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MINISTRY OF AVIATIONAEf?ONAUKAL UESEARCHCOUNC/L

CUUUENT PAPH?S

The Damping of Structural Vibrations

G.G. Parfift cd fl. Lumbeh,Physics Dept.,

lmpeficd College

LONDON: HER MAJESTYS STATIONERY OFFICE1962

PRICE 4s. 6d. NET

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C.P. No.596

The Damping of Structural Vibrations- By -G. G. Parfitt and D. Lambeth,Physics Dept.,Imperial College

September, I!360Contents--

1. Introduction . . . . .* . . . . . . . . . .I .I General . .1.2 Damping in sol& . . . . . . . . l . . .1.3 The effect of da*qxi~~ tr~~tmo~~ on*~tru~&.lre~'1.4 Work in other l.aboratories . . . .

2. Theory of the Backed Damping Layer . .3* Work at Imperial College . . . .3.1 Experimental measuremen-&' . . . .Jo2 Simfle damping layers . . . . . .3.3 The backed damping layer . . . .3.4 The backed layer in practice . .3.5 Cuts in a backed layer . . . . . .3.6 Deep backing layers . . . . . .3.7 Spaced layers . . . . . . ..,4. Summary and Conclusions . . . . . . . .

APPENDIX - The Design of a Damping Tape

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--1. Introduction----.--

Page. . 1. . I. .. .. .

i. . 7. . 8'* . 8. .. . ;*. 9. . 11. . 12. . 12. . 14. . 15

I.1 The application of layers of heavily-damped materials to metalpanels to deaden them acoustically is a practice of long standing in themotor-car industry, and in the last decade particularly the use of suchdamping media has spread rapidly with the increasing urgency of problemsof vibration, noise and metal fatigue, above all in the aeronautical field.The most comprehensive attempt to develop a suitable material forapplication as a unif'orm layer to a metal sheet was that of Oberst and hisco-workers1,2j3. They appear to have been the first to point out publiclythat such a material needed to have not only a high damping but also a highelastic modulus. Par if the damping material is mechanioally weak comparedwith the panel, then by definition only a small propor tion of the tOhI-

elast ic/

'A.1' REPORTPreviously issued as A.R.C.22,250.

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-2-elastic energy of deformation is stored within the material. The fractionof this which is dissipated by the material can then be barely significantcompared with the total energy. For this reason a layer of, say, clothfirmly cemented to a panel adds very little damping.

Oberst developed a filled high-polymer material havk-g the desirablehigh modulus and damping. The commercial form of this now available inGreat Britain, termed Aquaplash, appears to be the best available material forstraightforward damping layers.in semi-liquid form, It is normally sprayed on to panels, etc.,though it can be trowelled on or first set into sheets andcemented on. This material forms a convenient standard of reference for thejudgemen-t of other damping treatments.The present work represents an attempt to see whether behaviour moreadvantageous than that of a simple damping layer can be obtained from a layerbacked by an additional sheet of metal, or from various other compositetreatments of similar character. The work has been largely directed towardstreatments which can be applied to an existing structure to modify itsproperties, and has been less concerned with methods whereby damping oould be'built into' a new structure. This report, which is a final one summarisingwork oarried out under contract to the Ministry of Supply (now Ministry ofAviation) is written with special reference to the damping of aircraftstructures, but the techniques discussed have wider application. Two interimreports have been submitted previously, dated August, 1958 and October, 1959.The present report embodies the main results given earlier. By request ofthe oontraoting authority, it also contains (Section 1.3) a brief discussionof the part to be played by damping treatments in aircraft and otherstructures in minimising the effects of noise, v ibration and fatigue. This

may be omitted by anyone familiar with the problems involved.1.2 Damping in solids

It is convenient at this point to define the terms whioh will beused in discussing the mechanical properties of damping layers. For *sinusoidal vibrations of sufficiently small amplitude, most solids behave ina substantially linear, viscoelastic manner, and their mechanical propertiescan be then adequately represented by a so-called complex elastio modulus5.Thus, if Young's Modulus is in question, the ratio of complex tensile stressamplitude to complex tensile strain amplitude is equal to the quantity Ex,where9 = EL + jl$ . . . (1.1)

EL is here the real or *conventional' elastic modulus, while Ea representsinternal damping. EL and I$., are independent of amplitude if this is smallbut are in general functions of the frequency f or (0/2x.Writing E = q + js = EL(l+$lE) . . . (1.2)

defines qE = s/T?,, the damping factor for tensile strains of the medimFor shearing strains, there is an analogous complex rigidity modulus.. . . (113)

where qG is the shear damping coefficient.When any elastic member is strained, it is often convenient todefine its stiffness Sx, i.e., the ratio of the force applied to thedisplacement produced across it. If the deformation involves strains ofpurely tensile or shear types, or of other simple definable type, *iscomplex stiffness

S = St+jE$ = Si (1 + jtl) . . . (1.4)

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has thetensileproduce

ing coefficient 7 = Ss/Si as that of the relevantshear or other modulus. *If such an elastic member is attached to a mass M (F ig.1) toa simple one-degree-of-freedom oscillator, of natural angular

frequency w , then the ratio of the damping existing at frequency 6~ tothat require 8 to damp the system critically is termed the damping rat20 6.6 is in fact equal to one half of the damping factor appropriate to theresonance frequency.

