Harris C.M., Piersol A.G. Harris Shock and vibration handbook

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MATERIAL DAMPING AND SLIP DAMPING 36.27

force is applied, conditions are represented by point O″ which corresponds to point O

in Fig. 36.12. The initial elastic phase during which there is no slip is represented by

O″P″ (note that the normal pressure σ may change during this phase). The phase dur-

ing which slip occurs only over part of the interface is represented by the curved line

P″B″; it corresponds to PB in Fig. 36.12.After slip has progressed over the entire inter-

face, the normal force vs. relative-displacement relation is linear. This phase is repre-

sented by the curve B″C″ in Fig. 36.13 and by BC in Fig. 36.12.When the exciting force

has reached its maximum value, a second nonslip phase C″D″ ensues. This is followed

by slip along the curve D″E″F″ until the exciting force reaches its maximum negative

value. As the exciting force completes its period, there is a nonslip phase F″G″ fol-

lowed by slip along G″C″. The lengths C″D″ and F″G″ are equal and the curves G″C″

and D″F″ are congruent (F″ corresponds to C″ and D″ corresponds to G″).

If the point in question is at an element of area dx dz of the xz interface, the energy

dissipated in slip is proportional to the area enclosed by the slip curve. Because of the

congruence of the curved portions of the diagram and the parallelism of the linear

portions, this area can be expressed in terms of the total slip and the pressures at two

instants during the loading cycle. Integrating over the entire interface,

=−µ[σ(E″ ) +σ(Q″ )] ∆ s

tot

dx dy (36.24)

In this expression, the parameters σ(E″ ) and σ(Q″ ) and the total slip ∆ s

tot

are func-

tions of x and z. They are the normal stresses at points E″ and Q″ in Fig. 36.13,

located midway between the vertical lines G″F″ and C″D″. Since the pressures σ are

always compressive (negative) and the total slip is always taken as a positive quan-

tity, the negative sign is required to ensure a positive energy dissipation. Equation

(36.24) is of little engineering value in itself because the stresses are functions of F

as well as of x and z. In many of the problems which are of design interest, however,

the shear on the interface is produced primarily by the exciting force and not by the

initial clamping pressure. Conversely, the clamping pressure is not greatly affected

by the addition of the time-varying exciting force. Under these circumstances, the

slip curve of Fig. 36.13, like the force-displacement curve of Fig. 36.12, is symmetric

about the point O″. Points P″ and Q″ then coincide, and the mean ordinate of the

slip curve is that corresponding to point O″. Then Eq. (36.24) reduces to

=−4µ σ(O″ ) ∆s

max

dx dz (36.25)

where σ(O″ ) is the clamping stress corresponding to zero exciting force. It may be

determined by any of the well-known methods of stress analysis. In most cases,

σ(O″ ) can be determined without any reference to the existence of an interface. The

term ∆ s

max

represents the arc length of the maximum relative displacement, the so-

called “scratch path.” It is a function of the maximum value of F

as well as of posi-

tion on the interface. It may be inferred from Eq. (36.25) that energy dissipation due

to interface shear is small both at very low clamping pressures and at very high ones.

In the former case, σ(O″ ) = 0; in the latter case, ∆ s

max

= 0. It follows that, for any dis-

tribution of clamping pressure, there is an optimum intensity of clamping force at

which the energy dissipation due to interface shear is a maximum. The maintenance

of this optimum pressure is essential to the utilization of this form of damping. From

the shape of the force-displacement curve OPBC shown in Fig. 36.12, it is evident

that systems in which interface shear damping plays a significant role behave like

softening springs.This means that instability and jump phenomena may occur at fre-

quencies below the nominal resonant frequency.

In the case of plane stress, the thickness of the material is t and Eq. (36.25)

becomes

8434_Harris_36_b.qxd 09/20/2001 12:28 PM Page 36.27

=−4µt σ(O″ ) ∆s

max

dx (36.26)

The slip can be related to stress through Hooke’s law:

∆s = E

−1

(∆σ

) dx (36.27)

This indicates that any discontinuity in displacement is associated with a discontinu-

ity in the component of stress parallel to the interface. These displacement disconti-

nuities due to slip are members of a class of generalized dislocations whose existence

has been demonstrated theoretically.

