Vibration damping of structural elements

Material Properties 241.7 Fundamentals of Vibrationof-Freedom Systems with Viscous Damping 452.3 Forced Vibrations of Single-Degree- of-Freedom Systems with Damping 48 v... vi CONTENTS2.

Vibration Damping

of Structural Elements

Vibration Dalllping

of Structural Elelllents

C.T.Sun Y.P.Lu

Prentice Hall PTR

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Material Properties 241.7 Fundamentals of Vibration

of-Freedom Systems with Viscous Damping 452.3 Forced Vibrations of Single-Degree-

of-Freedom Systems with Damping 48

v

vi CONTENTS

2.4 Transient Vibrations of

Single-Degree-of-Freedom Systems with Damping 542.5 Free Vibrations of Two-Degree-of-Freedom

Systems with Viscous Damping 662.6 Forced Vibrations of Two-Degree-

of-Freedom Systems with Damping 772.7 Transient Vibrations of Two-Degree-

of-Freedom Systems with Damping 832.8 Vibrations of Multi-Degree-of-Freedom

Systems with Damping 89

Chapter3 Dampingof Fiber-Reinforced

Composite Materials 1053.1 Introduction 1053.2 Damping of Aligned Discontinued

Fiber Composites 1063.3 Damping of Aligned Off-Axis

Discontinuous Fiber Composites 1203.4 Damping of Randomly Oriented Short-

Fiber Composites 1283.5 Damping of Laminated Composites 1373.6 The Influence of Fiber-Matrix Interface

Beam Structures 1614.3 Other Vibration Analyses of Three-

Layered Damped Beam Structures 1744.4 Vibration Characteristics of Constrained

Three-Layered Damped Beam Structures 186

Plate Structures 1995.3 Vibration Solutions Using A General

Purpose Finite Element Computer Program 2105.4 Vibration Characteristics of Constrained

Three-Layered Damped Plate Structures 226

Damped Ring Structures 2396.3 Vibrations of Continuously Constrained

Damped Ring Structures 2526.4 Vibration Solutions Using General Purpose

Finite Element Computer Program 2726.5 Continuously Versus Discontinuously

Constrained Damped Ring Structures 276

Chapter 7 Vibrations of Constrained Damped

7.1 Introduction 2817.2 Effectiveness of Damping Treatments

of Constrained Beam-Damped CylindricalShell Structures 282

7.3 Vibrations of Constrained Beam-Damped

Cylindrical Shell Structures 2847.4 Vibration Characteristics of Beam-Damped

Cylindrical Shell Structures 2937.5 Vibrations of Constrained Beam-Damped

Titanium Cylindrical Shells 2957.6 Vibrations of Constrained Damped Shell

Structures with Attached Mass Segments 3037.7 Vibrations of Constrained Damped Shell

Structures with Curved Elements 312

Chapter 8 Continuous and Discontinuous Constrained

Viscoelastic Material on Structural Elements

8.1 Introduction 3188.2 Finite Element Formulation of Structures

with Constrained Viscoelastic Damping Layer 3218.3 Derivation of Elastic Stiffness Matrix

of Plate Structures 3278.4 Derivation of Elastic Stiffness Matrix

of Beam Structures 3328.5 Derivation of Mass Matrix 3348.6 Evaluation of Loss Factors 3368.7 Constrained Viscoelastic Damping

Materials on Structural ElementsWith Prestress Effects 349

Er years, vibration damping as atechnology has been well received by various industries, however, veryfew books exist on this subject In 1985, Nashif, Jones, and Henderson

published an excellent book entitled Vibration Damping which provides

practical and detailed information on the research and developmentwork ranging from the fundamentals of vibration damping to design as-pects in many practical applications in this subject area Our intent isthat the contents of this book be considered as a continuation of andcomplementary to the above work rather than competitive in the techni-cal subject of vibration damping