-3-

For purely viscous damping, which is easy to analyse theoretically,?l is proportional to frequency. This case is however rather rare inpractical materials, in which q is usually more nearly independent offrequency. This is often termed hysteretic damping, though in fact itsmechanism may still be essentially viscoelastic.1.3 The effect of damping treatment on structures

The undesirable effects of vibration in structures are manifold,but can all be mitigated to a greater or lesser degree by the addition ofabsorbers of vibrational energy. The main phenomena of practical concerncan be distinguished as follows:(i) Transmission o f vibration along long structural membersIn the simple but in practice rather rare case where purelyprogressive mechanical waves are involved, the wave amplitude in a uniformbeam whose elastic properties are linear with respect to amplitude and whosedamping factor is q decreases exponentially with distance. For flexuralwaves such as are usu~,lly of interest, it can be shown that the decrease is

approximately in the ratio e T!* over a distance of one wavelength.Since qE is very rarely likely to exceed say 0.5, when e Td* = 0.46.any attempt to localise vibration by damping a structure is likely to bepracticable only for frequencies high enough for the flexural wavelength tobe considerably smaller than the dimensions of the structure.

(ii) Transmission of sound through panels--M--w- m.mpAcoustic pressures applied to walls or panels cause them to vibrate,and their motion causes sound waves to be re-radiated on the further side,.3-.e., the sound is (partially) transmitted. The intensity radiated is acomplex function of the system geometry and wavelengths in general. In thesimplest case, when the panel vibrates in phase over a region large comparedwith the wavelengths in air, the intensity is proportional to the square ofthe panel w. This is then the quantity to be minimised. A problemclosely related to ihe foregoing and to case (i) is that of radiation ofsound by a panel excited by vibration transmitted to it through a structurefrom a remote source. It is again necessary to minimise panel velocity inthe simple case.

(iii) Fatigue of structural membersIn tiis case which is of pressing importance in the aircraftindustry, it is necessary to minimise the stresses presen t within structuralmembers and at joints, rivet holes, etc. .To examine the effect of damping on the quantities of interest incases (ii) and (iii), it is necessary to consider a simple mdiel of astructural member. (The discussion which follows is drawn primarily fromstudies of the excitation of aircraft struotures by the random noise fieldsproduced by jet engines, notably by J. W. Miles in the U.S. and members of

the/

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-4-the Aeronautics Department of the Southampton Universi$y. A useful andauthorLtative summary of relevant aspects of this work has been publishedvery reoently by D. J. Mead ofthe present surve$. Southampton 6, and justifies the brevity of

Broadly speaking, two types of vibrational resonance may ocour inan aircraft structure. The first involves the struoture as a whole, or atleast large sections of it. The pattern of nodes spreads over large partsof the structure, which behaves as a roughly uniform elastic shell. !Thesecond type is characteristically the resonance of a single panel, stringersection or other small localised element of the structure. The prinoiplesof response reduction by damping are essentially the same in each case,though the treatments will necessarily be on the more massive scale in thefirst case. In the case of excitation by noise fields of largely randomstructure (i.e., having little phase correlation between the pressures atany two points unless these are relatively close) and relatively shortwavelengths, the first type of mode may not be of such great practicalimportance.NOW when a force varying in space and time is applied to thesurface of a panel, the response is determined not only by the frequency andpanel stiffness and mass, but in a complex way by the spatial distributionof the forces and of the nodal pa ttern of the manif'old modes of resonanceof the panel. However for the present purpose it will be sufficient toconsider a localised point force acting on a single-degree-of-freedomosoillator.Such a system, shown in Fig.1, consists of a mass to which the

driving force is applied, supported from a rigid foundation by a spring ofcomplex stiffness (& + jSs) (see Eqn.l.4). It will be supposed forclarity that Si(~/a), thou& th and % are substantially independent.of frequencyis is not essential. If the applied foroe is-sinusoidalin time and of amplitude F, Ihe amplitude x of the displacement of themass, the amplitude V o f the velocity of the mass and the amplitude Ftof the foroe transmitted to the foundation are given by. . . (1.5)

OFv = -----I2 I . . . (1.6)

. . . (1.7)

where z = (Sl -u?M+jE&) . . . (1.8)The resonance frequency ~0o/2xis given by

. . . hY1is given as a function of frequency for two values of the damping

= &a/s1 in Fig.2. !Chemaximum values of x/F and of F.tboccur when w = w. and are given byF Fx = -- = --- . . . (1.10)

= @?. F ... (1.11)qs*

Wt-4

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-5 -Both quantities are seen to be inversely proportiona l to damping factor;xO is inversely proportional to stiffness, and Fto largely independentof it.