If Eq. (36.27) is substituted in Eq. (36.26), the

energy dissipation can be expressed in terms of stress alone:

=−4µE

−1

t 

σ(O″ )





(∆σ

)

max

dx′



dx (36.28)

The computation of energy dissipation per cycle D

is the first step in the predic-

tion of the dynamic amplification factor to be expected in service. For interface

shear damping, an elementary theory permits the dynamic amplification factor to be

estimated even though the system behavior is nonlinear. The technique employs an

averaging method. Denoting the displacement corresponding to the exciting force

by the symbol v,

v = v

cos ωt and F

= F

cos (ωt +ϕ) (36.29)

where v

is the peak dynamic displacement, F

is the peak exciting force, and ϕ is the

loss angle. One relationship between these quantities is obtained by making the

average value of the virtual work vanish during each half-cycle of the steady-state

forced vibration:



π/ω

[mv + kv − F

] cos ωt dt = 0 (36.30)

In this integration, the stiffness k changes as slip progresses across the interface. If

the hysteresis loop of Fig. 36.12 is replaced by a parallelogram, only two phases, elas-

tic and fully slipped, need be considered. Denoting the stiffness (i.e., the ratio of

exciting force to displacement) in the unslipped condition by the symbol k

and the

reduced stiffness in the fully slipped condition by the symbol k

, the phase angle ϕ

and the dynamic amplification factor A may be related by Eq. (36.30) to the dura-

tion of the elastic phase t′:



1 −



(ωt′+sin ωt′ ) =π





(36.31)

where A is the conventional dynamic amplification factor, i.e., A = v

. The dura-

tion of the elastic phase is given by the first of Eqs. (36.29) with v = v

− 2v

, where v

is the displacement at which slip first occurs. Then eliminating t′ from Eq. (36.31):



1 −





1 ++cos

−1



1 − 2



− (36.32)

Equation (36.32) gives the relation between phase lag ϕ and amplification factor A.

A second relationship between these quantities is found from the consideration that

the energy dissipated during each half cycle of forced motion must be D

/2:



π/ω

dt =

⁄2D

or sin ϕ= (36.33)



πF



mω



cos ϕ



cos ϕ



mω



36.28 CHAPTER THIRTY-SIX

8434_Harris_36_b.qxd 09/20/2001 12:28 PM Page 36.28

Equations (36.32) and (36.33) serve to determine the dynamic amplification factor

A, after D

has been estimated. Conversely, they serve to estimate the amount of

energy which must be dissipated per cycle to produce a given reduction in the ampli-

fication factor by interface shear. A detailed analysis of the response to a parallelo-

gram hysteresis loop has been made.

Hysteresis loops other than parallelograms

also have been studied.

At resonance, ϕ=90° and

A = A

= (36.34)

In general, the energy dissipation does not increase as rapidly as the square of the

peak exciting force; consequently, the resonance amplification factor decreases as

the exciting force increases. As a result, structures in which interface shear predom-

inates tend to be self-limiting in their response to an external excitation.

The foregoing discussion is based on the premise that changes in the exciting

force do not materially affect the size of the contact area.There is an important class

of problems for which this assumption is not valid, namely, those in which even the

smallest exciting force produces some slip. An example of this type of joint is the

press-fit bushing on a cylindrical shaft. If the ends of the shaft are subjected to a

cyclic torque, part of this torque is transmitted to the bushing. Each part of the com-

pound torque tube carries a moment proportional to its stiffness. Transmission of

torque from the shaft to the bushing is effected by slip over the interface. The length

of the slipped region grows in proportion to the applied torque. There is no initial

elastic region such as OP or O″P″ in Figs. 36.12 and 36.13. If the peak value of the

exciting torque is not too large, the fully slipped region BC or B′C′ in Fig. 36.12 never

occurs. In these cases, Eqs. (36.31) to (36.34) are not applicable because there are no

assignable constant values of k

and k

. A variety of simple cases of this type which

occur in design practice have been analyzed. They include the cylindrical shaft and

bushing in tension and torsion, and the flexure of a beam with cover plate.

