20 Damping Theory Randall D. Peters

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Mercer University

20.1 Preface .................................................................................. 20-2

20.2 Introduction ........................................................................ 20-4

General Considerations of Damping † Specific

Considerations † The Pendulum as an Instrument for the

Study of Material Damping † “Plenty of Room at

the Bottom”

20.3 Background ......................................................................... 20-12

Terminology † General Technical Features † Active vs.

Passive Damping † Magnetorheological Damping †

Portevin – LeChatelier Effect † Noise †

Viscoelasticity † Memory Effects † Early History of

Viscoelasticity † Creep † Stretched Exponentials †

Fractional Calculus † Modified Coulomb Damping Model †

Relaxation

20.4 Hysteresis — More Details ................................................ 20-19

20.5 Damping Models ................................................................ 20-20

Viscous Damped Harmonic Oscillator † Definition

of Q † Damping “Redshift” † Driven System †

Damping Capacity † Coulomb Damping †

Thermoelastic Damping

20.6 Measurements of Damping ................................................ 20-23

Sensor Considerations † Common-Mode Rejection †

Example of Viscous Damping † Another Way to Measure

Damping

20.7 Hysteretic Damping ............................................................ 20-27

Equivalent Viscous (Linear) Model † Examples from

Experiment of Hysteretic Damping

20.8 Failure of the Common Theory ........................................ 20-29

20.9 Air Influence ....................................................................... 20-30

20.10 Noise and Damping ............................................................ 20-31

General Considerations † Example of Mechanical 1=f Noise †

Phase Noise

20.11 Transform Methods ............................................................ 20-34

General Considerations † Bit Reversal † Wavelet

Transform † Heisenberg’s Famous Principle

20.12 Hysteretic Damping ............................................................ 20-36

Physical Basis † Ruchhardt’s Experiment †

Physical Pendulum

20.13 Internal Friction .................................................................. 20-41

Measurement and Specification of Internal Friction †

Nonoscillatory Sample † Isochronism of Internal Friction

Damping

20.14 Mathematical Tricks — Linear

Damping Approximations ................................................. 20-43

Viscous Damping † Hysteretic Damping

20-1

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20.15 Internal Friction Physics .................................................... 20-44

Basic Concepts † Dislocations and Defects

20.16 Zener Model ........................................................................ 20-45

Assumptions † Frequency Dependence of Modulus and

Loss † Successes — Models of Viscoelasticity † Failure of

Viscoelasticity

20.17 Toward a Universal Model of Damping ........................... 20-48

Damping Capacity Quadratic in Frequency † Pendula and

Universal Damping † Modified Coulomb Model —

Background † Modified Coulomb Damping Model —

Equations of Motion † Model Output † Experimental

Examples † Damping and Harmonic Content

20.18 Nonlinearity ........................................................................ 20-58

General Considerations † Harmonic Content †

Nonlinearity/Complexity and Future Technologies †

Microdynamics, Mesomechanics, and Mesodynamics †

Example of the Importance of Mesoanelastic Complexity

20.19 Concluding Remark ............................................................ 20-65

Summary

This introductory chapter synthesizes the many, though largely disjointed attributes of friction as they relate to

damping. Among other means, events selected from the history of physics are used to show that damping models

have suffered from the inability of physicists to describe friction from first principles. To support fundamental

arguments on which the chapter is based, evidence is provided for a claim that important nonlinear properties have

been mostly missing from classical damping models. The chapter illustrates how the mechanisms of internal friction

responsible for hysteretic damping in solids can lead to serious errors of interpretation. Such is the case even though

hysteretic damping often masquerades as a linear phenomenon. One attempt to correct common model deficiencies

is the author’s work toward a “universal damping model,” that is described in Section 20.17. Section 20.17 is

developed in a “canonical” damping form. It shows the value of a direct, as opposed to an indirect, involvement of

energy in model development. To keep a better perspective on how the treatment of damping is likely to evolve in the

future, the last section of the chapter addresses some of the remarkable complexities of damping that are only

beginning to be discovered. The manner in which technology has played a role in some of these discoveries is

addressed in Chapter 21.

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