The situation is rather different when the driving force has asubstantially random waveform instead of a sinusoidal one, so that itsFourier spectrum covers a wide range of frequencies. The broken line inFig.2 might for instance represent the spectral density curve of excitingpower f(W) and tie response of the system is obtained by integratingthe product of the spectral density function and the response function

WV l It is plausible that if the damping is small, the region aroundresonance will be the only one making a substantial contribution to theintegrated product, and in this case the response is essentially anoscillation of the -panel at the resonance frequency , with a randomly-varying amplitude. This is borne out by experiments on aircraft panels7.The R.M.S. value of the amplitude is given by

where f(ti ) i.3 the magnitude of the applied spectral density f,unctionw =OJ0. 'Similarly the maximum transmitted force has R.M.S. amplitudewhen

It will be noticed that for sinusoidal excitation, the displacementis inversely proportiona l to the damping parameters Sa or ?J. However,for random excitation, it is the square roots of these quantities which appearin the denominator. This is due to the fact that for the random case,increased damping does not merely reduce the response at the resonancefrequency as l/q but increases the resonantbandwidth ?W ,and so rendersthe system su:ceptible to resonant excitation over a wider prequency range*The factor 3 appears in the numerator of Eqns.1 .I2 and I.13 for this reason*Thus damping ?s a less effective variable for random excitation than forsinusoid&. The real stiffness Sl appears explicitly in Eqn.l.12 and also

Using Eqn.l.9 gives It must be borne inmind of course that the change of resonance frequency due to change ofstiffness will in genera l change f(W ), i.e., will shift the resonance intoa region of either more or less power ul excitation, and this effect mayoverride the relatively small changes in Si and Mi brought about by mostdamping treatments. The changes in f(&) ) may be quite rapid as w varies,since a condition of "coincidence V betwee: the trace wavelength of so&d wavesof the resonant frequency arriving at the panel from a given direction andthe flexural wavelength on the panel may be created or removed.

The panel velocity, which was stated to influence sound transmission,is obtained from Eqn.l.12 by multiplying by w. and so is proportionaltol/(4 MS ?I&).less'so. Damping and mass increase are beneficial, stiffness increAseIt should be borne in mind however that There passenger comfort inan aircraft is concerned, high-frequency noise makes a particularly importantcontribution to the total subjective effect, and that many panels may haveonly high-order resonances in this range. Their corresponding resonantresponse, which is all that is susceptible to improvement by damping, willthen be weak in any case. The moss of the panel tends to be the controllingfactor here.

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The last and perhaps the most important quantity to be consideredis the stress, both in the panel and its fastenings. Eqns.l.9 and I.13 forrandom vibration show Ft,Lmeasuring the stress on the fastenings, to beproportional to (St + S~)'/~' S1 M4, 05 since ,Y$

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-7-has been started,probably independently, in several laboratories. !l!heseare known to include those of the Aeronautical Engineering Departmentiof theUniversity of Minnesota (under Prof. B. J. Lazan) and the University ofSouthampton (unde r Prof. E. J. Richards), the firm of Bolt, Beranek andNewman Inc., Cambridge, Mass. (E. M. Kerwin and others) and the AcousticsInstitute of the U.S.S.R. Academy of Sciences. The basic theory of acomposite layer was laid down by Kerwin and his colleagues and is brieflydiscussed in Section 2. Other work is referred to where relevant throughoutthe report.2. !I'heory of the Backed Damping Layer----

When bending waves travel through a three-layer plate in which thecentral layer is relatively soft, (Fig.4.), the nature of the deformation willapproximate to one or other of the two types sketched in Fig.5 and termed(5b) pure bending and (57) shear bending. In the latter oase the centrallayer undergoes a shearing action while the outer layers bend. The theoryof bending wa s in such a system has been developed by Kerwin and hisco-workers10~T'~'2 who showed that when the wavelength is long, thedeformation is of type (b)transition towards type (c j

and that as the frequency is increased, a steadyocours~ Thus at low frequency the effectivebending stif'fness of the plate has the high value appropriate to a thickplate, and decreases with wavelength towards the lower value correspondingessentially to the combined stiffnesses of the two outer layers bendingsimultaneously and with the same radius of curvature . If the outer layersare of metal, which has low mechanical losses, the damping of the compositeplate arises essentially from the shear distortion of the central lqer, andin fact passes through a broad maximum in the transition region (Fig.6).The controlling factor is the socalled "shear parameter" g, defined by.*. (2.1)

where G =a shear modulus of damping layer*a = thickness It It t133 = Young's modulus of baoking layerI-!5 = thickness n It 1t