Another important case in which the smallest exciting force may produce slip

arises in the contact of rounded solids. If these are pressed together by normal forces

along the line joining their centers, a small contact region is formed. Subsequent

application of a cyclic tangential force produces slip over a portion of the contact

region even if the peak tangential force is not great enough to effect gross slip or

sliding.This situation has been analyzed and verified experimentally.

3, 36

REFERENCES

1. Alfrey, T., Jr.: “Mechanical Behavior of High Polymers,” Interscience Publishers, Inc., New

York, 1948.

2. Lazan, B. J.: “Damping of Materials and Members in Structural Mechanics,” Pergamon

Press, New York, 1968.

3. Johnson, K. L.: “Contact Mechanics,” Cambridge University Press, 1985.

4. Lazan, B. J.: “Fatigue,” chap. II, American Society for Metals, 1954.

5. Cochardt, A. W.: J.Appl. Mechanics, 21:257 (1954).

6. Lazan, B. J.: Trans. ASME, 65:87 (1943); Pisarenko, G. S.: “Vibrations of Mechanical Sys-

tems Taking into Account Incompletely Elastic Materials,” 2d ed., Kiev, 1970 (in Russian).

7. Von Heydekampf, G. S.: Proc. ASTM, 31 (Pt.II):157 (1931); Jones, D. I. G., and D. K. Rao:

ASME Des. Div. Pub. DE, 5:143 (1987).



πF

MATERIAL DAMPING AND SLIP DAMPING 36.29

8434_Harris_36_b.qxd 09/20/2001 12:28 PM Page 36.29

8. Lazan, B. J.: Trans. ASM, 12:499 (1950).

9. Maxwell, B.: ASTM Bull., 215:76 (1956).

10. Nowick,A. S.: “Progress in Metal Physics,” vol. 4, chap. I, p. 29, Interscience Publishers, Inc.,

New York, 1953; Tschan, T., et al.: Proc. Eurosensors V, Sensors and Actuators, A: Physical,

32, n.1-3:375 (1992); Kalachnikov, E. V., and P. N. Rostovstev: Instr. Exp. Tech, 32:1241

(1990).

11. Podnieks, E., and B. J. Lazan: Wright Air Development Center Tech. Rep. 55-284, 1955.

12. Lazan, B. J.: J.Appl. Mechanics, 20:201 (1953).

13. Lakes, R. S.: “Viscoelastic Solids,” CRC Press, New York, 1999.

14. Zener, C.: “Elasticity and Anelasticity,” University of Chicago Press, Chicago, Ill., 1948.

15. Wert, C.: “The Metallurgical Use of Anelasticity” in “Modern Research Techniques in

Physical Metallurgy,” American Society for Metals, Cleveland, Ohio, 1953.

16. Jones, D. I. G.: J. Sound and Vib., 140:85 (1990).

17. Fay, J. J., et al.: Proc. ACS Div, Polymetric Mat’ls. Sci. and Eng’g., 60:649 (1989).

18. Fujimoto, J., et al.: J. Reinf. Plastics and Composites, 12:738 (1993).

19. Chang, M. C. O., et al.: Proc. ACS Div. Polymetric Materials Sci. and Engg., 55:350 (1986).

20. Weibo, H., and Z. Fengchan: J.Appl. Polymer Sci., 50:277 (1993).

21. Cochardt, A.: Scientific Paper 8-0161-P7, Westinghouse Research Labs., W. Pittsburgh, Pa.,

1956.

22. Person, N., and B. J. Lazan: Proc. ASTM, 56:1399 (1956).

23. Torvik, P.: Appendix 72fg, Status Rep. 58-4 by B. J. Lazan, University of Minnesota, Wright

Air Development Center, Dayton, Ohio, Contract AF-33(616)-2802, December 1958.

