20.2 Introduction

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20.2.1 General Considerations of Damping

The etymology of the word “damping” is difficult to determine. It is obviously allied with the word

damper, commonly defined as a “device that decreases the amplitude of electronic, mechanical,

aerodynamic or acoustical oscillations,” used for centuries, for example, to describe the sound attenuator

pedal on the piano. Perhaps the German word dampfen (to choke) has had an influence in the evolution

of the word. One can only wonder if water, as a moistening agent, played any role. Certainly, liquid water

is important to some cases of energy dissipation in oscillators. Moreover, friction determined by the

viscosity of a fluid (gas or liquid) is an important type of damping. A curious piece of history, in the

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celebrated work of Stokes, is why his expression “index of friction” did not take precedence over our

modern word, viscosity. Peculiar terminology is also encountered to describe damping, such as the

engineering device known as a dashpot, which is a mechanical damper. The vibrating part is attached to

a piston that moves in a liquid-filled chamber.

We will see that the number of adjectives used to describe various types of damping is extensive. This

multiplicity of terms to describe the loss of oscillatory energy to heat is no doubt an indicator of the

complexity of damping phenomena in general. We will attempt (i) to identify similarities and differences

among various types of damping, while (ii) explaining some of the physics responsible for the

characteristics observed. Conceptual ideas and techniques of both theory and experiment will be

provided, targeting the lowest level of sophistication for which semimeaningful results can be obtained.

The reader should be aware that a “grand-unified” theory of damping does not exist, nor is it likely that

one will ever be created.

Damping causes a portion of the energy of an oscillator, otherwise periodically exchanged between

potential and kinetic forms, to be irreversibly converted to heat, sometimes by way of acoustical noise.

Whether by suitable choice of materials during design of passive equipment, or by using feedback in active

control of a sophisticated system, control of damping is important since mechanical vibrations can be

detrimental or even catastrophic. An oft-quoted example of catastrophe is the Tacoma Narrows bridge,

which collapsed in high winds on November 7, 1940. Like the vibration of a clarinet reed, this disaster is

probably best described by the term negative damping, which can drive parametric oscillations.

The optimal amount of damping for a given system might fall anywhere in a wide range from great to

extremely small, depending on system needs. The engineering world frequently wants oscillations to be as

close to critically damped as possible. Physics experiments, such as those searching for the elusive

gravitational wave (centered at the Laser Interferometer Gravitational Observatories, or Laser

Interferometer Gravitational Wave Observatories [LIGO], in the United States; GEO600 in Germany

[involving the British]; VIRGO in Italy [with the French], and TAMA in Japan), want damping in some

of their components to be as small as possible. Frequency standards the world over require very small

damping to insure high precision for timekeeping.

For the specific components of a system, a successful design frequently requires identification of the

specific mechanisms primarily responsible for the dissipation of energy. Even after identifying the

dominant sources, the theoretical difficulty of their treatment can also range from great to small,

depending on the type of damping. For dashpot fluid damping, adequate models have existed for

decades. For material damping, on the other hand, theories of internal friction are numerous and largely

lacking in self-consistency.

The fundamental mechanisms responsible for damping are in most cases nonlinear; however, the

oscillator’s motion can itself be approximated in many cases by a linear second-order differential equation.

If the potential energy is quadratic in the displacement, then the undamped linear equation of motion is

that of the simple harmonic oscillator, because its solution is a combination of the sine and cosine

(harmonic) functions. This undamped equation comprises the sum of two terms, one being a displacement

and the other term an acceleration. The constant parameter multiplying each term of the pair depends on

the nature of the system. For example, in the case of a mass – spring oscillator, the acceleration is multiplied

by the mass, and the displacement by the spring constant. Thus, the equation corresponds to Newton’s

Second Law applied to a Hooke’s Law (idealized) spring. In an electronic L –C oscillator, the

“displacement” corresponds to the charge on the capacitor (divided by C) and the “acceleration”

corresponds to the second time derivative of the capacitor’s charge (multiplied by inductance L).

The usual means to describe damping, which is always present with oscillation, is to add a velocity

term to the aforementioned displacement and acceleration. Although the damping could derive from

several causes, there is usually a single dominant process. For example, the damping of current in a seriesconnected

resistor, inductor, capacitor (RLC) circuit may depend mostly on Joule heating in the resistor

R, in spite of the fact that there must also be energy loss in the form of radiation. Thus, the equation of

motion includes a first-time derivative of the capacitor’s charge (current) multiplied by R, in accord

with Ohm’s law.

