
Figure 1.1 (Left) View of Tacoma Narrows Bridge deck undergoing largeamplitude torsional vibration due to air flow in gorge, and (right) sideview of torsional mode showing violent motion of cable stays
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Earthquakes produce another form of ‘bad’ vibration, which can have devastating
effects (see Figure 1.2). It has been said that ‘earthquakes don’t kill people, structures
do’ because it is rare that an earthquake will harm someone directly. In most
earthquakes, the vibrations of large surrounding structures (e.g., buildings, highway
overpasses and houses) are responsible for the majority of injuries and deaths. Figure 1.2
shows representative pictures of the severe type of damage that was sustained by a
highway overpass (left) and bridge support member (right) when they oscillated
excessively during the Northridge, CA earthquake of 1994. Most but not all structural
damage of this type in earthquakes is caused by shear failures during repeated side-to-
side oscillations of reinforced concrete members. Later in the course, large structures
like these will be modeled and analyzed, and the interactions between a structure like the
overpass shown below and its foundation/support will be studied to attempt to understand
how those interactions mitigate or worsen the effects of an earthquake.
Engineers of so-called ‘smart structures’ have been working for decades, and
continue to work, to design and build structures that have enough intelligence and power
to not only withstand but to respond to earthquakes and other forms of environmental
excitations in order to suppress as much of the resulting vibration as possible. In fact, it
Figure 1.2 Damage to a highway overpass (Left) and support member (Right)during the 1994 Northridge, CA earthquake (courtesy Prof. F. Seible, UCSD)
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has been shown that this type of damage due to earthquakes can be largely mitigated by
implementing the kinds of design modifications for vibration suppression that are
pictured in Figure 1.3. For example, a friction pendulum bearing is shown on the left and
an elastomer bearing is shown on the right of that figure. Both of these mechanical
subsystems are designed with the appropriate vibration characteristics (i.e., ‘resonant
frequencies’) so as toisolatecivil infrastructure from large seismic oscillations just as car
suspension systems are designed to isolate passengers from road inputs and engine
vibrations. In that sense, these two isolation system can be thought of as mechanical
‘filters’, which bypass mechanical energy that would otherwise destroy the isolated
structure. In effect, the bearings pictured in Figure 1.3 block much of the energy from
the seismic oscillations thereby protecting the isolated infrastructure. There are also
many examples of passive and active isolation systems for civil infrastructure and smaller
scale mechanical systems like rotating machinery. Specific examples of vibration control
systems like these and the fundamental nature of isolation systems in general are
discussed in detail later in the course. By examining a specific type of vibration isolation
system like one of the infrastructure vibration suppression devices shown below, general
conclusions about the entire class of isolation systems can be drawn.
Figure 1.3 (Left) A friction pendulum bearing and (Right) an elastomeric bearingfor isolating civil infrastructure from earthquake seismic oscillations to reducedamage caused by earthquakes (courtesy Prof. F. Seible, UCSD)
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There are also many other not-so-famous, everyday examples of unwanted vibrations.
For example, when transportation systems (e.g., airplanes, trains, ships and automobiles)
vibrate excessively, passengers inside can become uncomfortable or even sick especially
when the frequencies of oscillation correspond to so-callednatural frequenciesof the
human body and its organs. Natural frequencies are frequencies at which mechanical
systems ‘want’ to vibrate; these systems can therefore be forced to vibrate very
efficiently at their natural frequencies. In fact, it is well known that the fundamental
resonant frequency of the human intestinal tract (i.e., approx. 4-8 cycles of oscillation
each second) should be avoided at all costs when designing high performance systems
like manned fighter aircraft and reusable launch aerospace vehicles because sustained
exposure to vibrations at those frequencies can cause serious internal trauma to
passengers (refer to Leatherwood and Dempsey, 1976 NASA TN D-8188).
In a similar way, low frequency undulations below one cycle per second on cruise
ships often make passengers sea sick. To counteract these oscillations, luxury cruise
ships are designed with automatic ballast systems through which water (i.e., weight) is
transferred from one side of the ship to the other to suppress the oscillations through a
shift in momentum. This transfer of fluid momentum acts to suppress the oscillations.
