20.18 Nonlinearity

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20.18.1 General Considerations

Electrical nonlinearity is the type with which most engineers are familiar. It is the very basis for common

nondigital forms of communication, such as that of frequency modulation type. A popular form of radio

amateur communication is one in which the carrier and one of the two normal sidebands of a signal are

suppressed before going to the antenna. At the receiver, the carrier is “regenerated” before going to the

demodulator. The demodulator required for ultimate transduction by speaker is also a nonlinear device.

Nonlinearity of mechanical type is encountered throughout nature. The human ear, for example, is not

linear, but rather characterized by both quadratic and cubic nonlinearities. If an intense, pure low

frequency (inaudible) sound of frequency f is present with a higher frequency audible one of frequency F,

then one typically hears (in addition to F) tones at F ^ f due to the quadratic nonlinearity and F ^ 2f

due to the cubic nonlinearity.

Very high frequency acoustics (ultrasound) is employed for studies of elasticity. The quasi-linear

features of ultrasonic propagation have been the basis for measuring second-order elastic constants

(determined by velocity of propagation) and internal friction (by attenuation of the beam, i.e., damping).

A commonly employed ultrasonic technique that has been used to study both linear and nonlinear

phenomena is the pulse-echo method. By using a thin specimen and extending the pulse width, the

overlapped signal can add constructively or destructively and, in the former case, resonance is

approached as the width gets very large (Peters, 1973). The pulse-echo method was the basis for this

author’s Ph.D. dissertation (“Temperature dependence of the nonlinearity parameters of copper single

crystals,” The University of Tennessee, 1968). The distinguished career of his professor, M.A. Breazeale,

has focused on ultrasonic harmonic generation as a means to determine the shape of the interatomic

potential of solids (Breazeale and Leroy, 1991). A longitudinal wave distorts because of the anharmonic

potential (acoustic equivalence of optical frequency doubling with lasers in a KdP crystal). In like

manner, phonon – phonon interactions are possible only because of nonzero elastic constants of order

higher than second (second-order constants determining the harmonic potential). Because phonon –

phonon interactions are part of damping, there must be consequences, at least for some cases, from

nonlinear damping terms.

The unifying theme for this chapter is that damping is fundamentally nonlinear, in spite of the fact that

linear approximations have prevailed in modeling and, for many purposes these linear models appear to

be acceptable (Richardson and Potter, 1975). In their paper, Richardson and Potter state that “… an

equivalent viscous damping component can always be derived, which will account for all of the energy

loss from the system. Thus, in measuring the modal vibration parameters for the linear motion of a

system, we don’t care what the detailed damping mechanism really is.”

Although their statement may be true for steady state, it is not expected to be true for the transient

processes that lead to steady conditions of oscillation. As demonstrated elsewhere in this chapter,

mixtures of different damping types are common among oscillators, and only with viscous or hysteretic

damping is the Q independent of amplitude. Other cases may result, for example, from the decay being a

combination of hysteretic damping and amplitude-dependent damping. An example used to illustrate

this combination was an outgassing pendulum oscillating in vacuum. Similarly, a long, “simple”

pendulum, oscillating in air, is found to require a pair of terms — viscous damping and “fluid” damping

(Nelson and Olssen, 1986). In the Nelson and Olsson experiment, the drag was found, because of the size

of the Reynolds number, to involve both first- and second-power velocity terms. Their case can,

incidentally, be treated by the modified Coulomb, generalized damping model of this document.

The presence of either amplitude-dependent damping or Coulomb damping is expected to play a role

in determining what modes of a multibody system are actually excited by external forcing. Concerning

the latter, Coulomb friction is the basis for exciting chaotic vibrations in mechanical systems (Moon,

1987). Without the nonlinear friction, the excitation would be impossible. In similar manner (although

chaotic motion may be present but not in an obvious way), friction from rosin on a violin bow is used to

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play the violin. Still another example of similar physics is the “singing rod” that was mentioned elsewhere

as exhibiting thermoelastic damping.