24. Whittier, J. S., and B. J. Lazan: Appendix B, Prog. Rep. 57-6, Wright Air Development Cen-

ter, Dayton, Ohio, Contract AF-33(616)-2803, December 1957.

25. Hinai, M., et al.: Trans. Japan. Inst. Met., 28:154 (1987).

26. De Batist, R.: ASTM Spec. Tech. Pub. 1169, 45-59, American Society for Testing and Mate-

rials, Philadelphia, 1992.

27. Zhang, J., et al.: Acta Met. et Mat., 42:395 (1994).

28. Goodman, L. E., and J. H. Klumpp: J.Appl. Mechanics, 23:241 (1956).

29. Lazan, B. J., and L. E. Goodman: “Shock and Vibration Instrumentation,” p. 55, ASME,

New York, 1956.

30. Pian, T. H. H., and F. C. Hallowell: Proc. First U.S. Nat’l. Cong.Appl. Mechanics, June 1951,

p. 97.

31. Fluegge, W.: “Viscoelasticity,” Blaisdell Publishing Company, a division of Ginn and Com-

pany,Waltham, Mass., 1967; Lee, E. H.:“Viscoelasticity,” in W. Fluegge (ed.),“Handbook of

Engineering Mechanics,” McGraw-Hill Book Company, New York, 1962; Lesieutre, G. A.:

Int’l. J. Solids and Structures, 29:1567 (1992).

32. Bogdanoff, J. L.: J. Appl. Physics, 21:1258 (1950).

33. Caughey, T. K.: J. Appl. Mechanics, 27:640 (1960).

34. Rang, E.: Wright Air Development Center Tech. Rep. 59-121, February 1959.

35. Goodman, L. E.:“A Review of Progress in Analysis of Interfacial Slip Damping,” in “Struc-

tural Damping,” ASME, New York, 1959.

36. Deresiewicz, H.: “Bodies in Contact with Applications to Granular Media,” in G. Herr-

mann (ed.), “R. D. Mindlin and Applied Mechanics,” Pergamon Press, New York, 1974.

36.30 CHAPTER THIRTY-SIX

8434_Harris_36_b.qxd 09/20/2001 12:28 PM Page 36.30

CHAPTER 37

APPLIED DAMPING

TREATMENTS

David I. G. Jones

INTRODUCTION TO THE ROLE

OF DAMPING MATERIALS

The damping of an element of a structural system is a measure of the rate of energy

dissipation which takes place during cyclic deformation. In general, the greater the

energy dissipation, the less the likelihood of high vibration amplitudes or of high

noise radiation, other things being equal. Damping treatments are configurations of

mechanical or material elements designed to dissipate sufficient vibrational energy

to control vibrations or noise.

Proper design of damping treatments requires the selection of appropriate

damping materials, location(s) of the treatment, and choice of configurations which

assure the transfer of deformations from the structure to the damping elements.

These aspects of damping treatments are discussed in this chapter, along with rele-

vant background information including:

●

Internal mechanisms of damping

●

External mechanisms of damping

●

Polymeric and elastomeric materials

●

Analytical modeling of complex modulus behavior

●

Benefits of applied damping treatments

●

Free-layer damping treatments

●

Constrained-layer damping treatments

●

Integral damping treatments

●

Tuned dampers and damping links

●

Measures or criteria of damping

●

Methods for measuring complex modulus properties

●

Commercial test systems

37.1

8434_Harris_37_b.qxd 09/20/2001 12:26 PM Page 37.1

MECHANISMS AND SOURCES OF DAMPING

INTERNAL MECHANISMS OF DAMPING

There are many mechanisms that dissipate vibrational energy in the form of heat

within the volume of a material element as it is deformed. Each such mechanism is

associated with internal atomic or molecular reconstructions of the microstructure