This type of vibration suppression system using ballast in a cruise ship is different from
the infrastructure vibration isolation systems shown above in Figure 1.3. In the case of
the cruise ship, the wave forces that cause the ship to sway act directly on the ship and so
the ballast is designed to absorb that energy; however, in the case of the infrastructure
shown in Figure 1.3 the support bearings do not even allow the vibrational energy to
enter the system in the first place. Pills, which effectively numb the inner ear thereby
isolating the human sense of balance from the ship’s oscillations, are also handed out to
passengers at the beginning of a cruise to help sea-goers avoid becoming sea-sick.
Of course, if an aircraft wing, or some other portion of an aircraft like a fuselage
panel, vibrates at large enough amplitudes for an extended period of time, there can be
more serious problems that just motion sickness. If oscillations continue for an extended
period of time, the structure itself could eventually fail due to fatigue, just as the Tacoma
Narrows did. Fatigue of the structure could potentially cause an aircraft, for example, to
crash resulting in serious injuries and/or fatalities. The devastating results of a corrosion
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fatigue failure in Aloha Airlines flight #243 are shown in Figure 1.4. This failure
occurred because corrosion in the overlapping aluminum fuselage panels near the rivet
locations on the skin of the aircraft introduced cracking, which subsequently weakend the
panels and compromised their ability to respond to vibrations and static pressurization
cycles after hundreds of flights. As multiple cracks near the rivets joined together to
produce a catastrophic failure of the fuselage, the front panel of the fuselage tore away
nearly completely and one stewardess was killed. Luckily, the pilots were able to land
the plane in spite of the damage to the fuselage. Fatigue failure can often be devastating
and is the most common type of failure in mechanical systems. This type of failure is
caused partially by vibrations of the structural components.
Figure 1.4 Corrosion fatigue failure in Aloha Airlines flight #243 due to‘link-up’ of multiple cracks near the rivets of the overlapping panels
Other common types of unwanted vibrations may not cause injury or sickness in
humans but are nonetheless very costly for engineers in the automotive and related
industries. For example, brakes often squeal when the rotor comes into contact with the
braking pads as the brake pedal is pushed. These oscillations usually produce ‘noise’, or
unwanted sound, which makes it difficult for manufacturers to sell new and used vehicles
and to maintain their warranty policies on new vehicles. Other types of unwanted
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vibrations in automobiles include excessive bouncing, rolling and pitching of the body
and passengers during normal driving operations in ride due to either road inputs (i.e.,
potholes, bumps) or the reciprocating engine; excessive vibrations of the steering wheel
due to engine vibration or oversteering of radial tires; and squeak and rattle vibrations
due to rubbing or some other type of interaction between components in the vehicle (e.g.,
dashboard and windshield). It is well-known that consumers associate excessive noise
and vibration with poor vehicle quality, so it is essential for engineers in the automotive
industry to be able to predict and measure noise and vibration behavior for the purpose of
reducing the overall noise and vibration levels in automobiles. Consequently, there are
entire teams of engineers in the major car companies and their supplier organizations that
are dedicated to studying and minimizing noise and vibration in automobiles.
Although all of the examples of unwanted vibrations given so far are engineering
examples, there are many other types of non-engineering systems which undergo
oscillations at inopportune times. For example, a plot of the Dow Jones Industrial
Average, which was taken from the CNNTM website on September 12, 2001, is shown in
Figure 1.5. Recall that the Dow Jones Average, which is derived from the values of stock
in a certain group of companies who are listed with the New York Stock Exchange, is an
indicator of the overall health of the United States economy. A healthy economy often
leads to a consistent increase in the Dow Average whereas a struggling economy will
often lead to consistent drops in the average. In Figure 1.5, oscillations are seen to occur
after an initial drop in the average. Later on in the course after mechanical vibrations are
discussed, possible reasons for this interesting oscillatory behavior will be explored. For
example, it will be postulated that optimistic and pessimistic traders interact during
trading thereby causing the oscillations in Figure 1.5 to occur. The important thing to
note here is that the same basic mechanisms that cause vibrations to occur in mechanical
systems can also elicit oscillations in socioeconomic and other types of systems as well.
