20.3 Background

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20.3.1 Terminology

The large number of mechanisms capable of energy dissipation has resulted in a host of adjectives to

describe damping phenomena in mechanical systems. They include (nonexhaustive list): viscous, eddy

current, Coulomb, sliding, friction, structural, fluid, thermoelastic, internal friction, viscoelastic,

material, solid, phonon – phonon, phonon – electron, and hysteretic. For present purposes, damping

types will be grouped according to one of the following three categories: (i) fluid (including viscous),

(ii) Coulomb, and (iii) hysteretic. Although hysteretic damping has come to be associated in engineering

circles with a particular form of material damping in solids, it should be noted that all forms of

damping involve hysteresis (for which the Greek meaning of the word is “to come late”). In a plot of

periodic stress vs. strain, which is a straight line for displacements of a nondissipative, idealized

substance, hysteresis causes the line to open into a loop. The size of the loop — more specifically the area

inside this hysteresis loop — is a measure of the amount of nonrecoverable work done per cycle because

of the damping. An actual force of friction is not readily recognizable in those cases that are labeled

“internal friction.” The word friction is used in a generic sense, meaning any process responsible for

conversion of the oscillator’s coherent motion into incoherent thermal activity.

With each of viscous, eddy current, and Coulomb damping, a force external to the oscillator is

responsible for the dissipation of energy. The external force is associated respectively with (i) laminar

fluid flow, (ii) induced currents, and (iii) surface friction. The surface friction case is not necessarily

the trivial textbook presentation involving a coefficient of kinetic friction and a normal force. The

cases just mentioned, along with thermoelastic damping; which is of internal rather than external

origin, are much easier to treat theoretically than other cases. Viscous damping and eddy current

damping (over the full range of the motion) are adequately described by a velocity term, which

yields a linear equation of motion. Coulomb damping, however, is not proportional to velocity, but

rather depends only on the algebraic sign of the velocity. The equation of motion is consequently

nonlinear. Additionally, and unlike most other forms, Coulomb damping is not exponential. The

turning points lie along a straight line when the motion is plotted vs. time. Similarly, if eddy current

damping exists only over a small part of the motion, the decay is linear rather than exponential

(Singh et al., 2002).

20.3.2 General Technical Features

Historically, viscous damping has been the model of choice because the resulting equation of motion is

mathematically attractive and, for the RLC circuit, the form is appropriate. For mechanical oscillators,

it is not generally appropriate, since viscous damping amounts to some part of the system moving

in an external Newtonian fluid that removes energy because of a friction force that is proportional

to velocity.

The defects responsible for material damping, such as dislocations, are also responsible for creep, so

that high strength and high damping tend to be incompatible attributes. Magnesium alloys tend to be

better than many other metals in this regard. Hardness of a material is neither a prerequisite for toughness

nor for small damping, as recognized by those familiar with the mechanical properties of cast iron.

On a different scale, defects determine “how things break”; concerning which Marder and Fineberg

have stated the following:

the strength of solids calculated from an excessively idealized starting point comes out

completely wrong; it is not determined by performance under ideal conditions, but instead by

the survival of the most vulnerable spot under the most adverse of conditions. (Marder and

Fineberg, 1996)

Three famous scientists are primarily responsible for the highly popular viscous damping model of

the simple harmonic oscillator; they are Lord Kelvin (Thomson and Tait, 1873), G.G. Stokes and

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H.A. Lorentz. Stokes is best known for his equations of fluid dynamics that also include the name

Navier. Stokes’ Law, which describes the terminal velocity of a raindrop, was developed through his

treatment of the damping of a pendulum. Not only does his law provide a basis for the simplest

approximation for damping of a macroscopic oscillator, it was used by Robert Millikan to determine the

charge of the electron. It should be noted that harmonic oscillation in a fluid (even at low Reynolds

number) is much more complicated than steady-flow viscous friction. This topic is treated in Chapter 21,

Section 21.9.

The first individual to use the term “simple harmonic oscillator” was probably Lord Kelvin. Such an

oscillator is a key tool of experimental physics and also the foundation for much of theoretical physics. It

is the basis for communication via electromagnetic waves and even esoteric theories of superfluids and

superconductors.

Much of the underpinnings of theory involving harmonic oscillation derive from the work of Hendrik

Anton Lorentz (1853 – 1928). Lorentz is well known for a variety of classical physics contributions, such

as (i) the transformation of special relativity associated with Einstein and (ii) the force law for the

acceleration of charged particles, both of which bear his name. Before the existence of electrons was

proved, Lorentz proposed that light waves were due to oscillations of an electric charge in the atom. For

his development of a mathematical theory of the electron, he received the Nobel Prize in 1902. The

importance of his contributions is further realized by noting that it is common practice to describe the

lineshape of atomic spectra by the term Lorentzian. The Lorentzian is equivalent to the resonance

response of the driven viscous-damped simple harmonic oscillator.