or with thermal effects. Only one or two mechanisms may be dominant for specific

materials (metals, alloys, intermetallic compounds, etc.) under specific conditions,

i.e., frequency and temperature ranges, and it is necessary to determine the precise

mechanisms involved and the specific behavior on a phenomenological, experimen-

tal basis for each material specimen. Most structural metals and alloys have rela-

tively little damping under most conditions, as demonstrated by the ringing of sheets

of such materials after being struck. Some alloy systems, however, have crystal struc-

tures specifically selected for their relatively high damping capability; this is often

demonstrated by their relative deadness under impact excitation. The damping

behavior of metallic alloys is generally nonlinear and increases as cyclic stress ampli-

tudes increase. Such behavior is difficult to predict because of the need to integrate

effects of damping increments which vary with the cyclic stress amplitude distribu-

tion throughout the volume of the structure as it vibrates in a particular mode of

deformation at a particular frequency. The prediction processes are complicated

even further by the possible presence of external sources of damping at joints and

interfaces within the structure and at connections and supports. For this reason, it is

usually not possible, and certainly not simple, to predict or control the initial levels

of damping in complex built-up structures and machines. Most of the current tech-

niques of increasing damping involve the application of polymeric or elastomeric

materials which are capable (under certain conditions) of dissipating far larger

amounts of energy per cycle than the natural damping of the structure or machine

without added damping.

EXTERNAL MECHANISMS OF DAMPING

Structures and machines can be damped by mechanisms which are essentially exter-

nal to the system or structure itself. Such mechanisms, which can be very useful for

vibration control in engineering practice (discussed in other chapters), include:

1. Acoustic radiation damping, whereby the vibrational response couples with the

surrounding fluid medium, leading to sound radiation from the structure

2. Fluid pumping, in which the vibration of structure surfaces forces the fluid

medium within which the structure is immersed to pass cyclically through narrow

paths or leaks between different zones of the system or between the system and

the exterior, thereby dissipating energy

3. Coulomb friction damping, in which adjacent touching parts of the machine or

structure slide cyclically relative to one another, on a macroscopic or a micro-

scopic scale, dissipating energy

4. Impacts between imperfectly elastic parts of the system

POLYMERIC AND ELASTOMERIC MATERIALS

A mechanism commonly known as viscoelastic damping is strongly displayed in

many polymeric, elastomeric, and amorphous glassy materials. The damping arises

37.2 CHAPTER THIRTY-SEVEN

8434_Harris_37_b.qxd 09/20/2001 12:26 PM Page 37.2

from the relaxation and recovery of the molecular chains after deformation. A

strong dependence exists between frequency and temperature effects in polymer

behavior because of the direct relationship between temperature and molecular

vibrations. A wide variety of commercial polymeric damping material compositions

exist, most of which fit one of the main categories listed in Table 37.1.

TABLE 37.1 Typical Damping Material Types

Acrylic rubber Natural rubber Polysulfone

Butadiene rubber Nitrile rubber (NBR) Polyvinyl chloride (PVC)

Butyl rubber Nylon Silicone

Chloroprene (e.g., Neoprene) Polyisoprene Styrene-butadiene (SBR)

Fluorocarbon Polymethyl methacrylate Urethane

Fluorosilicone (Plexiglas) Vinyl

Polysulfide

Polymeric damping materials are available commercially in the following cate-

gories:

1. Mastic treatment materials

2. Cured polymers

3. Pressure sensitive adhesives

4. Damping tapes

5. Laminates

Some manufacturers of damping material are given as a footnote.* Data related

to the damping performance is provided in many formats. The current internation-

ally recognized format, used in many databases, is the temperature-frequency

nomogram, which provides modulus and loss factor as a function of both frequency

and temperature in a single graph, such as that illustrated in Fig. 37.1.

1,2

The user

requiring complex modulus data at, say, a frequency of 100 Hz and a temperature of

50°F (10°C) simply follows a horizontal line from the 100-Hz mark on the right ver-

tical axis until it intersects the sloping 50°F (10°C) isotherm, and then projects verti-

cally to read off the values of the Young’s modulus E and loss factor η.