A = wavelength of flexural waves in the plate.We define also

EL = Youngts modulus of bottom layer (panel)*A = thickness 11 I? t1va = shear damping factor of damping layer.

g is the abscissa of Fig.6. The maximum damping is reaohed when1

g Z ~~~~s.~~~~ . . . (2.2)(I + q$

and if it is assumed that (E3H3/EiHI) is fairly small, then the maximumdamping factor of the composite plate is approximately:

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-8-H is the distance of the centre of the backing layer from the neutrals%%face of the composite plate. If, further, the added layers are thincompared with the main panel, so that so + Hi/2, and if also qi

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-Y-top curve in Fig.7. Its damping was considerably higher however, being0.17 at 21 c/s and 0.38 at 72 C/S. Higher-order resonances were toohighly damped to be observable. The bending stiffness of the strip wasincreased 2.9 times by the rib.

3.3 The backed damping layerThe next experimental results to be described were aimed atconfirming the theory of the backed damping layer briefly discussed inSection 2. Figs.9 and 10 show respectively the measrd bending stiffnessand the damping of a sandwich of 0.08 in. of Aquaplas grade F 102 B, betweensteel strips each 0.0625 in. thick. The stiffness measurements show clearlythe transition in stiffness with increasing frequency, agreeing well with thecalculated values. The damping measurements show less satisfactoryagreement with theory, being some 2 - 3 times too large. This may be partlydue to the stated inadequacy of the theory for rather stiff damping layers.

A more probab le alternative is that the values of the mechanical constants ofAquaplas used to derive the calculated dampings were inacourate. A valueof 0.2 for q in Aquaplas was obtained from measurements with an unbackeddamping layer and used for the computation. More recznt measurementshowever have given values up to 0.38, which would largely remove thediscrepancy. It is worth noting that if the bar vibrated in pure bending(Fig.5b) the damping would be more than 100 times less than was measured.Another set of measurements was made with a layer of fairly softrubber on the standard l/8 inch aluminium strip and backed by an 0.022 in.aluminium foil. The material constants of the rubber were obtained byvibrating a small weighted strip of the rubber and observing its longitudinal

resonance with a microscope under,stroboscopic illumination. The shearconstants were then obtained by assuming Poisson's ratio + 0.5.The results for the backed layer are shown in Fig.11; along withthe theoretical curve. A peaked curve is obtained of about the correctheight, though at a somewhat higher frequency than predicted.Most of the curves for backed layers in this and other sectionsare rather flatter than the theoretical curve of Fi.g.6. Possible reasonsfor this are end effects in the test strip (the phenomenon discussed inSection 3.5) and variation of the constants of the damping material withfrequency. Whatever the reason, the effect is advantageous in practice inthat the damping achieved is less frequency-se lective than theory implies.At present two firms in Great Britain are kncwn to manufacture"damping tapesfi - i.e., backed damping layers. Johnson and Johnson Ltd.produce (to special order) their Permacell "Acoustimat", consisting of afabric strip impregnated with adhesive some 0.010 - 0.012 in. thick andbacked by an aluminium foil about 0.005 in. thick. A curve for -Ihis tapeon a steel strip was given as Fig.7 of the previous report, and other tapesusing this damping layer are discussed in Section 3.4. The MinnesotaMining and Manufacturing Co. Ltd. has very recently produced a range(Series 428) of tapes having a thin adhesive layer on aluminium foils ofthickness 0.0055, 0.008 and 0.012 in. No tests have yet been made on thesetapes but some data supplied by the manufacturer yield points on Fig.8lying near or rather below the line for the various Permacell tapes.

3.4 The backed la.yer in practiceUsing the formulae of Section 2 for backed layers and those ofOberst' for unbacked layers, a theoretical comparison was carried out betweenthe two systems of damping, assuming the most suitable of available materialsto be employed in each case, and using the added weight of the dampingtreatment as a basis of comparison. Some considerations relating to thedesign of a damping tape are set out in Appendix I. It was concluded

(see report No.2) that for the same added weight the backed layer Wuldgive/

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- -lo -give somewhat the greater damping if' used in its optimum frequency range.(Fig. 6). Kervtin and Ross'3 have also predicted backed layers to besuperior, at least for mass increases up to 4% on aluminium panels.