This book is intended as a reference book for aerospace, cal, civil, and acoustical engineers It should also serve as a valuable ref-erence work for graduate students, professors, and researchers in thearea of Aerospace Engineering, Mechanical Engineering, and Engineer-ing Mechanics This book consists of eight chapters The first threechapters address the fundamentals of vibration damping, properties ofviscoelastic damping materials, vibrations of discrete damped systems,and damping of fiber-reinforced composite materials Chapters four toseven present vibrations of damped structures for beams, plates, rings,and shells In chapter eight, a finite element numerical method is pre-sented to solve vibration problems of beam and plate structures with apartialiy attached damping treatment on the surface of the structures.The effect of initial loading is also included

mechani-ix

Originally, a chapter to address the vibrations of damped cal shells with curved elements was planned A series of outstanding pa-pers on this specific subject were published by Dr Michael El-Raheband his colleague Mr Paul Wagner However, the research work wasperformed for their specific purposes and no numerical solutions on thevibrations of shell structures without incorporating the enclosed fluidmedium was available We regret that these materials are unable to beincluded in this book Instead, we have added paragraphs pertaining tothis subject topic at the end of Chapter 7 which deals with vibrations ofconstrained damped cylindrical shell structures.

cylindri-It should be pointed out that this book deals with vibration ing characteristics of structures or systems employing damping materi-als We use the properties of the viscoelastic materials in the vibrationanalysis, but the detailed analysis regarding their material behaviorswill not be our primary concern The book addresses the vibration damp-ing of structural elements, and is not a materials oriented book

damp-This book emphasizes analyses in the presentation of dampedstructural systems, their validations and verifications This is done be-cause the authors feel that analyses are the tools which not only enable

us to better understand the complicated physical phenomena, but alsocan help calculate the physical quantities which are useful in practicalapplications One might be critical of the fact that there are not enoughtables and figures which may be used directly and readily for designpurposes Our reply is that much of this information for damped beamstructures may be found in the book by Nashif, Jones, and Henderson.Additionally, because information pertinent to damped structures otherthan beams may not be available and because the vibration characteris-tics of damped structures depend strongly on the realistic (not assumed)properties of the damping materials employed as well as the geometricalparameters of the structures considered, we strongly believe that thepresentation of "design data" should be reduced to· a minimum unlessthe geometrical and particularly the damping material parameters of agiven damped structural system are specified Since it is technically dif-ficult to develop and to manufacture viscoelastic materials, the dampingmaterial properties can not be assumed for materials not commerciallyavailable

Though vibration damping technology is multi-disciplinary, the searchers and practitioners have, however, formed a rather close-knitcommunity In preparing this book, we would be remiss if we did not ac-knowledge the fact that Dr Lynn Rogers, currently a vibration dampingconsultant and formerly with the Flight Dynamics Laboratory of the Air

Force Wright Aeronautical Laboratories, has for years not only devotedhis efforts in promoting vibration technology tirelessly, but also has or-ganized meetings and workshops on a regular basis to provide a forumfor the exchange and dissimulation of the latest state-of-the-art technol-ogy His professional and enthusiastic efforts in advancing vibrationdamping technology certainly deserve our recognition

We are very happy to take advantage of this opportunity to knowledge the support and encouragement provided by the managersand individuals of our respective organizations in the Department ofAerospace Engineering, Mechanics and Engineering Science at theUniversity of Florida and at the Carderock Division, and the NavalSurface Warfare Center (formerly David Taylor Research Center).Particularly, we would like to thank Dr J.M Bai at the University ofFlorida, Dr Bruce Douglas, Director of Research, and Mr A J.Roscoe,

ac-at the Carderock Division for their assistance, support, and invaluablecomments Finally, the support and cooperation from the staff ofPrentice Hall Publication Company, especially our editor Mr MichaelHays in all phases of the production process are also acknowledged

C T Sun, Gainesville, Florida

Y P Lu, Annapolis, Maryland

Conventional structural designs are often unacceptable in copingwith modern problems of structural resonance caused by the complexnature of the dynamic environments and the requirements of design ob-jectives, including low noise, light weight, long life, and increased relia-bility One approach to solving the high vibration and noise levels ofstructural resonance problems is to incorporate a high-energy-dissipat-ing mechanism into the structure during fabrication.

Structural damping refers to the capacity of a structure or tural component to dissipate energy or to its capacity for removing fromthe structural vibration some of the energy associated with that vibra-tion This removed energy may be converted directly into heat andtransferred to connected structures or to the ambient media The mech-anism of structural damping is a complicated physical phenomenon Inaddition to controlling the amplitude of resonant vibrations, dampingalso has other effects, such as modifying wave attenuation and soundtransmission properties through structures, reducing structural fatigue,and increasing structural life

struc-In the past decades, it has been known that by utilizing layers ofviscoelastic shear damping material, the dissipation of energy within