APPLIED DAMPING TREATMENTS 37.3

* Manufacturers of damping materials and systems, from whom information on specific materials and

damping tapes may be obtained, include:

Antiphon Inc. (U.S.A.) Leyland & Birmingham Rubber Company (U.K.)

Arco Chemical Company (U.S.A.; www.arco.com) MSC Laminates (U.S.A.)

Avery International (U.S.A.; www.avery.com) Morgan Adhesives (U.S.A.; www.mactac.com)

CDF Chimie (France) Mystic Tapes (U.S.A.)

Dow Corning (U.S.A.; www.dowcorning.com) Shell Chemicals (U.S.A.; www.shellchemicals.com)

EAR Corporation (U.S.A.) SNPE (France; www.snpe.com)

El duPont deNemours (U.S.A.; www.DuPont.com) Sorbothane Inc. (U.S.A.; www.sorbothane.com)

Farbwercke-Hoechst (Germany) Soundcoat Inc. (U.S.A.; www.soundcoat.com)

Flexcon (U.S.A.; www.flexcon.com) United McGill Corporation (U.S.A.;

Goodyear (U.S.A.; www.goodyear.com) www.unitedmcgillcorp.com)

Goodfellow (U.K.; www.goodfellow.com) Uniroyal (U.S.A.; www.uniroyalchem.com)

Imperial Chemical Industries (U.K.) Vibrachoc (France; www.vibrachoc.com)

8434_Harris_37_b.qxd 09/20/2001 12:26 PM Page 37.3

ANALYTICAL MODELING OF COMPLEX MODULUS BEHAVIOR

It is very convenient to be able to mathematically describe the complex modulus

properties of damping polymers, not only in the form of a nomogram as just

described, but also by algebraic equations which can be folded into finite element

and other computer codes for predicting dynamic response to external excitation

(see Chap. 28). Such models include the standard model, comprising a distribution of

springs and viscous dashpots in series and parallel configurations

for which the

complex Young’s modulus E* (and equally the shear modulus G*) can be described

in the frequency domain by a series such as

E* =



n = 1

(37.1)

or a fractional derivative model

for which the series becomes

E* =



n = 1

(37.2)

where a

, b

, and c

are numerical parameters, which may be real or complex, the β

are fractions of the order of 0.5, and α

is a shift factor which depends on tempera-

ture. Both models work, but Eq. (37.1) will usually require many terms, often 10 or

more, to properly model actual material behavior, whereas Eq. (37.2) usually

requires only one term for a good fit to the data. The shift factor α

is determined as

a function of temperature for each material from the test data, and is usually mod-

eled by a Williams-Landel-Ferry (WLF) relationship

1,5

of the form

+ b

(i f α

)



1 + c

(i f α

)

+ b

(i f α

)



1 + c

(i f α

)

37.4 CHAPTER THIRTY-SEVEN

FIGURE 37.1 Temperature-frequency nomogram for butyl rubber composition.

8434_Harris_37_b.qxd 09/20/2001 12:26 PM Page 37.4

log [α

] = (37.3)

or by an Arrhenius relationship

1,5

of the form

log [α

] = T



−



(37.4)

where C

and B

are numerical parameters, the temperatures T and T

(the reference

temperature) are in degrees absolute, and T

is a numerical parameter related to the

activation energy.The behavior of each specific polymer composition dictates which

expression is most appropriate, and simple statistical methods may be applied for

determining “best estimates” of each parameter in these equations.

BENEFITS OF APPLIED DAMPING TREATMENTS

When the natural damping in a system is inadequate for its intended function, then

an applied damping treatment may provide the following benefits:

Control of vibration amplitude at resonance. Damping can be used to control

excessive resonance vibrations which may cause high stresses, leading to prema-

ture failure. It should be used in conjunction with other appropriate measures to

achieve the most satisfactory approach. For random excitation it is not possible to

detune a system and design to keep random stresses within acceptable limits

without ensuring that the damping in each mode at least exceeds a minimum

specified value. This is the case for sonic fatigue of aircraft fuselage, wing, and

control surface panels when they are excited by jet noise or boundary layer

turbulence-induced excitation. In these cases, structural designs have evolved

toward semiempirical procedures, but damping levels are a controlling factor and

must be increased if too low.