Whatever combinations of normal modes are initially excited in a linear system are the only ones that

can exist thereafter. Such is not the case, however, for many systems and, since nonlinearity is required for

mode coupling, there must be nonlinearity in the equations of motion. There is no question about the

existence and importance of elastic nonlinearity. Indeed, thermal expansion would be impossible in the

absence of higher order elastic constants. The importance of nonlinear damping remains yet to be

quantified, since models to include it have been few in number. For those who have found it

advantageous to include the oldest and simplest type of nonlinearity in a damping model — Coulomb

damping (sliding friction) — the improvements realized by their choice are unlikely to cause them to

revisit the problem and try to solve it in terms of a viscous equivalent linear approximation.

There are many examples of damping of a single type other than viscous. In their efforts to improve the

knowledge of the Newtonian gravitational constant G ¼ 6:67 £ 10211 Nm2=kg2 (approx.), Bantel and

Newman (2000) discovered a pure form of amplitude-dependent damping of internal friction type. They

did their experiments at liquid helium temperature (4.2 K) and noted the following: “A striking feature

noted in our data is the linearity of the amplitude dependence of Q21 for the three metal fiber materials,”

and also “Linearity implies that Q may depend on frequency but not on amplitude, while in fact Fig.1

displays a significant amplitude dependence (and hence nonlinearity) of internal friction in all fibers

tested.” They also considered the temperature dependence of damping and note that there are two

independent contributions in Cu – Be. One is linear and temperature-independent and the other

amplitude-dependent and independent of temperature. Finally, it is worth noting their statement, “…our

results are strongly suggestive of some kind of ‘stick – slip’ mechanism …,” which lends strong support to

the modified Coulomb internal friction damping model of the present document.

Repetition is felt to be warranted — such systems cannot always be reasonably described by an

equivalent viscous form! For a case of amplitude-dependent Q; the equivalent form has no meaning

unless the amplitude is fixed, i.e., it oscillates at steady state. Unfortunately, the evolution of the system to

steady state is expected to depend on the damping form(s). Surely a model (not yet realized) that predicts

what modes survive is worth much more than one which only characterizes the modes after they have

reached steady state. The author and Prof. Dewey Hodges of Georgia Tech’s Aerospace School are

planning projects to try to develop such predictive capability. The present state of the art applied to

structures suggests that a truly predictive model cannot ignore damping nonlinearity.

As demonstrated by Bantel and Newman (2000), the mixture of damping types that can co-exist in a

system may change with temperature. Early experiments by Berry and Nowick (1958) also showed, as have

many investigators subsequently, that damping generally depends on aging. It is naive to believe that aging

would not also change the mix of damping types, when there is more than one type. Thus, an adequate

damping model must be able to easily accommodate several damping types that are simultaneously active.

A variety of engineering techniques have evolved to treat such problems. The most “successful” ones suffer

from the fact that an excessive number of parameters or coupled equations must be adjusted by trial and

error to yield decent agreement with experiment. This is reminiscent of the state of high-energy (nuclear)

physics before the standard model. The hallmark of physics success has always been simplification. As noted

by Albert Einstein: “All physics is either impossible or trivial. It is impossible until you understand it. Then

it becomes trivial.” It is hard to imagine, however, that certain damping physics could ever become trivial.

Nevertheless, the simplifying nature of better conceptual understanding is a goal to strive for.

One of the remarkable things about the majority of damping models has been the absence of a direct

consideration of energy in describing the dissipation process. After all, the most important quantity

transformed by the damping is energy, so its inclusion is natural.

20.18.2 Harmonic Content

When the damping is nonlinear, the waveform of the oscillator in free-decay contains harmonics.

The harmonic content is most obvious in the residuals (difference) after fitting a damped sinusoid to

the record, as shown in Figure 20.23.

Damping Theory 20-59

© 2005 by Taylor & Francis Group, LLC

Residuals are still present for the viscous case because the equation of motion was integrated

numerically and compared against the classic exponentially decaying sinusoid (solution to the equation)

that was used for fitting in all cases. There is always some degree of mismatch with the fit because of

rounding errors in the computer. In Figure 20.23, the fundamental is smaller for the viscous case because

the fit is inherently more perfect by about an order of magnitude in most of the “eye-ball” fits that were

performed by Excel after importation of the data.