It is easy to show how resistance in an electric network is responsible for damping; however, it is a

challenge to understand anelastic processes of mechanical damping in terms of viscosity. From

comments of his Ph.D. dissertation, it has been said that even Lorentz was never apparently satisfied with

the velocity damping term in his equation — not knowing just how to relate it to the underlying physics.

It is also clear from Stokes’ paper that he recognized the need for caution in the use of his law of viscous

friction. It appears that both Lorentz and Stokes were very careful compared with the carelessness with

which the viscous model has been employed by many individuals in recent years.

The failure of solids influenced by “hysteretic” damping to be adequately described by the methods of

viscoelasticity is not widely appreciated. It is unfortunate that too few people have expanded their view of

damping to include other important types, such as derive from the anelasticity of solids. It is important

in this work to recognize some subtle differences, for example, inelastic (not elastic) is not to be equated

with anelastic (other than elastic).

20.3.3 Active vs. Passive Damping

With improvements in cost/performance of electronics, active damping is increasingly popular. Using

force-feedback with integration/differentiation circuitry (opamps), a mechanical oscillator can sometimes

be tailored for a specific purpose. A sophisticated example of this technology is the broadband

seismometer that began to replace earlier version (passive) instruments roughly 35 years ago.

The Sprengnether – LaCoste spring instrument that was used for some of the experiments reported in

this document has been superceded by force-feedback units such as the Streckeisen STS-1 and STS-2.

In lieu of feedback, another way actively to influence the damping of a mechanical oscillator is to

connect the sensor to an amplifier having a negative input resistance. The seismometers marketed by

Lennartz Electronics in Germany use this in a patented technique to improve the performance of

ordinary, off-the-shelf electrodynamic geophones.

Active damping depends on the nature of the transfer function of the composite system (electronics

plus mechanical). The characteristics of the transfer function are determined by the location of its poles

and zeros in the complex plane. Seismometers operate nominally near 0.707 of critical damping. This is

done for two reasons: (i) the instrument is easier to adjust and (ii) the interpretation of

earthquake records is simpler. Of course, to increase damping is to decrease sensitivity because of the

fluctuation – dissipation theorem.

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The force-feedback technique is not practical for some situations, regardless of cost. Additionally, it

must be recognized that the method is not the answer to all problems, since electronics cannot

compensate for a poor mechanical design. The description of commercial products is in some cases

highly exaggerated, giving the impression that almost any sensor can perform flawlessly in this manner.

Some accelerometers have employed dithering to offset the effects of “stiction” in bearings. The dithering

was necessary because the potential energy function is not truly harmonic, being afflicted with the

consequences of nonlinear damping. Even with sensing schemes that do not use a bearing, the effects of

nonlinearity persist. In “Seismic Sensors and their Calibration” (Bormann and Bergmann, 2002), Erhard

Wielandt, in talking about transient disturbances in the spring of a seismometer, says the following:

Most new seismometers produce spontaneous transient disturbances, quasi-miniature earthquakes

caused by stress in the mechanical components.

In other words, internal friction from defects at the mesoscale cause behavior that is in some ways

similar to ordinary sliding friction, where the static coefficient is greater than the kinetic coefficient. The

postulate of Bantel and Newman is consistent with this idea (Bantel and Newman, 2000) when they refer

to their observations as being consistent with a “stick – slip” model of internal friction.

It is seen then that one must use a detector that responds faithfully to the signal around which the

servo-network functions. The linearity and sensitivity of that sensor are of paramount importance, since

the basis for force-feedback design is linear system theory. For some less-challenging cases, the design

approach is straightforward, since software packages like MATLAB (see Chapter 6 and Appendix 32A)

have built in functions to describe behavior.

20.3.4 Magnetorheological Damping

A recent approach to damping control, that is quite different from the servo-networks mentioned above, is

one that uses an magnetorheological (MR) fluid. It takes advantage of the large variation in viscosity of

certain compound fluids according to the size of an applied magnetic field. J. David Carlson (Carlson,

2002) describes how an MR sponge damper is activated during the spin cycle of a washing machine to keep

it from “walking out of the room.” The peak in the Lorentzian (resonance response) of the machine is

shown in his article to be substantially lowered by supplying current to the electromagnet of the damper.