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© 2005 by Taylor & Francis Group, LLC

Whether radiation is important for damping of the RLC circuit depends on the amount of coupling to

the environment. If the circuit communicates with a final amplifier connected to an antenna, then

radiation may become more important than Joule losses. The frequency of oscillation is a key parameter

in this case, and also for damping problems in general. Unfortunately for some common systems,

theoretical efforts to account properly for the effects of frequency have proven largely unsuccessful —

except for models of phenomenological type developed by empiricism.

20.2.2 Specific Considerations

The mass – spring oscillator is the textbook example of harmonic motion, for which one of the most

sophisticated mechanical oscillators ever built is the LaCoste version of vertical seismometer. Significant

portions of the experimental data presented in this chapter were generated with an instrument designed

around the LaCoste zero-length spring (LaCoste, 1934). The instrument used for this data collection was

part of the World Wide Standardized Seismograph Network (WWSSN) during the 1960s. The spring of

this seismometer is responsible for hysteretic damping of the instrument, rather than viscous damping as

commonly assumed. Contrary to popular belief, air damping is not important for this seismograph at its

nominal operating period, which is typically greater than 15 sec. Since every long-period pendulum

apparently exhibits similar behavior, we thus find strong synergetic evidence in support of an old (mostly

unheeded) claim that hysteretic damping (friction force independent of frequency) is universal (Kimball

and Lovell, 1927). Their claim in 1927 to have discovered a universal form of internal friction (damping)

is strengthened since the same behavior is seen in three distinctly different systems: (i) a mass – spring

oscillator (as demonstrated by Gunar Streckeisen, details given later); (ii) a pendulum whose restoration

depends on the Earth’s gravitational field (demonstrated by several independent groups); and (iii) a

rotating rod strained by a transverse deflection (1927 experiments of Kimball and Lovell).

The assumption of universality for hysteretic damping is a key point of this chapter. It will be shown

that the damping of even a vibrating gas column (Ruchhardt’s experiment to measure the ratio of heat

capacities) is likely also hysteretic. The models that are described represent a departure from common

theories of damping. Interestingly, the author’s model has similarities to ordinary sliding friction, as

given to us by Charles Augustin Coulomb. It effectively modifies the Coulomb coefficient of kinetic

friction to yield an effective energy-dependent internal friction coefficient. The energy dependence is

necessary to obtain exponential decay, as opposed to the linear decay of Coulomb damping. Just as with

conventional Coulomb damping, its form is nonlinear, involving the algebraic sign of the velocity. We

will see that the damping capacity predicted by the model permits an equivalent viscous form. Yet the

underlying physics is related to creep of secondary type as opposed to the primary creep of viscoelasticity.

It is this author’s opinion that much of the existing theory of damping is not the best means for

modeling dissipation. The difficulties arise from approximating oscillator decay with linear mathematics.

Although most individuals recognize the oft-stated caveat that viscous damping is an approximation to

the actual physics of dissipation, they do not recognize some of the many serious limitations of the

approximation. The situation is similar to the place in which we found ourselves at the beginning of the

era labeled “deterministic chaos.” The “butterfly effect” (Lorenz, 1972) has radically altered the thinking

of many, but only in relationship to large-amplitude motions of a pendulum, where the instrument is no

longer iscochronous because of nonlinearity. As an archetype of chaos, the pendulum must be rigid and

capable of “winding” (displacement greater than p) before chaos is possible. Nonlinearity is a

prerequisite for the chaos, but it is not sufficient, since there are many examples of highly nonlinear but

nonchaotic motions. For example, amplitude jumps of nonlinear oscillators, during a frequency sweep of

an external drive, have been known for many years. They were observed before chaos was recognized, in

systems like the Duffing and Van der Pol oscillators. Yet chaos, with its sensitive dependence on initial

conditions (responsible for the butterfly effect), was not contemplated at the time. As with most

significant advances, Lorenz’s discovery was by accident, as he modeled convection in the atmosphere.