1

2 C HAP T E R 1 Fundamentals of VibrationDamping

the vibrating members themselves can reduce noise and vibration els A combination of viscoelastic damping material and metal will pro-vide strength and rigidity but with a low response to vibration In thepast decades, there has been much research, development, training, andother activities accomplished nationally and internationally in the tech-nical areas related to vibration damping The science of damping tech-nology is a synthesis of this material and many engineering disciplines.This book will address all essential elements in the technical areas of vi-bration damping of discrete and structural systems

lev-1.2 SCOPE OF THE BOOK

This book consists of eight chapters covering the state-of-the-arttechnology in the area of vibration damping of discrete and continuousstructural systems

Chapter 1 provides an introduction to the fundamentals of tion damping, connecting all the essential and fundamental elementsrelevant to the technical subjects leading to vibration damping of struc-tural elements This chapter addresses such items as the classification

vibra-of damping, characterization of viscoelastic materials, effects of mental factors (temperature, frequency, cyclic strain amplitude, and sta-tic preload) on the viscoelastic material properties, fundamentals ofdamping material properties, and vibration control techniques such asdamping treatments

environ-Chapter 2 presents free vibrations, forced vibrations, and transientvibrations of discrete multi-degree-of-freedom systems with nonpropor-tional damping in either viscous or viscoelastic form Closed-form solu-tions as well as numerical resUlts for different cases are given The ef-fects of damping on vibrations of discrete systems are also presented.Vibration damping of composite materials and structures is dis-cussed in Chapter 3 Structures include aligned discontinued fiber com-posites, aligned off-axis discontinued fiber composites, randomly ori-ented short-fiber composites, and laminated composites The energyapproach and force balance model and the elastic-viscoelastic correspon-dence principle are used in the determination of loss factors Numericalexamples for various cases of composite materials are presented and dis-cussed

Chapters 4 through 7 present the vibrationi:Jof constrained dampedbeam, damped plate, damped ring, as well as damped cylindrical shellstructures Vibration theories, the associated analyses, analytical solu-

1.2 SCOPE OF THE BOOK 3

tions, experimental data, analytical/experimental results correlations,and vibration characteristics of each damped structural system aregiven and discussed

In Chapter 8, a finite element analysis employing offset techniquefor structures with constrained viscoelastic damping layer is formulatedand presented The system loss factors are evaluated using the finite el-ement numerical solutions The effects of using discontinuous (segment)constrained damping tape on structural vibrations are analyzed.Vibration solutions of damped beams and plates under prestress arealso determined using the offset finite element numerical method

For years, vibration damping as a technology has been well ceived by various industries, however, very few books exist on this sub-ject Nashif, Jones, and Henderson published an excellent book entitled

re-Vibration Damping (1985), which uses many practical applications to

provide useful and detailed information on research and developmentwork ranging from the fundamentals of vibration damping to design as-pects Our intent is that this book be considered as a continuation of andcomplementary to the above work rather than competitive in the techni-cal subject of vibration damping

It should be pointed out that this book deals with vibration ing characteristics of structures or systems employing damping materi-als We use the properties of viscoelastic materials in the vibrationanalysis, but a detailed analysis of their material behaviors will not beour primary concern The book addresses the vibration damping ofstructural elements, and is not a materials-oriented book

damp-This book emphasizes analyses in the presentation of dampedstructural systems, their validations and verifications This is done be-cause the authors feel that analyses are the tools that not only enable us

to better understand the complicated physical phenomena, but also tohelp calculate the physical quantities that are useful in practical appli-cations One might be critical of the fact that this book does not containenough tables and figures that may be used directly and readily for de-sign purposes Our reply is that much of this information for dampedbeam structures may be found in the book by Nashif and colleagues(1985) Additionally, because information pertinent to damped struc-tures other than beams may not be available, and because the vibrationcharacteristics of damped structures depend strongly on the realistic(not assumed) properties of the damping materials employed as well asthe geometrical parameters of the structures considered, we strongly be-lieve that -the presentation of "design data" should be reduced to a mini-mum unless the geometrical-and particularly the damping material-

4 C HAP T E R 1 Fundamentalsof VibrationDamping

parameters of a given damped structural system are specified Since it

is technically difficult to develop and manufacture viscoelastic als, the damping material properties cannot be assumed for materialsnot commercially available

materi-1.3 CLASSIFICATION OF DAMPING

Damping can be classified into many different types For example,comparing different materials A and B under exactly the same condi-tions (same boundary conditions, same geometrical dimensions), thesame magnitude of periodic forcing function with the same frequency ofexcitation, material A may oscillate longer (or shorter) with larger (orsmaller) amplitude than material B This is primarily due to the differ-ence in material properties The damping force due to internal molecu-lar friction in material A is less (or more) than the damping force due tointernal molecular friction in material B This kind of damping is called

material (or structural) damping.