20.3.5 Portevin – LeChatelier Effect

Physics, engineering, geoscience, and mathematics have all contributed greatly to a better understanding

of damping phenomena; however, there has been little cross-discipline exchange of ideas and lessons

learned. Some of the impediments to strong interdisciplinary programs derive from (i) the complexity of

damping problems in general and (ii) the tendency for physics and mathematics research to be, on the one

hand, less pragmatic and, on the other hand, highly specialized — focused on specific energy dissipation

mechanisms. A good example of (i) involves the PLC effect, discovered in 1923. Why physics mostly

ignored this early example of “dirty science” by two of their own number is not easily understood,

although the birthing of quantum mechanics around this time may have been a factor. Had history turned

in a different direction, perhaps we would already be able to explain from first principles the most

important, but still barely understood, form of noise known as 1=f ; or flicker, or pink noise. Even though

R.B. Johnson (well known for his discovery of white electronics noise in a resistor) was one of the first to

see this form of noise, it still is not explained from first principles — although recent discoveries suggest an

intimate connection with fractal geometry involving self-similarity. Such geometry is associated with the

mesoscale of materials where the grain, rather than the atom, is the basic element of statistical mechanics.

For alloys, the PLC effect appears to be, in some ways, what the Barkhausen effect is to magnetic

systems. In the case of ferrous materials, the noise which derives from the mesoscale has long been

recognized; however, similar noise of mechanical type has not been seriously studied. This oversight is

even more puzzling when one considers the admonition by G. Venkataraman, as recorded in the

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proceedings of a Fermi conference, for scientists to get involved in what he felt should become an

important new field (Venkataraman, 1982).

20.3.6 Noise

Noise is purposely discussed in this chapter (also see the chapters in Section IX of this handbook)

because it has been a, largely, missing component of efforts to understand the physics of damping.

A feel for the importance of noise to damping research is to be gleaned from a comment by

Kip Thorne in his foreword to the English translation of a book by V.B. Braginsky et al. (1985).

Mainly because of instrumental needs of the Laser Interferometer Gravitational Wave Observatories

(LIGO), Thorne writes,

The central problem of such experiments is to construct an oscillator that is as perfectly simple

harmonic as possible, and the largest obstacle to such construction is the oscillator’s dissipation. If

dissipation were perfectly smooth, it would not be much of an obstacle, but the fluctuation –

dissipation theorem of statistical mechanics guarantees that any dissipation is accompanied by

fluctuating forces. The stronger the dissipation, the larger the fluctuating forces, and the more

seriously they mask the signals that the experimenter seeks to detect.

This comment by Thorne suggests a frequently important impediment to dialogue between

engineering and physics — concern for different issues. LIGO is trying to minimize damping, whereas

many engineering problems are concerned with just the opposite — making the damping as large as

possible without compromising strength. More detailed discussions of noise are provided later.

20.3.7 Viscoelasticity

Within the world of polymers, damping is frequently described by the expression “viscoelasticity.” This

word, around which handbooks have been written (e.g., Lakes, 1998), is a combination of the two

words, viscous and elastic. We like to think of ideal fluids as being viscous in the manner described by

Newton. Likewise, ideal solids that obey Hooke’s Law (stress proportional to strain) are described as

elastic. Unfortunately, nature contains neither ideal solids nor ideal fluids. Real springs do not obey

Hooke’s Law, but rather are influenced by “anelasticity” (other than elastic) which gives rise to

hysteresis in the stress – strain relationship. Real fluids usually have some (if not near total) degree of

non-Newtonian character. Thus an envisioned “mixing” of fluid-like and solid-like character has

dominated the thinking of those who, through the decades, attempted to develop theoretical models

of damping.

It should be noted that the springs and dashpots used in models of viscoelasticity do not actually exist.

They serve as a phenomenological means for (hopefully) understanding the elementary processes which

their arrangement is designed to mimic. Consider, for example, high polymers, in which the interwoven

structure of the long-chain molecules is one of extensive mechanical interference. (One popular

visualization is that of an entanglement of a huge number of long, writhing snakes.) An increase of

temperature is met with overall length reduction (negative temperature coefficient of expansion for the

so-called entropy spring), which stands in stark contrast with metals. Such behavior is clearly important

to damping since, as noted by Gross years ago, “…thermal movement interferes with the orientation and

disorientation of the molecules and ultimately causes delay in the expansion and contraction of the

specimen” (Gross, 1952).