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Figure 1.5 Oscillations in the Dow Jones Industrial Average onSeptember 12, 2001 following the terrorist attacks on the U.S. TradeTowers due to an interchange between optimistic and pessimistic investors
1.1.2. Good vibrations
Even though unwanted vibrations often receive the most attention from engineers, the
public, and the press, there are just as many examples of ‘good’ vibrations in the natural
world and in engineered systems. Some of the most interesting and subtle examples of
beneficial vibrations are found in nature. As one example, consider the predatory
techniques of common orb web spiders like the one shown in Figure 1.6. These spiders
can actually be observed using vibration to their advantage to locate and restrain prey like
the Japanese beatle shown in the figure. The routine that this type of spider follows in
order to capture and restrain prey is based entirely on vibration. Initially, an orb web
spider will position itself at the center of its web. Once an insect strikes the web and
becomes caught, the spider tugs on the web thereby causing it to vibrate (refer to Witt,
Reed and Peakall, 1968). These vibrations can be thought of as ripples in a pond that
travel out towards the perimeter of the web and then back again to the spider at the
center. When an insect is caught and struggling in the web, the vibrations that return to
the spider are different than when the web is empty. Orb web spiders can actually be
observed walking out towards a trapped insect, tugging all the while to eventually locate
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and restrain the prey. Of course, the struggling insect produces vibrations of its own, in
addition to those that the spider creates, making it even easier for the spider to locate the
insect. Once the prey has been restrained, the spider returns to the center, or focal point,
of the web so that it can most effectively locate the next insect that becomes trapped. It is
reasonable to conclude that without vibrations, orb web spiders might starve to death
because they would not be able to locate their prey.
Figure 1.6 Orb web spider which uses vibrations to locate and restrain itsprey that becomes trapped in the web
There are also many other examples of how humans, and other species, use vibration
as a means to sense their environments just as orb web spiders do. For example,
vibrations in the form of compressional pressure waves of propagation in air make it
possible for humans to communicate with one another. The oscillations in pressure that
cause sound waves to propagate in the first place are created by vocal chords, or some
other type of pressure or flow ‘actuator’ (e.g., loudspeaker), and the resulting small
dynamic variations in pressure are then sensed by the undulating cilia within the human
ear. As humans age, the vibrational properties of the cilia change and these changes
often result in a partial loss of hearing especially in the higher frequency ranges. The
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possible reasons for this type of hearing loss due to aging will be modeled and analyzed
later in the course using the techniques that are developed.
Virtual reality video games and commercial simulators like the tank simulator used
by the Tank and Automotive Command in Warren, MI (see Figure 1.7), for example, also
use vibration as a means of exposing human subjects to background vibration
environments for recreation and training purposes. In fact, the Department of Defense is
using simulations increasingly to train their troops so that they can react to scenarios,
which might otherwise be too costly or dangerous to re-create in real military exercises.
Also, it is well known that the vibrations that humans experience while driving a car or
riding a bike, in addition to visual stimuli, enable them to effectively and safely maneuver
their vehicles because the vibrations communicate important information about the
terrain over which they drive. When a change in the road profile is sensed, the driver
reacts accordingly to avoid collisions and other mishaps. It can therefore be concluded
that without vibrations, humans could not drive their vehicles effectively and safely
because the information that the vibrations convey would not be received by the driver.