Another type of damping encountered in a vibrating system is troduced through the surrounding medium in which the vibration takesplace For example, a vibratory structural system will oscillate much

in-longer in the air than in water This kind of damping is called viscous

damping. The viscous damping force depends on the property of the rounding medium and the velocity of motion In general, one can classifydamping into two basic categories: nonmaterial damping and materialdamping

sur-1.3.1 Nonmaterial Damping

1.3.1.1 Viscous damping The viscous damping force is afunction of the velocity of vibration, and is due to fluid resistance duringoscillation Generally, their mathematical description is quite compli-cated and not suitable for vibration analysis Fortunately, a simplifiedversion of a damping model was developed so that manageable math-ematical solutions can be achieved for engineering purposes This vis-cous damping model, designated by the dashpot, is very easy to use inanalyzing engineering vibration problems

In this simplified viscous model, the damping force Fd is assumed

to be linearly proportional to the velocity of a particle moving in fluidmedium Thus the viscous damping force is expressed by the equation

where c is a proportional constant called the coefficient of viscosity, and

dx / dt is the velocity of the particle relative to the fluid One of the major

objectives of this book is to analyze the effect of material damping on thevibration behaviors of structural elements by using a constrained vis-coelastic damping layer Since the linear viscous damping model is fa-miliar to most engineers and scientists and is mathematically manage-able for analytical purposes, vibrations of systems with viscous dampingwill be discussed Under some conditions, there exist similarities be-tween viscous damping and viscoelastic damping This is also the reasonwhy the vibrations of discrete systems with viscous damping are dis-cussed first in Chapter 2

For systems with viscous damping force Fd' the energy dissipatedper cycle of vibration Wdis equal to

In general, W d is a function of many factors, such as frequency, tude, and temperature of the surroundings

ampli-For a spring-mass system with viscous damping under a sinusoidal

forcing function F =Fo sinrot, the response of the steady-state motion isgiven by

which represents an ellipse in Fd-x plane as shown in Fig 1.1(a) The

area enclosed by the ellipse is equal to 1tcr0X2,which is the energy

dissi-pated per cycle of vibration W d. If, however, we consider the total force

by adding the lossless spring force kx to the viscous damping force Fd'

and plot the result in (F d +kx) and the x plane, the curve is a rotated el~lipse as shown in Fig 1.1(b) It can be shown that the area enclosed bythe rotated ellipse is the same as before, that is, 1tCr0X2.For linear vibra-tion, the curve is an ellipse, and for the nonlinear case, the curve is aclosed but not elliptical curve This curve, ellipse or nonellipse, is usu-

ally called a hysteresis loop The area enclosed by this loop is equal to

the energy dissipated per cycle of vibration This representation forms to the Voigt model, which consists of a dashpot in parallel with aspring

con-Damping can also be defined in terms of energy dissipation and thepeak potential energy as

The Coulomb damping force is assumed to be independent of the tive velocity of motion between the two surfaces The sign of the damp-ing force is always opposite to that of the velocity For a system withCoulomb damping, the decay in response amplitude per cycle of vibra-tion is a constant value 4Fe / k where k is the spring constant The mo-tion will stop when the spring force is insufficient to overcome the staticfriction force.

rela-The concept of Coulomb damping is usually applied in structuraljoints Damping force is introduced from slipping at the joint, and thisgives rise to energy dissipation at the joint Coulomb damping can also

be applied for the two-layer beam The response will become nonlinear

as soon as the midplane shear stress exceeds JlN. The nonlinear problem

is very complex and the analysis is quite difficult and tedious More tailed discussions and applications of Coulomb damping can be found inPlunkett (1981), Schlesinger (1979), and Earles (1966)

de-1.3.1.3 Other nonmaterial dampings. Many nonmaterialdamping mechanisms have been very useful in engineering applications,

one of which is acoustic radiation damping The damping force is the

force acting on the mass due to the acoustic medium, which is

deter-8 C HAP T E R 1 Fundamentalsof VibrationDamping

mined from the solution of motion of the acoustic medium Once it is termined from the wave equation, the equation of motion for the re-sponse can be analyzed

de-The other well-known nonmaterial damping device is the linear air

pump. The linear air pump's nearly airtight volume is placed adjacent tothe vibrating system The entrapped air is alternately compressed andrarefied by the motion of the structure This motion will produce a pres-sure increment that is proportional to the structure motion In order tohave energy dissipation, many small holes (or leaks) are built into thepanel In the presence of leaks, the pressure increment will be changed.More detailed discussion about the applications of linear air pumpingcan be found in Ungar (1964)