20.3.8 Memory Effects

In this same article, Gross is one of the first to mention “memory” properties of creep. He describes a

by-then old demonstration in which a “firmly suspended metal or plastic wire is twisted first in one

direction for a long time and then in the other direction for a short time. Immediately after release,

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the deflection will be in the direction of the last twisting, but it decreases rapidly. Presently, a reversal

occurs, and the wire begins to turn in the other direction, corresponding to the first twisting — the

memory of the recent short-term handling has been obliterated by that of the more remote but longer

lasting and therefore more impressive one!” Perhaps this old demonstration (sometimes today called the

anelastic after-effect) is not so startling to those familiar with more modern shape-memory-alloys, which

are expected by many to play increasingly important roles in the applied science of damping.

20.3.9 Early History of Viscoelasticity

Those who provided seminal influence in the development of the theories of viscoelasticity during the

19th century were some of the most famous names in physics, like Maxwell and Kelvin. Maxwell is best

known for the electromagnetic equations associated with his name. He is far less known for two other

significant contributions: (i) kinetic theory of gases and (ii) viscoelasticity — both of which are important

to theories of oscillator damping. Maxwell’s interest in the problem of viscoelasticity is first documented

in a paper during his teen years, titled “The Equilibrium of Elastic Solids.” Through his development with

Boltzmann of the kinetic theory of gases, Maxwell showed a counterintuitive property of the viscosity of a

gas. The viscosity does not decrease significantly as the pressure is reduced, until the mean free path

between collisions of the molecules begins to approach dimensions of the chamber holding the gas.

Important even to modern innovations such as MEMS oscillators, his surprising prediction was quickly

verified by experiment. Maxwell’s model of viscoelasticity combines a purely elastic spring with a purely

viscous dashpot (fluid damper in which the friction force is proportional to the velocity).

Kelvin, probably the first to include a viscous damping term in the equation of motion of the simple

harmonic oscillator, developed a similar model of viscoelasticity. Each of the two models is usually

represented in literature (without original references) as containing a single spring and a single dashpot.

They differ in that one connects the pair in series (Maxwell), while the other connects them in parallel

(Kelvin – Voigt).

Both the Maxwell model and the Kelvin – Voigt model have been found by engineers to be less useful

than the standard linear model (SLM) of anelasticity, largely advanced in the 20th century by Clarence

Zener (Zener, 1948). In the three-component Zener model, a spring is connected in series with a parallel

combination of spring and dashpot. Curiously, Zener is widely associated with electronics because of

the common diode named after him, but fewer people know of his work in anelasticity. No doubt,

his understanding of anelasticity helped him to better understand the complex processes at work in

his diode.

20.3.10 Creep

The prevailing models of anelasticity appear frequently in the literature, but mostly in relationship to

primary creep. Some of the papers exceptional to this rule are those by Berdichevsky (Berdichevsky et al.,

1997). Recent work of a more heuristic type has shown that the equations of viscoelasticity are also able to

accommodate secondary creep, in which the decay of strain rate with time has disappeared (Peters,

2001a, 2001b).

The importance of creep (and relaxation) physics to damping warrants some discussion. When a

sample is subjected to a constant stress, the strain evolves through three phases of creep: (i) primary,

(ii) secondary, and (iii) tertiary. An example of the first two of these phases is shown in Figure 20.2.

In the primary stage, the sample is deformed by anelastic processes involving defects of the crystalline

structure. Influence of the disordering mechanisms is progressively reduced as the sample undergoes

work hardening (such as pinning of dislocations). Work hardening would result in a purely exponential

creep, in the absence of thermal effects which strive to undo the hardening (via diffusion processes).

(At zero Kelvin, the creep would eventually cease, if described by a single time constant.) In the secondary

stage, a balance between work hardening and thermal softening is attained, in which the strain vs. time

has converted from exponential to linear. This balance cannot continue forever, if the stress is larger than

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a threshold associated with failure (the elastic limit), and thus a final complex fracture of the sample

finally occurs as the sample passes through the tertiary stage. Although one might want to divorce the

issues of tertiary creep from considerations of damping, there is clearly a link between damping and

failure, involving defects. We will return to this point later.

In Figure 20.2, creep resulted from the instrument having been severely disturbed (relocated

accidentally by plumbers working in the building). As with any long-period mechanical oscillator, it is

necessary for this instrument to stabilize after a major rebalancing. Primary creep is seen to have endured

for about 5 h and what is labeled secondary creep in the figure does not continue indefinitely with the

implied constant rate. Thus, the instrument will typically stabilize after one or several days, when the

period of oscillation is of the order of 20 sec.

The total amount of creep in Figure 20.2 deserves mention. In the indicated 14.2 h, the mass of the

seismometer moved a vertical distance of only 0.25 mm, which can be ascertained from the ordinate axis

using the sensor calibration constant of 2000 V/m.

