20.1 Preface

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The sheer volume of published material on the subject is a testament to the difficulty of selecting topics

for inclusion in a chapter on damping. Viscoelasticity alone is the basis for several voluminous

engineering handbooks. The present chapter is purposely different from similarly titled chapters of other

reference books. There is little repetition of well-known and proven classical methods, for which the

reader is referred to excellent other sources, such as de Silva (2000) and Chapter 19 of the present

handbook. They provide solution techniques for many of the routine problems of engineering. The goal

of the present chapter is to provide assistance with problems that are not routine, problems that are being

encountered more frequently as technology advances. It is thought that this goal is best served by

revisiting fundamental issues of the physics responsible for damping.

Once a multibody system has come to steady state, its damping treatment can be far less formidable

than its description during approach to steady state. When dealing with limit cycles involving

aeroelasticity and joints in helicopters, nonlinearity has a profound influence on the transient behavior.

Attempts to model it have been largely unsuccessful, forcing the empirical selection of elastomers to

reduce the vibration. (In the old days hydraulic dampers were used; Hodges 2003). At a much lower level

20-2 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

of sophistication, our understanding is quite limited on some common phenomena, such as the negative

damping character of sound generated by a violin or a clarinet. Historically, when technology “hit the

wall” because of too much theoretical handwaving, it became apparent that fundamental assumptions

needed to be examined. In physics, a complete alteration of conventional wisdom was sometimes

necessary, one of the best examples being the events that gave birth to quantum mechanics. Hopefully,

from the multitude of seemingly disparate (but assumed by the author to be connected) observations

which follow, the purpose for the architecture of this chapter can be partially realized. The enormous

complexity of damping in general makes it unrealistic to hope for complete success.

Physics played a prominent role in developing the classical foundations of damping, starting in the

19th century. Subsequently, engineers uncovered many features of the subject that physicists never even

thought about. In recent years, however, physics has been circumstantially forced to reconsider damping

fundamentals. With the advent of personal computing, and an increased awareness of the importance of

nonlinearity, new discoveries point to serious limitations of the classical foundation. The field of

mechanics was severely limited until it began tackling problems of nonlinearity (not of damping type),

and became concerned with previously ignored features giving unique system properties. Just as these

unique properties could only be solved by techniques more sophisticated than the equations of linear

type, there is mounting evidence that nonlinear damping may be the key to understanding some

bewildering engineering cases.

It is important to try to identify the major mechanisms responsible for energy dissipation. This is

easier said than done, since a host of different friction processes are usually at work. Moreover, the

description of friction from first principles remains a daunting task. Thus we are forced to work with

phenomenological models. There are also conflicts of nomenclature, with a given word meaning two

different things from one profession to another. Thus, much of this chapter will attempt to define

carefully terms while focusing on the physics, the treatment of which follows naturally along the lines of

historical developments.

Engineers tend to be interested in higher frequencies and higher amplitudes of vibration than are

scientists. A perfect damping model would be unconcerned with such differences of application; however,

such a model is far from being realized. Because small-amplitude, long-period (low and slow) oscillations

provide a valuable means for studying many processes of damping in general, much of this chapter

focuses in that direction.

From the multitude of choices available to writers on the subject of damping, this author has selected a

single (hopefully) unifying theme — nonlinear damping, especially as found in low and slow oscillations.

Because it is a field still in its infancy, many of the ideas that follow are more speculative than one would

prefer; however, they deserve discussion because of their perceived importance. To this author’s

knowledge, damping has not been previously treated in the manner of this chapter. Concerning the

earliest relevant paper (Peters, 2001a, 2001b), the following was indicated by oft-cited Prof. A.V. Granato:

“I don’t know of anyone thinking about internal friction along the lines you have mentioned.”

There are two important elements to the unifying theme of nonlinear dissipation: (i) the influence of

nonlinear damping on multibody systems in their approach to steady state, and (ii) the close connection

between damping and mechanical noise. When vibration decay is not exponential because of

nonlinearity, there are significant ramifications and they are only beginning to be appreciated.

The novel features of this chapter are possible because of dramatic improvements in both

sensing and data collection/analysis in the last decade. Demonstrating that a decay is not purely

exponential requires both (i) a good linear sensor and (ii) the means to study readily long-time

records when the damping is small (high Q). The first prerequisite has been met through the use of

this author’s patented fully differential capacitive sensor. The second has been realized with the

availability of good, inexpensive analog-to-digital (A/D) converters having user-friendly, yet powerful

Windows-based software. In addition to the “preview” software that comes with Dataq’s A/D

converter, a proven means for identifying nonexponential decay has been the analysis of records

imported to Microsoft Excel. Details of these novel methods will be provided in the various sections

that follow.

Damping Theory 20-3

© 2005 by Taylor & Francis Group, LLC

There are many examples in the engineering literature of nonlinear damping; even Coulomb damping

is nonlinear because the friction force involves the algebraic sign of the velocity rather than the velocity

itself, as in linear viscous damping. What has been realized for the first time in the course of writing this

chapter is the following. As will be shown in the subsequent material, a decay process is not usually a pure

exponential. Whatever the reason for a pure exponential, whether fundamentally linear (viscous) or

nonlinear (hysteretic present model), the quality factor Q for such a pure exponential decay is constant.

When there is a second mechanism, such as amplitude-dependent damping (even if it is the only

mechanism), the Q now becomes time dependent. This is significant to mode coupling for the following

reason. When a pair of modes couple because of elastic nonlinearity (a process that is impossible

assuming linear dynamics), the strength of the coupling is proportional to the product of the individual

amplitudes of the pair.

Consequently, variability in Q can influence the evolution to steady state. It is a factor in determining

which modes ultimately survive and/or dominate. Moreover, the distribution of the modes which remain

depends on initial conditions, including the intensity of excitation.

Long ago, musicians learned to deal with nonlinearity, due in part to properties of the ear that are

responsible for aural harmonics. A pair of purely harmonic signals can beat in the ear to produce a

“sound” that does not exist when sensed with a linear detector. For example, consider a strong and

undistorted 500 Hz signal sounded simultaneously with a pure 1003-Hz sound. The ear will hear a 3-Hz

beat due to the superposition of the ear’s aural second harmonic of the first with the fundamental of the

second. However, there’s more to this story. Conductors call for fortissimo and pianissimo sounds, not

only because of the ear’s nonlinearity, but also because of nonlinearities inherent to musical instruments.

For example, it is easy with a good microphone and LabView (see Appendix 15A) to demonstrate that the

timbre of stringed instruments is intensity dependent. Not only is the mix of harmonics, as displayed in a

fast Fourier transform (FFT) power spectrum, different according to volume, but their distribution also

changes with time.

Noise is not typically treated in an engineering discussion of damping; however, mechanical noise is an

important part of the technical material included in this chapter. Believing that there is a great deal of

connectivity among vibration, damping, and noise, evidence will be provided in support of a premise —

that the most important and least understood form of internal mechanical damping (material ¼

hysteretic ¼ “universal”) is closely allied with the most important and least understood form of noise

(1=f ¼ flicker ¼ pink). If this premise is true, then the foundations of damping physics need

reconstruction on several counts. Evidence in support of the premise will be provided through tidbits

of experimental discoveries from a host of independent investigations. It is hoped that the unusual and

lengthy introduction that follows will be beneficial in this regard. Historical elements serve to synthesize

the many parts and are offered without apology. Following the introduction, some practical and novel

equations of damping will eventually be provided. Even if readers find little identification with the

philosophies that birthed them, it is hoped they will at least carefully examine the equations that are

presented here in Section 20.17 for the first time.

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