To set against this, the backed layer must in principle bedesigned for each individual application so that its effective frequencyrange coincides with that of the vibrations present, and the requiredincrease of damping or of weight is ob tained. To do this it is in generalnecessary to vary thicknesses (and possibly materials) of both backing anddamping layers, and so the two layers must be separately available,A case in which this is possible is tiat of the Permacell tape(see foregoing Section). The adhesive-impregnated fabric strip whichforms the damping layer here is available separately under the namePermacell 45. It is therefore possible to add several layers of this to apanel and cover with a layer of aluminium foil of suitable thickness.Some experiments were therefore undertaken to examine thispossibility. One set of specimens was made up to check that the frequencyof maximum loss could be reduced by ticreasing the thickness of bacUng anddamping layers (Eqns.2.1 and 2.2). This appeared to be the case, but theresults were not conclusive quantitatively owing to the very high dampingof the thicker treatments which made observation djf'ficult.Fig.12 shows some further results using superposed layers of thedamping material. Treatments were made up consisting of 2 and of .!I- ayersof adhesive tape backed by aluminium strips 0.013 and 0.021 in. thickrespectively. The curves for these are those of Fig.12 marked "comparable

single tape'. !L'he shapes are rather complex, thou& still show a generallypeaked character.An alternative way of employing multiple layers which-would bemore convenient in practice would be to superimpose several damping tapeseach with a damping layer and backing of a standard thickness. The curvesof Fig.13 show results for up to four superposed layers of damping tape,each consisting of a layer of Permacell 45 and an O.Ol5 in. aluminiumbacking. The shift to lower frequencies and higher dampings expected in aIn Fig.12 theingle tape of increasing dimensiona is in fact evident.results for 2 and 4 layers of such damping tape are replotted for comparisonwith the measurements for comparable 'simple' treatments of similar weight.

It is seen that for some 30s mass increase, four distinct damping tapesk?ive about the same maximum damping as one of four times the thicknessthough this maximum is somewhat conjectural in each case owing to thedifficulty of observing such heavily damped resonances).increase, the 'simple' For 15 - 1% masstapes. tape gives rather more damping than two superposedThe shapes of the damping-frequency curves are in all cases rathercomplex9 but the curves for multiple layers are shifted somewhat to higherfrequencies compared with single layers of similar oonstitution.

The theory of Kerwin et al for single damping tapes has beenextended to cover multiple tapes. It is extremely complex and notsusceptible to explicit solutions for a general oase. A specimen resulthowever is that for two identical superposed tapes thti in comparison withthe panel, the damping frequency curve is generally similar to that for asingle tape of doubled dimensions, having a virtually identical maximumloss but at a frequency 1.7 to 2 times higher, depending on the tapedamping. !&is is broadly cons istent with the experimenta l results, whichdid not adequaUngar and Ross t sly fulfil the condition that the tapes should be thin.have very recently reported making a similar calculationwhich showed that behaviour of multiple tapes was similar to that of asingle tape having the same total metal thickness but the same dampinglayer as any one of the layers. This agrees substantially with the abQveresult.

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- II -As a practical conclusion therefore it seems reasonable toregard multiple layers of a standard tape as sufficiently similar inperformance to that of single tapes of greater thickness to justify theirconsiderab le advantage in convenience and availability.It is tempting to try to use the greater stiffness available ina damping material such as Aquaplas to minimise the quantity required andhence the weight. This can in principle be done by having the dampingmaterial in the form of small "islands" under the backing instead of acontinuous layer. It is doubtful however if' the advantage in weight wouldoffset the difficulty of application.

3.5 Cuts in a backed layer (See also Addendum on p-16)Commercial damping tape with a thin foil backing tends to becomesomewhat wrinkled during storage and application, an effect which would beexpected to reduce the extensional stiffness of the backing on which themaximum damping has been shown to depend (Eqn.2.4). A test was accordinglymade but yielded the at first surprising result that wrin.kJes actuallyimproved the damping on the low-frequency side of the damping peak.Some systematic measurements were therefo re carried out in whichan unwrinkled damping tape was weakened by cuts across it dividing itsuccessively into 3, 4, 8, 16 and 32 sections along the 36 in. length of thetest strip. The results are shown in Fig.14. The initial finding above isconfirmed, in that for a number up to 8 sections (7 cuts) the low-frequencydamping increases. For the larger numbers of cuts damping tends to reaoh amaximum at a frequency which rises as the length between cuts decreases.