Figure 1.7 (Left) Driver ride motion simulator at Tank and AutomotiveCommand in Warren, MI; (Right) turret gunner motion simulator at TACOM
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Although person-to-person communication in many different types of species is
achieved via low to medium frequency propagating sound waves (10 to 20 kHz),
vibrational wave frequencies far above the range of human hearing are exploited in many
engineering applications included medical diagnostics. For example, expectant mothers
are usually examined at least once during the term of their pregnancy using ultrasound to
determine if any risks are anticipated for them or their fetus during pregnancy and/or
delivery. During the ultrasound procedure, high frequency sound waves (>20 kHz) are
sent through a wetting gel into the mother’s womb. These waves are then reflected by
different parts of the fetus in slightly different ways. By processing the reflected waves,
a two and sometimes even a three-dimensional sonogram image of the fetus can be
rendered. For example, Figure 1.8 shows a three dimensional black-and-white sonogram
of a fetus at seven months of development. This procedure is non-invasive and so does
not pose any serious threat to the fetus or the mother during the examination but does
provide essential information about the pregnancy.
Engineers also use vibrations to determine if machines (e.g., automobiles,
refrigerators, lawn mowers) are operating properly in the same way that doctors use
ultrasound to diagnose the health of patients. For example, a ‘purr’ or a ‘humm’ is
usually a sign that machinery is working properly whereas a ‘clank’ or a ‘clunk’ is a sign
that the machinery is not working as it was intended to work. Due to this association
between vibration/sound characteristics and process quality, trained technicians in
manufacturing environments are often able to identify if there is a problem with a given
piece of machinery simply by listening to it. For example, if a part is loose on a piece of
machinery, then the sound that the machinery produces will contain higher frequencies
than if the part is properly tightened. In fact, an entire area of technology involving
condition-based maintenancehas spun out of the idea that mechanical systems that are
not operating properly can be monitored to decide when it is time to service them. By
monitoring the vibrations of a lathe, for instance, it can be determined when the cutting
tool needs to be replaced or when there is part-to-tool misalignment.
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Figure 1.8 Three dimensional ultrasound image of a fetus usingpropagating high frequency compressional sound/vibration waves
As an example of how condition-based maintenance can be implemented, Figure 1.9
shows a typical ‘black-box’ on the left containing instrumentation for monitoring
vibration response signals from a lathe, which is also shown in the figure. The plot at the
right shows a set of ‘healthy’ vs. ‘unhealthy’ cutting response signals in addition to a
response signal during idle. These response signals are used to ‘diagnose’ if the machine
tool is working properly. The particular machine tool shown in Figure 1.9 would be
taken out of service after approximately 1700 cycles due to the tool breakage that is
evident in the right-hand side plot. It is much less expensive to make decisions about
service and maintenance based on the actual health of the machine rather than on some
fixed time schedule. Furthermore, this condition-based approach to maintenance is based
on the idea that ‘if it’s not broken, don’t fix it’ and is therefore much more cost-efficient
than a purely time-based approach in which machinery is scheduled for service and
maintenance every so many cycles, miles, etc. It is widely believed that operating costs
of all types of mechanical systems from machinery on the shop floor to commercial and
military aircraft can be reduced substantially by examining the vibrations that these
systems experience on a day-to-day basis and then scheduling service and maintenance
based on whether or not those vibrations indicate that the system is operating within
specifications.
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In summary, Table 1 contains a short but representative list of detrimental and
beneficial vibrations that occur in engineering-related applications. These notes can be
used to understand how vibrations can and should be tuned through design to provide
better engineering performance.
Figure 1.9 (Left) ‘Black box’ for monitoring the vibrations of a machinetool lathe in a manufacturing facility; (Right) Unusual vibrations indicatethat a tool needs to be replaced or that a misalignment exists between thetool and the part during the cutting operation.
Detrimental vibrations Beneficial vibrations
Machine tool chatter involves excessivevibrations in the cutting tool and producespoor cutting accuracy in machined parts
Unusual vibrations in a machine tool areused to schedule service and maintenancethereby reducing the operating costs
Large amplitude vibrations in ductilemechanical components will result infatigue after a sufficient number of cycles
Vibrations can be used in a sievingoperation to sort desirable materials fromundesirable materials
Vibrations at or near resonant frequenciesof the human body or organs can result inserious trauma after sustained exposure
High-frequency ultrasonic vibrations areused to scan the wombs of expectantmothers to identify risks to the pregnancy
Vibrations in microlectronic fabricationfacilities make it difficult to etch and buildvery small micron and sub-micron parts
Vibrations are created in virtual realityvideo games and in simulators for trainingcommercial and military personnel
Table 1 Examples of detrimental and beneficial vibrations in various types of systems
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1.2. WHY DO VIBRATIONS HAPPEN?