The author’s confrontation with complexity that derives from mesoscale structures in metals was likewise

unexpected. “Strange phenomena” (as Richard Feynman would probably have labeled them)

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were encountered while using his patented fully differential capacitive sensor to study various mechanical

systems, mainly oscillators.

As with chaos, the pendulum may ultimately serve as an archetype of complexity. When operated at

low energy, especially through a combination of long period and small amplitude, the free decay of the

physical pendulum departs radically from the predictions based on linear equations of motion. Such

complexity can be easily demonstrated when the pendulum is fabricated from soft alloy metals. For

example, Figure 20.1 illustrates the decay of a rod pendulum constructed with ordinary (heavy-gauge

lead – tin) solder of the type used for joining electrical conductors (Peters, 2002a, 2002b, 2002c).

The “jerkiness” (discontinuities) in the record of Figure 20.1 is in no way related to amplitude jumps of

the type previously mentioned; rather, these are jumps of the Portevin – Le Chatelier (PLC) type

(Portevin and Le Chatelier, 1923). They are a fundamental, yet “dirty” phenomenon that physics has

chosen for decades to try and ignore (even though materials science and engineering took early note of

the PLC effect). The most obvious and profound thing that can be said about Figure 20.1 is the following:

the presence of PLC jerkiness means that the concept of a potential energy function is not really valid,

since the requirement for its definition is that a closed integral of the force with respect to displacement

must vanish.

No matter the form of hysteresis, which is the cause for damping, it disallows the curl of the force to be

zero, so that potential energy is never formally meaningful for a macroscopic oscillator (since there is

always damping). In those cases where the damping is essentially continuous (not true for the example of

Figure 20.1), the assumption of a potential energy function retains some computational meaning. For

oscillators influenced by the PLC effect, this is no longer true. The resulting properties are important to a

variety of technology issues, such as sensor performance, since noise is no longer the simple thermal form

predicted by the fluctuation – dissipation theorem (used to characterize white, i.e., Johnson, noise).

0.3

0.2 ( T = 4.4 s)

Sensor Output (V)

0.1

−0.1

−0.2

−0.3

−0.4

−0.5

−0.6

0

Time (S)

0 20 40 60 80 100

FIGURE 20.1 Free-decay of a rod pendulum fabricated from solder.

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© 2005 by Taylor & Francis Group, LLC

Practical means for dealing with systems influenced by “stiction” have been known by engineers for

decades. Because of metastabilities in the assumed potential function, the system is prone to latching

(stuck in a localized potential well). One means for mastering the metastabilities (unstick the part

designed to move) is to “dither” the system. The process has become more sophisticated in the last

decade, which saw a major growth of interest in stochastic resonance. In the definition by Bulsara and

Gammaitoni (1996):

A stochastic resonance is a phenomenon in which a nonlinear system is subjected to a periodic

modulated signal so weak as to be normally undetectable, but it becomes detectable due to

resonance between the weak deterministic signal and stochastic noise.

The phenomenon is related to dithering (Gammaitoni, 1995). It is a case where the signal-to-noise

ratio (SNR) can be increased by the counterintuitive act of raising the level of noise. Such a gain in SNR is

not possible with a harmonic potential.

In recent studies of granular materials, “tapping” has become a popular means to study behavior

that violates the fundamental theorem of calculus. Years ago, this author used tapping as a means to

accelerate creep in wires under tension. Evidently, hammering the table on which the extensometer

rested caused vibrational excitations of the wire that stimulated length changes of discontinuous PLC

type. Because of the broad spectral character of an impulse, various eigenmodes of the wire (see

Chapter 4) could be thus readily excited. After “hammering-down” under load, a silver wire could by

this same means be stimulated, after partial load removal, to exhibit length contractions. Since the

total number of atoms is fixed, the process must involve exchange of atoms between the surface of the

sample and its volume.

Extensometer studies of wires at elevated temperature have also displayed strange behavior.

A polycrystalline silver wire of diameter 0.1 mm and approximate length 30 cm was found to

exhibit large fluctuations when heated in air to within 100 K of its melting point, using a vertical

furnace (Peters, 1993a, 1993b). The large fluctuation in length at these temperatures (reminiscent

of critical phenomena and visible to the naked eye) may be associated with oxide states of the

metal, since the experiments were not performed in vacuum. When cycled in temperature,

fluctuations in the length of a gold wire were found to exhibit dramatic hysteresis. With influence

from Prof. Tom Erber of Illinois Tech University, it was postulated in (Peters, 1993a) that there

may be some mesoscale quantization of fundamental type responsible for the thermal hysteresis

(hysteron).