A test for harmonic content was performed on the seismometer (17-sec period) data displayed in

Figure 20.11 illustrating phase noise. The power spectrum of the residuals for that case is shown in

Figure 20.24.

The third harmonic is especially noticeable in this case. That the other harmonics are not so “cleanly”

displayed may result from the significant phase noise of the record.

Frequency (Hz)

15

10

5

0

0 0.1

−5

−10

0.2 0.3 0.4 0.5 0.6

FIGURE 20.24 Power spectrum of residuals, Sprengnether vertical seismometer free-decay, showing harmonic

content.

viscous modified Coulomb

Ref.--spectr. of pure sawtooth

time waveform

FIGURE 20.23 Harmonic differences between the residuals of the modified Coulomb damping model and the

classic viscous damping model. For reference purposes, a pure sawtooth is included in the figure.

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By looking at the FFT of residuals, rather than the experimental record itself, one finds evidence for a

combination of both mechanical and electronic noise. At lower frequencies, the noise (largely

mechanical) is approximately 1=f ; while at higher frequencies the noise (largely electronic) begins to be

more nearly “white” (frequency-independent) because of discretization errors of the resolution-limited

12-bit A to D converter.

In general, more spectral information can be gleaned from a consideration of the residuals than from

the experimental data alone, particularly as one looks for harmonic distortion of mechanical type.

Spectral “fingerprints” may prove ultimately useful in determining to what extent damping models of

engineering type need to be implemented in full nonlinear form as opposed to an “equivalent viscous”

form that is more convenient mathematically.

The importance of the harmonics observed in Figure 20.24 in determining system evolution is not

completely known. It was noted earlier that they are expected to influence the evolution of a

multibody system to steady state. Presently, it appears that they may serve to validate damping

models. From one model type to another, there can be significant differences in the spectral character

of the residuals, as shown in Figure 20.25. As compared with Figure 20.23, the fit with the modified

Coulomb (hysteretic case) model has been tweaked to reduce the fundamental somewhat, but the

odd harmonics remain significant. Observe that the spectrum of the residuals is almost the same

for this model and the simplified structural model (see de Silva, 2000, p. 354). This is true

even though the temporal variation of the friction force is dramatically different for the two, as seen

from the lower time traces that were used to obtain the residuals (which are too small to be seen in

the graphs).

From this author’s perspective, the simplified structural model is unrealistic, since the friction force,

given by f ¼ clxl sgnðx_Þ; vanishes for zero displacement (the absolute value of the displacement being

used to get the hysteretic form of frequency dependence). This is seen in Figure 20.26, which compares

hysteresis curves for several models. The modified Coulomb case shown is slightly different from

Equation 20.52 that was used to generate Figure 20.25; Figure 20.26 was generated with the Aprev shown

in Equation 20.10.

More studies of this type are obviously called for. The spectrum of residuals is a powerful means for the

study of damping physics, and it needs to be more widely employed.

FIGURE 20.25 Illustration of the spectral difference of the residuals for three different damping models. The

corresponding temporal records used to generate the spectra are also shown underneath each case.

Damping Theory 20-61

© 2005 by Taylor & Francis Group, LLC

20.18.3 Nonlinearity/Complexity and Future Technologies

Nonlinear damping models must improve if we are to overcome various technological barriers.

One barrier is in the area of civil engineering. One of the pioneers of finite element modeling (FEM) is

Prof. Emeritus Edward L. Wilson, of the University of California Berkeley. In Technical Note 19

(pertaining to “structural analysis programs”) — a document published by his company Computers

and Structures Inc — Dr. Wilson says the following:

Linear viscous damping is a property of the computer model and is not a property of a real

structure.

Expanding upon the statement, he notes:

the use of linear modal damping, as a percentage of critical damping, has been used to approximate

the nonlinear behavior of structures. The energy dissipation in real structures is far more

complicated and tends to be proportional to displacements rather than proportional to the

velocity. The use of approximate “equivalent viscous damping” has little theoretical or

experimental justification…the standard “state of the art” assumption of modal damping needs

to be re-examined and an alternative approach must be developed [in reference to Rayleigh

damping].