1.2.1. A mental experiment with a simple pendulum
It was stated at the beginning of these notes that vibrations are oscillations within
mechanical systems, or structural dynamic systems as they will sometimes be called.
Because these notes aim to model and analyze oscillations in order to develop a better
understanding of how mechanical, civil and aero-systems vibrate, it makes sense to
discover why oscillations occur in the first place. If the physical source of these
oscillations can be understood at a conceptual level, then many mistakes later on in the
course when differential equations are introduced to describe the oscillations can be
avoided. For example, if the physics of energy transfer during the oscillations are
examined now, then it will be clear later in the course why positive or negative signs
should be used in certain places in the differential equations that are derived.
To that end, consider the fact that although any structural system can be made to
oscillate when it is forced to do so externally by a sequence of potholes in a road, by
wind gusts during flight, or during a seismic event, say, the termvibration in engineering
is usually reserved for systems that canoscillate freely without being forced to do so by
externally applied excitations. In fact, oscillations in the absence of sustained excitations
should be expected in most mechanical systems. Having defined what vibrations are in
engineering applications, the two natural questions to ask next are why do vibrations
occur, and subsequently why do engineers care so much about them when they do occur?
A simple mental experiment will be discussed next to help answer both of these
questions, and then details will be given to support the conclusions that are drawn after
the experiment. Mental experiments like this can be very helpful in developing insight
into how structural dynamics systems behave, so these notes will use experiments on a
regular basis to motivate discussions on vibration.
The experiment to be conducted involves a simple pendulum, which includes a point
mass,M, hanging on an inextensible string of lengthL with a gravitational force of
magnitudeMg (whereg=9.81 m/s2) acting vertically downward on the mass, as shown in
Figure 1.10. The inextensible string assumption must be enforced here because without it
the kinematics and dynamics of the string become too complicated to investigate the
elementary vibration phenomena. Most engineers at one time or another have observed
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that if the simple pendulum is moved, or perturbed, somewhat away from its resting
position as illustrated in the figure, it will oscillate back and forth until it again
approaches that position.Vibrations, then, are said to occur as systems like the pendulum
oscillate in search of their resting positions. There are many interesting things to note
about this oscillatory behavior, which is ripe with terminology that must be defined
before moving forward in the course.
Figure 1.10 (Left) Idealized simple pendulum with gravity acting, and (right) a typical
free oscillation of the pendulum due to an intial perturbation
First note that a ‘resting’ position of the pendulum, like position (A) in the figure, is
also sometimes called an ‘equilibrium’ position, so these terminologies will be used
interchangeably in the notes. Position (A) is officially referred to as astatic equilibrium
position because the pendulum is at rest. In Position (B), the pendulum has been
perturbed to some initial angular position, sayoθ =20 deg, in the counterclockwise
direction and is held at rest there until it is released in Position (C). Also note that in
Position (B), the potential energy of the pendulum is greater than that at Position (A) by
M
θLMg
(A)Initial state of
pendulumθ=0 rad, θ=0 r/s
(B)Pertrubed stateof pendulum
θ > 0 rad, θ=0 r/s
(C)Pendululm is
releasedθ > 0 rad, θ < 0 r/s
(D)θ=0 rad, θ <0 r/s
(E)θ < 0 rad, θ <0 r/s
Heads backtowards theequilibrium position
(F)θ < 0 rad, θ=0 r/s
Will head backtowards theequilibrium positionand continue tooscillate until allenergy is dissipated
L(1-cosθ)
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an amount Mg times the change in height, ( )θcos1−L , or ( )θcos1−MgL . As the
pendululm swings higher, its potential energy becomes larger. Recall thatstiffnessis the
physical element that stores potential energy. In this case, stiffness exists in the
gravitational field between the pendulum mass and the mass of the earth. It is worth
pointing at this point that potential energy and kinetic energy will play important roles in
vibrations, so this should be kept in mind throughout this mental experiment. Once the
pendulum is released from Position (B), the oscillations begin.