1.3.2 Material Damping

It is generally true that all engineering materials dissipate energyduring cyclic deformation Some materials, such as rubber, plastics, andelastomers, dissipate much more energy per cycle of deformation thansteel and aluminum For conventional structural materials, the energydissipation per unit-volume per cycle is very small compared to certainhigh damping alloys, polymer matrix composites, and rubberlike materi-als (viscoelastic materials)

The hysteresis loops that may be used to quantify damping ties of materials are very useful in understanding damping For conven-tional structural material the hysteresis loop is very thin, unless thematerial is stressed into its plastic range Within the elastic range theloop is not easily observed For high damping alloys and viscoelastic ma-terials, however, very large hysteresis loops are easily seen For nonlin-ear damping, the hysteresis loop is a closed but not elliptical curve

proper-1.3.2.1 High damping alloys High damping alloys are notusually the best structural materials This is due to the fact that mosthigh damping alloys have inferior stiffness, strength, corrosion resis-tance, and thermal properties, and are only used in structures for somespecial situations Most high damping alloys are also highly nonlinear,and the experimental results of damping (loss factors) only exist for cer-tain materials In general, the loss factors of high damping alloys de-pend on a number of parameters, such as mode of vibration of interest,strain amplitudes, and temperature One of the well-known high damp-ing alloys is called Sonoston and is available commercially The loss fac-tors '111' 112' and 113correspond to modes 1, 2, and 3 of Sonoston beams

1.3 9

are presented in Fig 1.2 as functions of temperature and strain tude (Bowie, Nachman, and Hammer, 1971; Weissman and Babington,1966) From this figure, it is observed that damping is reduced as thetemperature increases beyond about 185°F (85°C)

ampli-1.3.2.2 Composite materials Fiber-reinforced composite terials are used extensively in the aerospace, automotive, and sports in-dustries Chapter 3 will present in detail the evaluation of the damping

ma-10 C HAP T E R 1 Fundamentalsof VibrationDamping

properties of continuous and discontinuous randomly oriented shortfiber composites, and laminated composites in terms of the constituentmaterial properties, fiber volume fraction, fiber orientations, and thestacking sequence of the composites The effect of interphase will also bediscussed Usually damping of composite materials is higher than that

of conventional structural metallic materials such as steel and minum Damping properties, like stiffness properties of composite mate-rials, are highly directional This means that the loss factor along thefiber direction is in general different from the loss factor along the direc-tion normal to the fiber We also find out that damping in randomly ori-ented short fiber composites is higher than unidirectional fiber compos-ites Therefore, randomly oriented short fiber composites can be used as

alu-a dalu-amping lalu-ayer It is alu-also importalu-ant to know that fibers with a highdamping property, such as Kevlar fibers, will enhance damping as well

as stiffness of the structural elements In general, the packing geometry

of fibers has little iirlluE!Dceon damping property Finally, the dampingproperty of composite materials depends slightly on the temperature,strain rate, and amplitude of vibration, and it is not as highly dependent

as viscoelastic materials

1.3.2.3 Viscoelastic materials It has been pointed out beforethat one of the major objectives of this book is to apply viscoelastic mate-rials to improve damping characteristics of structural elements Theproperties of viscoelastic materials that will affect their performancewill be presented in detail in section 1.4

1.4 CHARACTERIZATION OF VISCOELASTIC MATERIALS

A viscoelastic material sometimes is called material with memory.This implies that a viscoelastic material's behavior depends not only onthe current loading conditions, but also on the loading history There are

a number of different methods to characterize a viscoelastic material'sconstitutive relations Within the scope of linear theory, the followingmethods will be presented

1.4.1 Convolution Integral

12 C HAP T E R 1 Fundamentals of VibrationDamping

1.4.2 Force-Extension Equations of Three Basic Models

As its name implies, viscoelasticity is a generalization of elasticityand viscosity The elastic element can be modeled by a linear spring,and the viscous element by a dashpot Therefore, the viscoelastic model

is a combination of a linear spring and a dashpot In the following, wediscuss briefly three basic models

