PRACTICAL EXPERIENCES IN VIBRATION - TUC · PRACTICAL EXPERIENCES IN VIBRATION Lecture Notes By Douglas E. Adams, Ph.D. Assistant Professor of Mechanical Engineering School of Mechanical

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

Vibrations 11/25/02 Purdue University, Mech. Eng.

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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

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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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PRACTICAL EXPERIENCES IN VIBRATION - TUC · PRACTICAL EXPERIENCES IN VIBRATION Lecture Notes By Douglas E. Adams, Ph.D. Assistant Professor of Mechanical Engineering School of Mechanical