Noise control. Damping is very useful for the control of noise radiation from

vibrating surfaces, or the control of noise transmission through a vibrating sur-

face. The noise is not reduced by sound absorption, as in the case of an applied

acoustical material, but by decreasing the amplitudes of the vibrating surface. For

example, in a diesel engine, many parts of the surface contribute to the overall

noise level, and the contribution of each part can be measured by the use of the

acoustic intensity technique or by blanketing off, in turn, all parts except that of

interest. If many parts of an engine contribute more or less equally to the noise,

significant amplitude reductions of only one or two parts (whether by damping or

other means) leads to only very small reductions of the overall noise, typically 1

or 2 dB.

Product acceptance. Damping can often contribute to product acceptance, not

only by reducing the incidence of excessive noise, vibration, or resonance-

induced failure but also by changing the “feel” of the product. The use of mastic

damping treatments in car doors is a case in point. While the treatment may

achieve some noise reduction, it may be the subjective evaluation by the cus-

tomer of the solidity of the door which carries the greater weight.

Simplified maintenance. A useful by-product from reduction of resonance-

induced fatigue by increased damping, or by other means, can be the reduction of

maintenance costs.



−C

(T − T

)



+ T − T

APPLIED DAMPING TREATMENTS 37.5

8434_Harris_37_b.qxd 09/20/2001 12:26 PM Page 37.5

TYPES OF DAMPING TREATMENTS

FREE-LAYER DAMPING TREATMENTS

The mechanism of energy dissipation in a free-, or unconstrained-, layer treatment is

the cyclic extensional deformation of the imaginary fibers of the damping layer dur-

ing each cycle of flexural vibration of the base structure, as illustrated in Fig. 37.2.

The presence of the free layer changes the apparent flexural rigidity of the base

structure in a manner which depends on the dimensions of the two layers involved

and the elastic moduli of the two layers. The treatment depends for its effectiveness

on the assumption, usually well-founded, that plane sections remain plane.The treat-

ment fiber labeled yy is extended or compressed during each half of a cycle of flex-

ural deformation of the base structure surface, in a manner which depends on the

position of the fiber in the treatment and the radius of curvature of the element of

length ∆l, and can be calculated on the basis of purely geometric considerations. One

fiber in particular does not change length during each cycle of deformation and is

referred to as the neutral axis. For the uncoated plate or beam, the neutral axis is the

center plane, but when the treatment is added, it moves in the direction of the treat-

ment and its new position is calculated by the requirement that the net in-plane load

across any section remain unchanged during deformation. The basic equations for

predicting the modal loss factor η for the given damping layer loss factor η

and for

predicting the direct flexural rigidity (EI)

as a function of the flexural rigidity E

of the base beam are well known.

1,7

The simplest expression relating the damping of a structure, in a particular mode,

to the properties of the structure and the damping material layer is

= (37.5)

where η is the damped structure modal loss factor, η

is the loss factor of the damp-

ing material, E

is the Young’s modulus of the damping material and E

is that of the

structure (e = E

), and h

and h

are the thicknesses of damping layer and struc-

ture, respectively (h = h

To calculate η, the user estimates η

and E

at the frequency and temperature of

interest (from a nomogram), then calculates h and e, and then inserts these values

into Eq. (37.5). Change thickness (h) or material (e) if the calculated value of η is not

eh(3 + 6h + 4h

+ 2eh

+ e

)



(1 + eh)(1 + 4eh + 6eh

+ 4eh

+ e

)



37.6 CHAPTER THIRTY-SEVEN

FIGURE 37.2 Free-layer treatment. (A) Undeformed. (B) Deformed.

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