In Position (C), the angular velocity of the pendulum,θÿ , is negative (clockwise
direction) because the pendulum is swinging back towards its equilibrium position. It
swings to the left towards (A) and not to the right because all systems, including the
pendulum, seek out their lowest energy states, and recall that in Position (B) the
pendulum has more potential energy than in Position (A), so it makes sense that the
pendulum should try to return to its lower energy equilibrium position. Also note that at
Position (C) the pendulum has lost some of its potential energy but has gained kinetic
energy in the amount of 2
2
1Mv , where θÿLv = is the tangential velocity, because the
pendulum mass is now moving with a nonzero angular velocity. Recall thatinertia is the
physical element that stores kinetic energy. In this mental experiment, the angular
velocity of the pendulum inn Position (D) is even larger than it was in Position (C) and
the pendulum’s potential energy is at a minimum. Note that although the pendulum is in
the same location in Position (D) as it is in Position (A), the pendulum does not remain
there because it still has kinetic energy, which causes the pendulum to overshoot its
resting position to the left until it reaches Position (E). At Position (E), the pendulum has
lost some of its kinetic energy because it has gained potential energy. Finally, at Position
(F), the pendulum turns around as its velocity goes to zero because all of the remaining
energy has been coverted to potential energy.
This oscillatory process continues on forever if there is no dissipation of energy or
eventually diminishes in different ways at different rates depending on the particular
pendulum being examined. Because most engineers would expect the pendulum
oscillation to decay as time progresses, it will be assumed here that the oscillation
diminishes. Various terminologies are introduced next, but an important conclusion can
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already be drawn based on the mental experiment in Figure 1.10:because all systems
move towards their lowest energy states, vibrations have to do with mechanical energy,
how it is transferred and how it is dissipated. In particular, note that the pendulum in
Figure 1.10 oscillates without the help of any external inputs that oscillate, and it is this
type of free oscillation(vibration) that will be of central importance in this course. It is
also important to understand how the pendulum responds to a force that varies with time,
but the free vibration response is a good place to start in these notes. Furthermore, this
interpretation of vibration in terms of energy exchange is an important conceptual tool
and will help immensely in deriving and understanding differential equations that
describe all kinds of engineered and natural systems.
Because the pendulum oscillates back-and-forth around its resting position (i.e.,
vertical downward position) and then eventually approaches that position as the
oscillations diminish, the oscilliatory motion of the pendulum is said to bestablein the
previous mental experiment (see Figure 1.11). Moreover, the stability of pendulum
oscillations is concerned with whether or not the pendulum returns to its equilibrium
position if the pendulum is perturbed away from that position. When the equilibrium
(resting) position of the pendulum being considered is the verticallydownwardposition,
then oscillations around that position are stable (top of Figure 1.11). If the pendulum is
perturbed away from the verticallyupwardequilibrium position (bottom of Figure 1.11),
then the resulting oscillations are said to be unstable because the pendulum does not
generally return to the vertically upward position but instead diverges away from that
position. For example, Figure 1.12 shows two plots with typical oscillations around the
pendulum’s stable equilibrium point (top) and away from the pendulum’s unstable
equilibrium point (bottom). Much more will be said about stability later in the course;
however, for now it is enough to note that vibrations can be either stable or unstable.