Marked on the curves for 3, 7, 15, and 31 cuts are the points at which thelength between cuts equals one quarter wavelength, in each case fairly neara point of maximum damping. For large numbers of cuts the low-frequencydamping decreases again.The phenomena involved here are not fully elucidated , but thefollowing explanation appears probable:If one end of a backing layer is pulled parallel to its length,the damping material beneath it near the end is strongly sheared. However,owing to stretching of the backing, this shear decreases away from the endand at a sufficient distance from the epd will have fallen to a negligible

valuedecreAses%?K%&" aracteristic length L over which the shearhas shown that this length is equal to (sHsEs/Gs)'.This would be of the order of one inch in the present layers. Thus theeffect of stresses applied to or removed from the backing at any point isfelt only over a distance of the order of L from that point. It followsfrom Eqn.2.1 that

and the condition for g $ 1 for maximum damping corresponds to X $ Z.KL.Now in the low-frequency region where g >> I, the system vibratesin pure bending (Fig.5b) with little shearing of the damping layer. cuttingthe tape will allow considerable shear strains to occur in the neighbourhoodof the cut, thus adding to the losses which occur on shearing. Withsuffioient cuts the system probably reaches a state in which shearingpredominates, such as is normally only reached at high frequencies (g < I).The damping factor reaches a value comparable with its normal peak value atg 1. However, if the length between cuts becomes comparable with orsmaller than L, shearing of the central layer decreases again and theshort "islands" of damping tape merely ride to and fro on the vibratingpanel without significant deformation.

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- 12 -At higher frequencies-(g c I), shear bending (Fig.$) is alreadyoccurring, and the effect of cuts may well be to release some of the shearstrains present. However the change should still not be very marked unlessthere a re several cuts in each wavelength, which implies a very close spacingindeed for high frequencies.These ideas appear to explain qualitatively the phenomena shown inFig.14. A detailed theory on these lines is being developed. D. J. Meadat Southampton University,in work as yet unpublished, has also pointed out thatcuts in a damping tape can be beneficial.This effect of cutting the tape has the very valuable result thatthe damping effect of a backed layer can be rendered substantially equal toits highest possible value over a very much wider range of frequencies thanis otherwise the case. This is achieved at no cost in weight or complexity.In fact it actually simpltiies the practical problems of application. Indamping a panel, it is not only unnecessary but actually disadvantageous tpuse a carefully-fitted uniform backing sheet. Instead strips of tape froma reel can merely be laid side by side. For vibrations in two dimensionsthe tapes themselves should be cut across. If less powerful damping isrequired, the panel can be partially covered by sets of tapes at right anglesto each other, the ability to cut them making the crossjngs no problem.An additional advantage is that it is permissible to set thefrequency of maximum damping near the upper end of the range of frequenciespresent, rather than near the centre. This allows a thi&ner and hencelighter da,mping layer.Until a quantitative theory is available it is not possible to saywhat the optimum cut spacing should be in any particular case. At presentsome four to eight cuts per wavelength in a region some two decades belowthe normal damping peak seems a not unreasonable figure.

3.6 Deep backing layersEqn.2.3 shows that for a given panel and damptig material, themaximum damping is proportional to sgO, and thus ticreases rapidly withthe distance between the centre of the backing layer and the neutral surface.One simple technique for increasing this distance which was tried out was tobend up the edges of the backing layer as shown in Fig.15. Typical measureddamping- frequency curves for these forms are shown in Fig.16. The dampjnglayer used was in bothcases a single layer of Permacell 45 tape. The curvesare more irregular than for plain damping tapes, but show the same featuresof a broad maximum, and a fall in da,mping at low frequencies which can bemitigated by cutting the backing at jntervals. A very stiff backing wouldof course not conform to a curved surface such as the side of a fuselagewithout being cut every few inches, so for ribbed or flanged backings cutsare doubly advantageous. Damping tapes with ribbed backings give a higherdamping for a &ven mass increase, as can be seen from Fig.8, but once againthis advantage is only obtained at the expense of an inand difficulty of application. qsease in complexityRoss, K.erwin and Ungar have proposed the

use of a very light, thick backing layer of rigid foamed plastic for thesame purpose of increasing SO, and this would provide the feature, usefulin some applications, of very good thermal insulation.3.7 Spaced layers

It was shown in Section I.3 that so far as minimising panel stresswas concerned, increasing the stiffness of the panel could be beneficial ingeneral. In the previous report, a damped channel-section stif'fener as inFig.17 was described and shown theoretically to provide a consi&erableticrease in stiffness along with a damping comparable with that given by asimple Aquaplas layer of the same added . weight.

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- 13 -The table below reproduces certain results of this investigation,extended now to show the change in resonance frequency and in panel stressdue to the stiffening.

wdFrequency and stress are supposed proportional

respectively to and to (S-i M-' +) xmazimum distance fromneutral surface as discussed in Section 1.3. The columns of the table fromleft to right represent the following ratios:

(1) the thickness of the Aqua.plas layer to the panel thickness(2) the total weight of the treatment to that of the panel(3) the bending stiffness to that of the panel(4) the resonance frequency to that of the panel(5) the peak stress in the panel for a given force after the

stiffness is added to that before but negleoting the effectw(6) the damping factor of the structure to that of the Aquaplasitself.

The stiffener considered consists of an aluminium ohannel ofdimensions shown on Fig.47 (cases e and f in the table deviate slightlyfrom these), considered as attached to a strip of aluminium or of steel).