Mechanical hysteresis resulting from mesoanelastic defect structures is evidently ubiquitous.

Piezotranslators, which are used as actuators in atomic force microscopes and other nanotechnology

applications, are afflicted with high levels of hysteresis when operating open loop. This behavior is

consistent with the anomalously large damping that was observed with a pendulum (reported elsewhere

in this document) in which there was a steel/PZT interface for the knife-edge.

Even the common strain gauge exhibits complex hysteresis behavior. The normally large hysteresis

that is observed in preliminary cycling of a gauge is typically reduced by significant amounts after

repeated cycling, a type of work hardening (if strained well below failure limits). It should be noted

that the hysteresis of all the discussions in this chapter is not to be confused with backlash (as in a

gear train).

All of these experiments are in keeping with the premise that mesoanelastic complexity determines the

nature of hysteretic damping. It is seen that there are a plethora of examples where strain (and thus

damping) of a sample is not simple, is not smooth, but more like the complex behavior of granular

materials. In the case of polycrystalline metals, the same grains that are made visible by methods of acid

etching decoration are evidently responsible for mesoscale (nonsmooth) internal friction damping. To

assume that damping is quantal at the atomic scale, rather than the mesoscale, is without experimental

justification. Nevertheless, this is a popular assumption with which estimates of the noise floor of an

instrument such as a seismometer is estimated, to calculate SNR.

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20.2.3 The Pendulum as an Instrument for the Study of Material Damping

Because of its early contributions to physics, which in those days was called natural philosophy, one

might be tempted to believe that the pendulum is only important to (i) the history of science or

(ii) teaching of fundamental principles. A single observation should be sufficient to resist this

temptation — (as already noted) the pendulum has in the last 30 years become the primary archetype

for the new science of chaos. Additionally, many of the data sets of this document, which show significant

and previously unpublished results, were generated with a pendulum.

To a student of elementary physics, the choice of a pendulum may seem unsophisticated. Yet, to the

author, who has spent 15 intense years trying to understand harmonic oscillators, the pendulum is the

most versatile instrument with which to understand damping. It has been central to the development of

science in general. It was studied by Galileo, Huygens, Newton, Hooke, and all the best-known scientists

of the Renaissance period. It served to establish collision laws, conservation laws, the nature of Earth’s

gravitational field and, most of all, it was the basis for Newton’s two-body central force theory. This

theory was foundational to the development of classical mechanics, which is central to all of physics and

engineering. Historian Richard Westfall has remarked: “Without the pendulum, there would be no

Principia” (Westfall, 1990).

In 1850, Sir George Gabriel Stokes published a foundational paper (Stokes, 1850). His treatment of

pendulum damping permitted the understanding, decades later, of a number of important phenomena

in physics and engineering. For example, his studies were foundational to the Navier– Stokes equations of

fluid mechanics. Moreover, viscous flow known as Stokes’ Law was the basis for Millikan’s famous oil

drop experiment that determined the charge of the electron.

Stokes noted in his paper that, “… pendulum observations may justly be ranked among those most

distinguished by modern exactness.” He also noted

The present paper contains one or two applications of the theory of internal friction to

problems which are of some interest, but which do not relate to pendulums. … the resistance

thus determined proves to be proportional, for a given fluid and a given velocity, not to the

surface, but to the radius of the sphere. … Since the index of friction of air is known from

pendulum experiments, we may easily calculate the terminal velocity of a globule [water] of

given size. … The pendulum thus, in addition to its other uses, affords us some interesting

information relating to the department of meteorology.