Although most engineers are more familiar with pendulums that eventually come to
rest after oscillating for a while, oscillations in special types of pendulums can continue
forever or at least for a very long time. Two sets of terminologies are used to
characterize whether or not systems sustain their oscillations. If the oscillations of the
pendululm decrease as it approaches its resting position, then the pendulum is said to be
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Figure 1.11 (Left) Stable oscillation of pendulum around its vertical downward
resting position, and (Right) unstable oscillation around vertical upward position
Figure 1.12 (Top) Stable oscillation of pendulum around its vertical downward
position, and (Bottom) unstable oscillation away from vertical upward position
0 0.5 1 1.5 2 2.5 3 3.5 4-20
-10
0
10
20
Time [sec]
θ[d
eg]
0 0.5 1 1.5 2 2.5 3 3.5 4-200
-100
0
100
200
Time [sec]
θ[d
eg]
Stableoscillation around thevertical downwardequilibrium position
Unstableoscillation aroundthe vertical upwardequilibrium position
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non-conservative. In other words, a non-conservative pendulum oscillates with smaller
and smaller amplitudes as time progresses because the energy that sustains the oscillation
is dissipated by physicaldampingmechanisms within the string and friction between the
pendulum and the surrounding medium (air). If the pendulum continues to oscillate
back-and-forth with the same amplitude, then it is said to beconservative. Figure 1.13
shows the angular responses of the simple pendulum for non-conservative (top) and
conservative (bottom) oscillations. Note that the amplitude of oscillation gradually
decreases for the non-conservative oscillation but remains constant for the conservative
oscillation.
0 0.5 1 1.5 2 2.5 3 3.5 4-20
-10
0
10
20
Time [sec]
θ[d
eg]
0 0.5 1 1.5 2 2.5 3 3.5 4-20
-10
0
10
20
Time [sec]
θ[d
eg]
Figure 1.13 (Top) Non-conservative oscillation of a simple pendulum around its
vertical downward equilibrium position, and (Bottom) conservative oscillation
The terms ‘damped’ and ‘undamped’ are often used by practicing engineers instead of
‘non-conservative’ and ‘conservative’ becausedampingis a physical phenomena, which
many types of systems exhibit due to energy dissipation. Damping is actually one of the
key research and development areas in the field of vibrations because engineers often
want to design and add special types of damping mechanisms to systems. The many
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forms of damping will be discussed later in the course, but for now it is enough to
mentionviscous damping, which is an idealized type of damping. For example, the loss
of energy per unit time in the pendulum increases as the angular velocity increases when
it is viscously damped. The mental experiment in Figure 1.10 has typical damped
oscillatory responses as shown in the top of Figure 1.13. As the pendulum oscillates
back-and-forth through its resting position, the pendulum loses some of its energy due to
its motion. It is reasonable to assume at this point in the notes that the rate at which the
pendulum loses energy increases with its angular velocity, i.e., it is losing the most
energy per unit time in Postion (D) and no energy in Position (F). As for conservative
systems, engineering examples of these types of systems are rare. There are, however,
many engineering applications in which the vibrating systems of interest are nearly
conservative because the oscillations last for a relatively long time so vibrations in
conservative systems are worth studying if only to understand how so-calledlightly
dampedsystems like large flexible space structures, for instance, behave.
1.2.2. Similarities in the oscillations of pendulums and offshore oil structures
Ultimately, engineers care about vibrations in systems because oscillations can be too
large or last too long thereby detracting from the performance of that system. Although
the discussion above surrounding the simple pendulum was academically useful because
most engineers can relate to the oscillations of a simple pendulum, there are many other
engineering-relevant structural systems in which vibrations occur that are similar to those
in the pendulum. More specifically, oscillations in most mechanical systems can be
attributed to two mechanical elements,inertia andstiffness, and how energy is exchanged
between them. For example, consider the offshore oil structure shown in Figure 1.14.
The right plot of the figure shows how the rig oscillates as incoming waves impinge on
the structure’s tower. The incoming force due to the waves is modeled with the time
function )(tu in Newtons, and the response displacement of the tower in one direction
only is modeled with the time function )(ty in meters. This so-calledsingle degree-of-
freedom(SDOF) model does not capture all of the important dynamics that can be
observed in a large structure like this, but the model will assist in understanding the
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fundamental behavior and response of the structure to wave forces1. The waves may
arrive from many different directions, in which case the structure could be displaced in
any of those directions; however, in some respects the response behaviors of the structure
in all of those directions are similar to the one that will be modeled next. In these notes,
the termdegree-of-freedom(DOF) will be used to refer tothe number of independent
coordinates that are needed to locate and orient all of the inertias in the mechanical
system of interest.