Lluminiumon steelpanelCLuminiumontluminiumanel

Aquaplas Added Stiffness Resonance Panelthickness weight ratio frequency Stressratio ratio ratio ratioa 1.0 0.14 7.4 2e5 0-30b 0.5 0.10 5.5 2.2 (L35c 1.0d CL5e 0f 0.18

6 IDampingratio

(J-350.200.60a450 ( < 0.01)o~l(o.l-0.2)

The results show that the stiffness and mass increase due to the,stiffeners can theoretically reduce panel stresses by some 3 - 5 times inthe cases considered, in addition of course to the reduction due to damping*Cases e and f in the table, representing an aluminium stiffeneron a l/8 in. aluminium strip with dimension ratios closely similar to thoseof Fig.1 7, were examined experimentally to check the validity of thetheoretical assumption that the system deflects by pure bending. The

experimental results are the bracke ted quantities in columns 3, 4. and 6.It is seen that the measured increases in stiffness and resonanoe frequenoy .are not far short of the calculated ones, and the damping inorease is fullyachieved. An attempt was made to observe the vibration of the stripstroboscopically to see whether any unde.sirable deformations of the stiffenerwere occurring (e.g., anticlastic bowing of the surface) , but the techniquewas insufficiently aoute to be conclusive. However the degree of agreementwith theory oan be taken as evidence of the absenoe of any major spuriousmodes.For comparison with the above results, it may be mentioned thatthe plain Aquaplas layers of Fig.7, representing mass increases of 10% and

34%/

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- IL+ -34% respectively, gave the relatively small stiffness ratios (cf., column 3of the table) of about I .I4 and I.&, the values being somewhat frequencydependent.

It thus appears that structures such as that doscribed, whilerather complex in practice may occasionally find application where themaximum possible mitiga.tion of vibrational stresses or amplitudes is calledfor.

It may be mentioned that the principle of spacing a dampingmaterial awa 2fTfrn,? vibrating member is being closely studied 51 otherTlaboratories 3 3 . It appears that the material can be used to itsmaximum possible effect in this way. However the treatments involved mustneoessarily be somewhat complex and for this reason as well as lack of timethis approach has not been followed further in the present investigation.The channel stiffener discussed in Section 3.7 could form a useful startingpoint for such a study however.4. Summary and Conclusions

The report describes a stuQ of backed damping layers and relatedtreatments for application to metal beams, panels, etc. to damp outmechanical vibration. Such systems are attracting attention in a number oflaboratories throughout the world. They represent a means whereby soft,highly-damped adhesive compounds can be employed as damping agents. Thefirst section includes by request a brief discussion of the file ofstructural vibration and anti-vibration treatments in aircraft strxtures.The report is couched in terms of the vibration of -panels or thin strips.The principles however apply equally to beams girders and other structuralmembers.The basio theory of a backed layer or damping tape is brieflysummarised in Section 2, and the more detailed design of a tape is discussedin the Appendix. The tape produces maximum damping in a particular frequencyrange determined by its dimensions and elastic constants. HoTever the peakon a damping-frequency plot is fairly broad, and in particular it has beenshown that by cutting the tape across at intervals a much flatter curve stillcan be obtained. This is achieved with no penalty in weight or cost andactually simplifies application of tapes. In principle it is necessary tavary the thickness of both backing and damping layers to obtain the requiredpeak damping at the required frequency. However if the tape is cut tominimise f requency discrimination and either the area of the panel coveredor the number of layers of tape superposed is varied to vary the peak damping,it should be possible to treat a wide range of panels w ith one or only a fewstandard patterns of tape. At least one firm now markets such a range inthis country.Fig.8 summarises the results for damping produced by the varioustreatments tested. Damping tapes gave rather greater damping for a givenmass increase than the best materials currently known for simple dampinglayers. The latter, applied by spraying or spreading,have of course obviouspractical advantages for application to curved or uneven surfaces.The performance of a damping tape can be further improved byeffectively thickening the backing for a given weight. One simple meanstested of doing this was to turn the edges of the tape up to form flanges.Foamed plastic backings have also been proposed. Other workers have proposedthe insertion of shear-stiff spacers between panel and tape to enhanoe itseffects. This somewhat complex treatment has not been examined here, thougha similar principle has been employed in a damped channel-section stiffener.This can provide good damping along with a substantial increase of stiffnesswhich is itself beneficial so far as reducing fa tigue stresses in the panelis concerned.Some further, mainly theoretical, study of outstanding points in thework is still in progress, and it is proposed in due course to produce anaddendum to the pesent report covering this.