(a) The Maxwell model If the two elements are combined in

series, it is known as a Maxwell model and is shown in Fig 1.3 where k

is the spring constant and c the viscosity When a force F is applied to

this model, the elongation is equal to the sum of the extensions in theelastic and viscous elements, that is,

provided we identify a as F and E. as the extension u The constants a and b can be related with the spring and dashpot constants by simple

comparison A more detailed discussion of the constitutive equations forviscoelastic material is documented by Rogers (1983)

The Maxwell, Voigt, and standard linear models are the basic

mod-els to describe the material behavior of most viscoelastic materials In

addition to the three basic models, there are some other models: (1) Thegeneralized Maxwell model in which a number of Maxwell models arecombined in parallel with a spring and a dashpot as shown in Fig 1.8,and (2) the generalized Voigt model, in which a number of Voigt modelsare combined in series with a spring and a dashpot as shown in Fig 1.9

1.5 EFFECTS OF ENVIRONMENTAL FACTORS 21

1.5.1 Temperature

Temperature is considered as the most important factor affectingthe properties of viscoelastic materials Temperature can be divided into{our different regions: the glassy region, the transition region, the rub-

berlike region, and the flow region The glassy region usually occurs at

room temperature In this region, the storage modulus E' decreases

slightly as the temperature increases and the loss factor increasessharply as the temperature increases This trend will continue into thenext region, the transition region, until the transition temperature Tgisreached At transition temperature Tg, the loss factor T\ reaches its max-

imum value while the slope of the storage modulus E' is almost a stant In the third region, the rubberlike region, both E' and T\ nearly re-

con-main constants This trend continues into the flow region in which the

values of E' decrease and T\ increases In engineering applications

usu-ally most structures will not go beyond the transition temperature Tg A

typical plot of the storage modulus E' and loss factor T\ of rubberlike

ma-terials is presented in Fig 1.11 The storage modulus E' could be as high

as 105GPa in the glassy region and as low as 10-2GPa in the rubber gion (Nashif et aI., 1985) The range of the transition region may varyfrom 20°C for an unfilled viscoelastic material to 200°C for a vitreousenamel The loss factor T\ is low in glassy region and reaches high value

re-at the transition temperre-ature In the rubberlike region, T\ usually variesbetween 0.1 to 0.3

1.5.2 Frequency

The effects of frequency on the storage modulus E' and the loss

fac-tor T\ again can be divided into four different regions: the glassy region,the transition region, the rubberlike region, and the flow region In gen-

eral, E' increases as the frequency increases However, the rate of

in-crease is small in the glassy and rubberlike regions, and the greatestrate of increase is in the transition region The loss factor T\ decreases asfrequency increases in the glassy region, reaches its maximum in thetransition region at the transition temperature, and increases in the

rubberlike region In the flow region, E' decreases and T\ increases as the temperature increases A plot of E' and T\ as a function of frequency

for the three regions is given in Fig 1.12 It should be emphasized thatthe frequency in Fig 1.12 varies from 10 Hz to 104 Hz Therefore, it

takes several decades of frequency to reflect the same change of E' and T\

.as a few degrees of temperature This illustrates that the effects of

fre-Fig 1.11 Variation of the storage modulus and loss factor with temperature.

(From Vibration Damping, Nashif, Jones, and Henderson, copyright ©

1985 John Wiley & Sons, Inc Reprinted by permission of John Wiley

&Sons, Inc.)

quency on the storage modulus and the loss factor are not as strong asthe effects of temperature

1.5.3 Cyclic Strain Amplitude

According to Warnaka and Miller (1968), the effects of cyclic strainamplitude on the damping behavior are very complex First, high dy-namic strain amplitude will cause more energy dissipation in the mater-ial, and high energy dissipation will increase the temperature of the ma-terial Therefore, the two effects on damping behavior are coupled In

the glassy region, the variation of E' and TI with strain amplitude is

small; E' decreases sharply and TI reaches its peak value in the tion region It should be mentioned that the temperature effects on the

transi-Fig1.12 Frequency dependency of modulus and loss factor for various types of

viscoelastic materials (From Vibration Damping, Nashif, Jones, and Henderson, copyright © 1985 John Wiley & Sons, Inc Reprinted by permission of John Wiley & Sons, Inc.)