In keeping with the observational tone of these notes, the discussion here will first
observe the displacement response of the tower in two different scenarios and then
examine the mechanics and mathematics necessary to describe these observations. First,
consider the scenario in Figure 1.14 for which there are no waves in calm seas. This
situation corresponds to the case in which)(tu is set to zero (i.e., N0)( =tu for all time
and gravity is assumed to be negligible compared to the other effects to be discussed);
this unforced scenario will be referred to as thefree responseproblem. Recall that the
only force acting in the simple pendulum experiment was gravity and that this force was
constant (i.e., did not vary with time). A mental experiment to examine the free response
of the offshore oil structure when the tower is displaced somewhat from its resting
position similar to in the simple pendulum is carried out next.
Figure 1.14 (Left) Offshore oil structure, and (Right) a typical oscillatoryphysical response displacement, )(ty , of the the structure to incomingwave forces, )(tu .
1 Note that wave forces are actually a nonlinear function of the incoming wave velocities,v, according to
)(ty
)(tu
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In order to displace the top of the tower to begin the experiment at Position (A) in
Figure 1.15, a force must be applied to the left at the top of the oil structure. When this
force is released and the tower is permitted to undergo motion, it will move in the
direction of its resting place because arestoring forcewithin the columns pulls the tower
back towards the straight up-and-down position. Recall that this position, as for the
simple pendulum, is the static equilibrium position of the tower. Also recall that just as
for the simple pendulum, the offshore oil structure in Position (A) in Figure 1.15 is
storing a certain amount of potential energy, which is larger than the potential energy
associated with the static equilibrium position. In the pendulum, this potential energy
was stored in the gravitational force field (or spring), and in the offshore oil structure, the
potential energy is stored in the ‘springiness’, or elasticity, of the structure itself.
As the tower moves back towards its resting positions through Positions (B), (C) and
then (D), the restoring force within the structure decreases and the velocity of the tower
of the structure increases. This increase in velocity of the tower does something
interesting to the resulting dynamical motion. If the increase in velocity is large enough,
the tower can actuallyovershootits resting position by a certain amount as it vibrates
back in that direction depending on how far the tower was displaced from its resting
position at the beginning of the experiment in Position (A). What determines whether the
tower will overshoot its equilibrium position? In the pendulum experiment, the amount
of dissipation determined how much kinetic energy was lost as a result of the motion and,
therefore, whether or not the pendulum would overshoot its resting position. The same
conclusion will be drawn in this case.
It is known from Newton’s first law and from everday experience that a massive
object moving at a certain velocity continues to move unless its motion is resisted by
other forces. The forces that resist the movement of the tower are related to the velocity
of the tower and its displacement. An increase in velocity as the tower moves back
towards its resting position causes an increase in the forces within the water that push
against the tower because the structural columns are moving more quickly through the
water. From the discussion of the simple pendulum, the term viscous damping was used
to describe this type of force, and experience swimming under water certainly supports a
Morrison’s equation, vbvvau += ÿ ; however, these nonlinear effects will not be considered until later.
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model for viscous forces that depend on velocity. Although these viscous forces tend to
slow the tower down, they may not be large enough to prevent the tower from
overshooting its resting position. If the tower overshoots its position as in Figure 1.15 at
Position (E), then the tower system is said to beunderdamped. If the tower does not
overshoot its static equilibrium position, then the tower is said to beoverdampedor
critically damped, and in order to explain the difference between these two types of
damping, mathematics will need to be invoked later in the course.
(A) (B) (C) (D)
(E) (F) (G) (H)
(I) (J) (K)
Figure 1.15 Free oscillation of offshore oil structure from largest negativedeflection (A) to the large positive velocity (D) to largest positivedeflection (F) and then back towards the starting position (K)
)0(y
Small pos. velocity Large pos. velocityLargest pos. velocity,zero deflection
Small neg. velocity
Initial neg. deflection,zero velocity
Large pos. velocityLargest pos. deflection,zero velocity
Large neg. velocity,zero deflection
Medium neg. velocity Small neg. velocityLarge neg. deflection,zero velocity
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