One of the hi-tech areas where modeling improvements are also sorely needed is that involving

miniaturized mechanical systems. For example, MEMS devices have already encountered some of

the “strange phenomena” of solid-state physics mentioned by Richard Feynman in his famous

1959 talk. To master or compensate for these phenomena, better understanding of the physics will

be necessary.

FIGURE 20.26 Comparison of hysteresis curves for some damping models.

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20.18.4 Microdynamics, Mesomechanics, and Mesodynamics

At least three different broad fields of research have focused on problems associated with the structural

defects that cause hysteresis. These are as follows.

20.18.4.1 Microdynamics

In the microdynamics world, the emphasis appears to have been primarily on “contact” friction. The 6th

Microdynamics Workshop held at the Jet Propulsion Laboratory in 1999 produced the following

statements (quoting Marie Levine’s Program Overview): (1) “We have demonstrated that microdynamics

exist. The next step is to qualify and quantify microdynamics through rigorous testing and analysis

techniques.” (2) Microdynamics is “defined as sub-micron nonlinear dynamics of materials, mechanisms

(latches, joints, etc.) and other interface discontinuities.”

In this workshop, it was noted that frequency-based computational methods cannot be used to model

quasi-static, transient, and nonstationary disturbances. One of the flight operations they have

recommended to minimize adverse effects of microdynamics is dithering.

20.18.4.2 Mesomechanics

Ostermeyer and Popov (1999) have the following to say about mesomechanics: “Real physical objects

inherently possess discrete internal structures. Great efforts are needed to formulate continuum models

of really granular bodies. The history of the last two centuries in a multitude of ways has been marked

by highly successful attempts at formulating and analyzing the continuum models of the discrete world.

In spite of great advances of continuum mechanics, a number of physical processes are amenable to

simulation within the framework of continuum approaches only to a very limited extent. Among these

are primarily all the processes whereby the medium continuity is impaired; i.e., those of nucleation and

accumulation of damages and cracks and failure of materials and constructions.”

Their paper speaks to one of the difficulties concerning granular materials that was mentioned earlier

in this chapter — that the potential energy cannot be defined in the common manner. They introduce a

temperature-dependent nonequilibrium interaction potential that is not constant in time due to the

relaxation processes occurring in the system.

20.18.4.3 Mesodynamics

The author of this chapter is singlehandedly responsible for the use of the term “mesodynamics” in

the context of mechanical oscillators. His research has been conducted independently of those

doing mesomechanics; he came only recently to know of the latter. Whereas mesomechanics seems

to have been largely concerned with failure, mesodynamics has been concerned with low-level

hysteresis. It is probably closely related to the aforementioned microdynamics, except that the latter

seems to have focused on surfaces (sliding friction), whereas mesodynamics is concerned with

internal friction.

A group of individuals using “mesodynamics” to describe some of their computational physics is part

of the Materials Science Division of Argonne National Laboratory. Their description of computational

theory includes: (i) atomic-level simulation (using molecular dynamics); (ii) mesoscale simulation, i.e.,

“mesodynamics” (using FEM); and (iii) macroscale (continuum) simulation (FEM). Like the author of

this chapter, they recognize that the mesoscale is not a continuum (meaning, for example, that the

foundation of viscoelasticity is, for many cases, on shaky ground). They employ “dynamical simulation

methods in which the microstructural elements (grain boundaries and grain junctions) are considered as

the fundamental entities whose dynamical behavior determines microstructural evolution in space and

time.”

At the Theoretical Division of Los Alamos National Laboratory, Brad Lee Holian has been modeling

mesodynamics via nonequilibrium molecular-dynamics (NEMD). In his paper, “Mesodynamics from

Atomistics: A New Route to Hall-Petch,” he notes that (i) the mesoscopic nonlinear elastic behavior must

agree with the atomistic in compression; and (ii) the mesoscale cold curve in tension represents surface,

rather than bulk cohesion, thereby decreasing inversely with grain size (Holian, 2003).