modulus and loss factor become small in the rubberlike region Thus the

dynamic strain amplitude effects on E' and 11in the rubberlike region

are more severe than in the other regions A plot of E' and 11 as a

func-tion of strain amplitude is shown in Fig 1.13

1.5.4 Static Preload

The effects of tensile static preload will increase the modulus anddecrease the loss factor A more detailed discussion of the effects of sta-tic preload on beams and plates with either a partially constraineddamping layer or fully constrained damping layer using finite elementapproach will be presented in Chapter 8

Fig 1.13 Variation of the storage modulus and loss factor with strain ampli-I

tude (From Vibration Damping, Nashif, Jones, and Henderson, right © 1985 John Wiley & Sons, Inc Reprinted by permission of John Wiley &Sons, Inc.)

copy-1.6 FUNDAMENTALS OF DAMPING MATERIAL PROPERTIES

gener-of damping testing and the basis for extending damping technology ceptually to distributed systems Common damping quantities are

con-• Loss factors of a structure made of a particular material, 11 (orl1n)'

are defined as a fraction of the system's vibrational energy that isdissipated per radian of the vibratory motion at resonances

• Specific damping capacity of the vibration system, '1', representsthe ratio between the vibrational energy dissipated per cycle andthe elastic-energy storage capacity of the system

• Decay rate of free vibrations, D, expressed in decibels per second

(dB/see), represents the rate of reduction of the vibration level ing exponential decay

dur-26 C HAP T E R 1 Fundamentals of Vibration Damping

Many of these parameters will be dealt with in Chapter 2 when the brations of discrete systems are discussed Other parameters, listedbelow, are also often used for the measures of the effectiveness of damp-ing of vibratory systems

vi-• T 60 denotes the reverberation time (in seconds), defined as thetime within which the vibration level of a system vibrating freely

at frequency O)n (Hz) decreases by 60 dB (i.e., the amplitude creases to 1/1000 of its initial value)

de-• Spatial decay, expressed in dB per wavelength, represents the duction in the steady-state vibration level with distance that occursalong a long beam vibrating in flexure

re-• Hysteresis loop, a plot of the amplitude of instantaneous force sus instantaneous displacement (or stress versus strain) duringsteady-state forced vibration, is equal to the energy dissipated inthe vibratory s):'stem

ver-• Nyquist diagram, plotting the real and imaginary components 01responses, is another measure of damping

It is noted that the loss factor and the damping capacity are fined directly in terms of the cyclic energy dissipation Also, none of themeasures of damping discussed above depend on how the energy is dis-sipated, and these measures make no reference to any damping mecha-nism within a cycle Most of the commonly used measures of dampingare defined on the basis of viscous damping For linear viscouslydamped systems, all of the measures of damping are independent of am-plitude, provided that the damping is small For vibration systems withother energy dissipation mechanisms, the damping properties are oftencharacterized in terms of an equivalent viscous damping coefficient

de-1.6.2 Material Property Measurements

To control vibration problems effectively in structures, it is tial to have accurate damping material properties for specific applica-tions There are various techniques for determining the dynamic proper-ties of damping materials The most notable methods are the resonantbeam test, the dynamic mechanical analyzer (DMA), and the resonanttest Other test procedures, such as the Rheovibron and the progressivewave technique, are also used The resonant beam technique for deter-mining damping properties of materials is the procedure endorsed bythe ASTM (American Society for Testing and Materials) and forms thebasis of ASTM Standard E756-80 The test setup, specimen selection cri-

essen-1.6 FUNDAMENTALS OF DAMPING MATERIAL PROPERTIES 27

teria, test procedures, test equipment, possible error sources, and vantages and disadvantages of each of these test methods are discussed

ad-in detail by Nashifand colleagues (1985) and Drake (1988)

1.6.3 Data Reductions and Presentations

Considering that Poisson's ratio is approximately constant for mostconventional materials, the property of metallic materials can be repre-sented by only one variable, either Young's modulus or the shear modu-lus However, for many rubberlike or linearly viscoelastic materials, thedynamic material properties, primarily Young's modulus (or shear mod-ulus) and the loss factor, are functions of frequency, temperature, strainamplitude, and prestress or prestrain Temperature and frequency aregenerally by far the most dominating variables