20.3.11 Stretched Exponentials

Systems typically demonstrate a more complex behavior than can be simply described by the SLM of

viscoelasticity. In 1847, Kohlrausch (1847) discovered that the decay of the residual charge on a glass

Leyden jar followed a stretched exponential law. The functional form that he discovered is often associated

with a broad distribution of relaxation times, and has been found to describe a remarkably wide range of

physical processes. To describe damping that is of the stretched exponential type, Kelvin chains or

Maxwell elements in parallel have been used. Although an improved fit to the data can be realized by this

means, the technique results in a high number of material parameters which have to be identified.

20.3.12 Fractional Calculus

A promising alternative to multiexponential decay models is to replace classical rheological dashpots by

“fractional” elements. It is claimed that with only a few parameters, material behavior of many

viscoelastic media can be described over large ranges of time and/or frequency (Hilfer, 2000). It may also

be possible with fractional derivatives to treat the discontinuities that are sometimes present in decay

(Asa, 1996). The disadvantages of fractional calculus are (i) the increasing computational/storage

requirements and (ii) the esoteric mathematics, which is alien to the training of most.

FIGURE 20.2 Example of creep in the spring of a vertical seismometer.

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20.3.13 Modified Coulomb Damping Model

Published here formally for the first time, with details described later, the heuristic “modified Coulomb”

model is an alternative to all of the aforementioned damping models. It is thought to be closely related to

secondary creep (Peters, 2001a, 2001b) and (like fractional calculus) accomplishes good fits with a small

number of parameters. Developed from energy considerations, its equations are expressed in canonical

form involving the quality factor Q:

20.3.14 Relaxation

Formally, relaxation is defined by the behavior of a sample subjected to a constant strain. Because of

the mechanisms just discussed in relationship to creep, the stress relaxes exponentially toward zero (in

the simplest approximation). In practice, the definition just given can be misleading since the word

relaxation is used to describe a host of processes in which some quantity decays exponentially in

time — for example, the relaxation of strain at constant stress in the Kelvin – Voigt model of

viscoelasticity.

Some of the viscoelastic models using dashpots and springs have been quite successful in the limited

regime of their applicability. For example, the Zener (Debye) model, which will be discussed again later,

has been used for years to describe a particular form of damping in solids, which derives from relaxations

associated with dislocations. Seminal experimental work of this type was conducted by Berry and Nowick

in the 1950s (Berry and Norwich, 1958). A well-known theoretical model to describe dislocation

damping was developed by Granato and Lucke (Granato and Lucke, 1956). The Granato model is that of

a vibrating string (bowed Frank – Read source), where the end points of the “string” are points on the

dislocation line that have been pinned. Recent theory shows that the Granato model is not always

adequate; that “dislocation interactions may alter substantially the dislocation component of the

spectrum observed during internal friction experiments.” (Greaney et al., 2002) (excellent introductory

material on this subject is to be found online at http://mid-ohio.mse.berkeley.edu/alex/rachel/rachel/

rachel.html).

Bordoni (1954) performed experiments that led to his observation of relaxation-type internal friction

processes where the acoustic attenuation is seen to peak at certain temperatures. The so-called Bordoni

peaks occur at low temperatures or at ultrasonic frequencies. These losses, which are maximum when

dislocation relaxations can take place in step with the driving frequency, were first observed in the FCC

metals: lead, copper, aluminum, and silver.

Dislocation damping as just described is characterized by a temperature-dependent relaxation that

exhibits Arrhenius behavior. By plotting the internal friction vs. reciprocal temperature, one may

estimate the activation energy of the process responsible for the damping. The following quotation

from the introduction of the Berry paper assists in defining some of the many expressions used

historically to describe damping:

Internal friction is often loosely described as the ability of a solid to damp out vibrations.

More strictly, it is a measure of the vibrational energy dissipated by the operation of specific

mechanisms within the solid. Internal friction arises even at the smallest stress levels if

Hooke’s Law does not properly describe the static stress – strain curve of the material. The

nonelastic behavior which Zener has called anelasticity arises when the strain in the material

is dependent on variables other than stress.

In a recent private communication, Prof. Granato has indicated the following:

Dislocations do follow the Zener (or Debye) form fairly well for the damping, but not for

the elastic modulus. This is because the response to a stress is given as a Fourier series. The

higher order terms in the series have little effect on the damping, but a strong effect on the modulus

at high frequencies. This makes the modulus fall off more slowly than with the reciprocal

frequency.

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