The last statement of this quotation speaks to some of the errors in the “common theory” of his

day. In similar manner, some of the common-to-physics damping models of today are erroneously

applied. Those who hold the viscous damping linear model in unwarranted regard, fail to

recognize the limitations under which it is valid. There are frequent misapplications for reason of

experimental deficiencies. We can all profit by taking seriously the following well-known words of

Kelvin:

When you can measure what you are speaking about, and express it in numbers, you know

something about it. But when you cannot measure it, when you cannot express it in numbers, your

knowledge is of a meager and unsatisfactory kind. It may be the beginning of knowledge but you

have scarcely in your thoughts advanced to the state of science. William Thomson, Lord Kelvin

(1824 to 1907)

Simple (viscous) flow of the Stokes’ Law type is possible only according to the restrictive conditions

that Stokes spelled out in his paper. We now specify those conditions for viscous flow according to the

nondimensional parameter given us late in the 19th century by Osborne Reynolds. Specifically, Stokes’

Law is valid only for Re ¼ rvL=h , 60 (approximately, for spheres); where r is the density of the

retarding fluid, v is the speed of the object relative to the fluid, L is a characteristic dimension of the

object, and h is the viscosity of the fluid. The requirement is not generally met for oscillators, and recent

experiments have shown that contributions to the damping from air drag proportional to the square of

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© 2005 by Taylor & Francis Group, LLC

the velocity cannot generally be ignored (Nelson and Olssen, 1986). This is just one example of how two

or more damping types must sometimes be folded into an adequate model of dissipation. A novel

method for combining all the common forms of damping in one mathematical expression is provided in

this document. Additionally, it is shown how to calculate analytically the history of the amplitude of

free-decay for such cases.

Considering the importance of Stokes’ work, it is surprising that some of his requests for further

experiments were apparently never seriously considered. On page 75 of his paper, one reads the

following: “Moreover, experiments on the decrement of the arc of vibration are almost wholly wanting.”

Having noted this, Stokes appealed to experimentalists to generate such data. In the 19th century,

collecting the data he requested would have been labor-intensive and therefore the experiments were

probably never attempted. Sensors and data processing of the modern age now make them

straightforward, but the pendulum has by now been viewed by too many as a relic rather than the

important instrument described by Stokes. Much of the author’s efforts have been directed at showing

that the pendulum is still an important research instrument. For example, one physical pendulum of

simple design was the basis for the generalized model of damping (modified Coulomb) that is here

presented. Another has been used to illustrate surprisingly rich complexities of the motion that results

from the ubiquitous defects of its structure (Peters 2002a, 2002b, 2002c). Thus studying the complex

motions of “low and slow” physical pendula could yield significant new insight into the defect properties

of materials — a field where relatively little first-principles progress has been made.

20.2.4 “Plenty of Room at the Bottom”

Richard Feynman gave a now-famous talk in 1959 titled, “There’s plenty of room at the bottom”

(presented at the American Physical Society’s annual meeting at CalTech). Drawing on observations from

biology, he spoke of a solid-state physics world involving “… strange phenomena that occur in complex

situations.” In the 44 years since Feynman’s prophetic comments, there have been spectacular

achievements in very large-scale integrated (VLSI) electronics, microelectromechanical systems (MEMS),

and even nanotechnology. Progress in the mechanical (including sensor) realm has been much slower

than in electronics; consequently, our present processing power far exceeds our acquisition (and

actuator) capabilities.

One of the major obstacles to miniaturization involves dramatic change to physical properties that can

occur as the size of a system shrinks below the mesoscale toward the atomic. For example, VLSI

electronics is already beginning to be impacted by quantum properties of the atom, as component size

continues to decrease in accord with Moore’s law (Moore, 1965). Among other things, Feynman

predicted that lubrication would no longer be “classical” at such a scale. On a related note, a paper by

Nobel Laureate Edward M. Purcell (Purcell, 1977) draws a striking contrast between our macroscopic

world and that of micro-organisms. At low Reynolds number, inertia becomes unimportant, and

mechanics is dominated by viscous effects. The adoption of a new paradigm will be necessary for

engineers to deal with these differences.

In the article “Plenty of room indeed” (Roukes, 2001), it is noted that there is an anticipated “dark

side” of efforts to build truly useful micro- and nano-sized devices. Gaseous atoms and molecules

constantly adsorb and desorb from device surfaces. This process is known to exchange momentum with

the surface, even permitting scientific study of the gas – solid interface (Peters, 1990). The smaller the

device, the less stable it will be because of adsorption/desorption. As Roukes has noted, this instability

may pose a real disadvantage in various futuristic electromechanical signal-processing applications

(Cleland and Roukes, 2002).