Damping Theory 20-63

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The complexity of mesodynamics, which this author has labeled “mesoanelastic complexity,” is

responsible for much of the aforementioned “strange phenomena.” To those familiar with the

Barkhausen effect and the PLC effect, they are less strange. It is thought that Richard Feynman, if he

were still alive, would identify with mesodynamics because of material in his three-volume series

(Feynman, 1970). For example, we have already noted his discussion of the Barkhausen effect, and he

included in its entirety a reprint of the Bragg – Nye paper on bubbles which show two-dimensional

defect structures such as dislocations, “grains,” and “recrystallization” boundaries after stirring (Bragg

et al., 1947).

Another famous individual, whose work related in an unexpected way to the material of this chapter,

was Enrico Fermi. In one of the first dynamics calculations carried out on a computer, he and colleagues

treated a chain of harmonic oscillators coupled together by a nonlinear term (Fermi, 1940). The

continuum limit of their model is the remarkable nonlinear partial differential equation known as the

Korteweg – deVries equation, whose solution is a soliton, used to advantage in optical fibers. Damping

of solitons, whether of the KdV type or the Sine Gordon (kink/antikink) type, is not to be described

by linear mathematics. Incidentally, the Sine Gordon soliton is used in modeling dislocations

(Nabarro, 1987). The earliest theory to describe dislocation damping using kink/anti-kink pairs was that

of Seeger (1956).

20.18.5 Example of the Importance of Mesoanelastic Complexity

As noted earlier in this chapter, once hysteretic

damping was finally recognized to be important to

the Cavendish experiment, better agreement with

theory and experiment was possible. Curiously,

Henry Cavendish may have been the first person to

encounter a “strange” phenomenon (which he did

not discuss) (Cavendish, 1798). In his first mass

swing to perturb the balance, which used a “fiber”

made of copper (silvered), there was an anomalously

small period of oscillation that was only

55 sec. The period reported for subsequent trials

was about 421 sec.

Whereas the Michell – Cavendish apparatus was

a torsion balance, the instrument of Figure 20.27

is a physical pendulum. The perturbing masses,

M, were hung from a bicycle wheel whose axle

was suspended from the ceiling. The long-period

pendulum was placed under a bell jar so that the

instrument would not be driven by air currents.

By rotating the wheel at constant angular

velocity, the driving force on the pendulum was harmonic. (In the figure, the position of each M

one-half period later are shown by the dashed circles.) Knowing the amount of damping, as

determined from large amplitude free-decay, it was easy to estimate the number of orbits of the bell

jar, at the resonance frequency of the pendulum, required to excite motion to a level above noise in

the sensor. Surprisingly, if it were initially at rest, no amount of drive by this means was able to get

the pendulum oscillating! The reason involves metastabilities of the defect structures. The potential

well is not harmonic (parabolic), but is rather modulated by “fine structure.” When located in a deep

metastability, the small gravitational force of the drive (in nanoNewtons) is not able to “unlatch” the

system. If the pendulum had been dithered (a practice used in engineering) this problem could have

been, at least partly, avoided. As it was, the pendulum rested on an isolation table of the type used in

optics experiments.

wheel axis for wheel rotation

bell

jar

M

M m

m

pendulum axis

FIGURE 20.27 Physical pendulum used in the late

1980s to try and measure the Newtonian gravitational

constant.

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More recently, a Hungarian research team has used a similar apparatus and postulated that the

anomalies of their experiment derive from gravity being other than prescribed by Newton (Sarkadi and

Badonyi, 2001). Although they claim that there is a “strong dependence of gravitational attraction on the

mass ratio of interacting bodies,” this author believes that additional experiments must be performed

before such a claim has merit. It may be that the anomalous behavior of their pendulum is instead the

result of mesoanelastic complexity, i.e., phenomena related to nonlinear damping.

The author’s most recent research on damping complexity is based on the premise that the

most important scale for the treatment of internal friction is the mesoscale, and not the atomic scale

(Peters 2004). Experiments to support this position center around a study of the SMA NiTinol.

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