In the resonant beam test, for instance, both the drive signal andthe response signal at a given point of the specimen are measured Withthis information, the resonance frequencies and the modal damping val-ues are calculated These values are then used to calculate the Young'smodulus (or the shear modulus) and the loss factor of the damping ma-terial In all test methods, intermediate response information is mea-sured and then the desired ultimate material properties are calculated.With various test techniques available, data reduction equations or for-mulae vary according to the test method employed and the test speci-men selected

It is important to note that the properties of damping materials arenot easily measured It is not uncommon that data analyzing the samematerial are scattered, indicating the deviation of test-to-test variationand material batch-to-batch variation Fig 1.14 shows a comparison ofdata curves (Young's modulus and loss factor) from the beam test, DMAtest, and the Rheovibron test for the 3M ISD-112 damping material(Drake, 1988)

The final result of damping material analysis is a temperaturenomogram, which expands the limited number of test res.ults to a graphfrom which the designer can obtain the damping material properties atany given combination of temperature and frequency The temperaturenomogram was developed by Jones (1978) and is considered a major ad-vance in the graphical presentation of complex modulus data The read-ings of storage (real) modulus, loss (imaginary) modulus, and loss factor

as functions of temperature and frequency from graphical presentationsare greatiy facilitated by its use In the nomogram, a vertical scale foractual frequency (Hz) is on the right and shear modulus (N /m2 or psi)

Fig 1.14 Comparisonof data curves fromthree test methods for

ISD-112mate-rials (From M L Drake, "Material Property Measurements, DataReduction, Data Presentation," Section 3.1 of the University ofDayton Research Institute Vibration Damping Short Course Note,1988.Reprinted by permissionof University of Dayton.)

as well as loss factor on the left The diagonal lines represent particulartemperatures

For illustration purposes, Fig 1.15 is a temperature nomogramwith some grid lines removed This nomogram can be read easily Theprocedure for reading this nomogram is as follows: Select a combination

of temperature and frequency, for example, 200 Hz and 59°F (15°C).Find the point for 200 Hz on the righthand axis Follow the point hori-zontally to the line for 59°F temperature At this intersection, draw avertical line Then read the modulus and loss factor values off the appro-priate graph, at the point of intersection with the vertical line In thisexample, modulus G (200 Hz, 59°F) is 150 psi and the loss factor is 0.89.This nomogram also shows a second example for the combination 40 Hzand 109°F (42.8°C) In this example, modulus Gis 20 psi and the lossfactor is 0.69

Fig 1.15 Typical temperature nomogram of polymeric material test data.

(From M L Drake, "Material Property Measurements, DataReduction, Data Presentation," Section 3.1 of University of DaytonResearch Institute Vibration Damping Short Course Note, 1988.Reprinted by permissionofUniversity of Dayton.)

1.7 FUNDAMENTALS OF VIBRATION CONTROL TECHNIQUES

The objective of enhancing damping in structural elements is tocontrol the response of the elements so that catastrophic failure due toexcessive deformation can be avoided This is particularly necessary atthe condition of resonance Without damping, the deflection of a struc-ture will increase to infinity at resonance Modifying the stiffness of thestructure is usually not a feasible solution, since changing the stiffnesschanges the natural frequency and sooner or later the condition of reso-nance will be reached Therefore, one of the effective ways to reduce theresponse of structural elements is to improve damping characteristics.However, it is well known that for most structural metallic materialsthe loss factor ranges from 0.0002 to 0.002 (see Table 1.1). For unidirec-tional composite materials, the loss factor is much higher than 0.002 butstill too low to meet design requirements Randomly oriented short fibercomposites may have adequate loss factors to satisfy design require-

ments, but their stiffness is too low, and at best may be used to serve as

a damping layer The loss factors for various materials are given inTable 1.1

Dynamic responses of structural elements depend on a number offactors If the range of the frequency of excitation is known, then we canadjust the values of the mass and the stiffness of the system This willchange the natural frequency so that the condition of resonance can beavoided However, if the condition of resonance cannot be avoided withthe given values of mass and stiffness, several different methods can beapplied to control the dynamic responses These vibration control tech-niques can be classified into two categories: passive and active The ac-tive vibration control technique is a relatively new research area, andwill not be addressed here Passive vibration control techniques includevibration absorber technique, vibration damper, discrete damping de-vices such as tuned dampers, and surface damping treatments

Passive vibration control is accomplished by applying some devicesexternally on the structural elements either to increase damping of thesystem or to alter the number of degrees of freedom of the system such