There is direct evidence, provided in the present chapter, that we need to be more concerned with

noise: (i) the evacuated pendulum where it is speculated that outgassing influenced its free-decay, and

(ii) the seismometer free-decay that showed both amplitude and phase noise and evidence for nonlinear

damping. Concerning (i), when the vacuum chamber pressure is reduced, the preexisting steady state

(normal rate balance between adsorption and desorption) becomes disturbed, so that there is a complex

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emission of gases from the surface of the pendulum. The emission is not likely to be spatially uniform,

but more like the jets seen on Halley’s comet when photographed by the Giotto spacecraft in 1986. In case

(ii), the noise is seen to derive from mesoanelastic complexity of the structure of the pendulum itself

rather than involving gases.

Miniaturized devices have the potential to serve as on-chip clocks, and the importance of phase noise

to clocks is well documented. There is another, more subtle issue that points out the importance of phase

noise. One of the best means for improving SNR is the technique of phase-sensitive detection, first

employed by Robert Dicke at Princeton to improve solar experiments. The performance of miniaturized

electromechanical sensors using “lock-in” amplifier methods may be influenced significantly by

mechanical phase noise.

Phase noise of miniaturized devices is still mostly speculative. In addition to the mechanism just

mentioned (adsorption/desorption), there is the matter that constitutes the theme of this chapter, defect

organization. It is not possible to grow materials without dislocations and/or other disturbances to

crystalline order, such as vacancies, interstitials, or substitutional impurities. Thus, “when mother nature

fills the vacuum she abhors, she rarely does so with perfection.”

Long before defects organize to the point of incipient failure (at much larger strains), they still

influence vibration. They may even be a primary source of 1=f noise. Electronic noise of 1=f type is known

to involve defects by means of trapping states, and these states derive from crystalline defects sometimes

involving the surface. The interaction of the surface and the volume of a solid are important. For

example, consider pure copper single crystals of the type used by the author in his doctoral work.

A practical joke suggested by Vic Pare (that we never conducted because of the cost of these samples)

would be to have a 98-lb weakling bend one by hand, then ask an NFL linebacker to straighten it back

out! The striking irreversibility is the result of work hardening as dislocations develop at the surface and

propagate into the bulk where they entangle.

In the case of polycrystalline materials, the memory features of hysteresis may be important according

to the method of their fabrication. Wires are typically produced by pulling through successively smaller

dies. This “swaging” may be conducive to the exchange of monolayer groups of atoms between the

volume and the surface during fluctuation length changes. The fluctuation – dissipation theorem does

not hold or, if it does, only in terms of larger entities than the atom. Thus, there are many yet-to-bequantified

elements of noise in the vibration of miniaturized devices. Feynman was right when he spoke

of strange phenomena of the solid state.

Technology of the future is expected to be confronted increasingly with damping problems that

must address issues of scaling — to deal with some factors discussed in this chapter, which, to the

author’s knowledge, have not been previously published. Until small (MEMS) oscillators become

more common to the engineering world, we must study the mechanisms responsible for their

damping by other than traditional means. One approach is similar to experimental techniques for the

verification of the kinetic theory of gases. As noted by Present in his textbook (Present, 1958), there

are two ways that Brownian motion can be studied: either (i) with small objects and an

unsophisticated detector, or (ii) with larger objects and a very sensitive detector. It is the latter that

provided some of our present knowledge of damping at the mesoscale. The fully differential capacitive

transducer, whose patent label is “symmetric differential capacitive” (SDC), is a robust new

technology that is sensitive, linear, and user-friendly (Peters, 1993a, 1993b). As with other sensitive

detectors that have been used to predict the properties of small objects by studying larger ones, smallenergy

studies of various macroscopic pendula are demonstrating some of the “strange phenomena”

of complex type predicted by Feynman.

From the author’s perspective, we of the physics community have been guilty of two significant errors:

(i) oversimplification of many problems by assuming a linear equation of motion based on viscous

damping, and (ii) losing sight of fundamental issues by working with inappropriate, overly complicated

damping models. The goal of this chapter is to assist progress toward a healthier balance between these

extremes. It is hoped that readers will be thus better equipped to identify, and then dismantle, some of the

impediments to the development of future technologies.

Damping Theory 20-11

© 2005 by Taylor & Francis Group, LLC

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