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- 15 -APFZNDIX

-!he Design of a Dmqxi.n~ Ji!!The wavelength h of flexural waves of frequency f in a thinstrip of thickness Hi, Youn.gts modulus I$ and density oL is given by

. . . (Al)

Assuming that the wavelength in the strip when covered by a damping tape issimilar to that without the tape, substitution of this relation into Eqn.2.1gives the shear parameter g as

. . . (A2)with symbols as in Section 2. It is convenien t to define the density andthickness ratios

Now suppose for simplioity that the backing material is the same as that ofthe strip, i.e., pi = p3, Ei = s etc. !The fractional weight Lnoreaae dueto the tape is

Substituting these quantities in Eqn.Aj gives

If the tape is to give maximum damping at this frequency then g + I, andsolution of Eqn.A.5 Gives. . . (A71

In order for this to be realwafaK 6 ---

4 . . . (~8)

l.e., for a given frequency panel and allowable weight increase there is amaximum value of aK and hence of the product paGs of density and shearmodulus of the damping layer allowing optimum performance. For the commoncase of a thin panel, the required value of shear modulus is quite low,comparable with that of a soft rubber for example. Thus damping tapes employa class of material quite differen t from that required in unbacked layers,which must have the highest possible modulus consistent with hi& internallosses.

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- 16 -If pG lies below the limit set by Eqn.A8, there are two valuesof aJ3, the ra-&osof damping layer wei&t to backing layer weight, givingoptimum damping. The smaller one will be chosen as it yields the largeroptimum damping, and is that oorresponding to the negative sign in Eqn.Ai'.

It lies in the range 0 < af3 s I. The maximum damping obtained (Eqn.2.4)is then given approximately by

The above argument applies exactly only to tapes thin oompared withthe panel and for 'rii

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- 17 -References

Title, etc*. Author(s)1 H. Oberst2 H. Oberst andG. W. Becker3 H. Oberst45 T. Alfrey6 D. J. Mead7 B. L. Clarkson8 D. H. D. Cooper

9 T. J. Mentel

10 D. Ross,E. M. Kerwin andI. Dyer11 E. M. Kerwin12 D. Ross,E. M. Kerwin andE. E. Ungar13 E. M. Kerwin andD. Ross14 J. S. Whittier

45 N. I. Naumkina?B. D. Tartakowskiand M. M. Efrussi

Acustica, Vol.2, 1952: Akustische Beihefte,No.4, p.181.Acustica, Vol.4, 1954, p.434.Acustica, Vol.6, 1956, p.144."Aquaplas", made by Revertex Ltd ., Harlow, Essex.The mechanical propert ies of high polymers.New York, 1948.Aircraft Engineering. March and April, 1960.J. of the Royal Aero. Sot., ~01.61, 1957, p.110.A suggested method of increasing the damping ofajroraft structures.A.R.C. R. & M.2398. Au&u+, 1946.Structural damping. Papers presented at ASMEAnnual Meeting, December, 19.59.Ed. by J. E. Ruzicka. American Sot. of Mech. Engrs.29 West 39th St., New York 18,(p.89).Report No.564 to Bolt, Beranek and Newman ic.,50 Moulton St., Cambridge 38, Mass, U.S.A.J. of the Acoustical Sot. of America, 2, 1959,p.9529p.49 of Ref.9.

Paper in 3rd International Congess of Aooustics,Stuttgart, September, 19%.Wright Air Dev. Centre Technical Rep. No.58-568.ASTIA Document No.214381.Soviet Physios-Acoustics.Vol.5, No.4, April - June, 1960.

ws

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FIGS.1 & 2.

3 E 0 2One -degree - of -freedom resonant system.

IO

Squareddisplacement

FIG. 2.

// I \ \ \ \1z 0.32 \1

/ 1 Iossible spectralknsity function9f exciting Iforce

I

Response function of resonator for a sinusoidal force.

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FIG. 3.

a.

i

Variution of random aq&tude with daabp iyq factor.

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FlGS4.8 !5,

Fig. 4.-METAL H3DAMPING MATERIAL 2METAL I Hl

Backed damping layer-

(a) Unbent

fb) FWC bending

(cl Shear bmdi

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FIG. 6.

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o- Aquaplas FJCU 8. upper curve 3d/o mass increase, Ithickness ratio Hz/H, - O*SS$ lower curve 100/o mass increase, H$i,*O4 44

Oa lDampinq

7

t - Supra thermoson. I74 O/b mass increase, Hz /H l = 057- Supra thermoson. I74 O/b mass increase, Hz /H l = 057

IO0Frequency c/sDampinq of simple layers.w- --

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FlG.8.

D ampingv

O*l

O-01Y. tota\ mass increase

A -Aquaplas FlO2B Th- Supra Thermoson layersSA - Aquaplas on spacer RA - Aqwpk~s rib.D- Damping tapes.2,3 & - Multiple layers of damping tape.RT - Ribbed damping tapes.

Dampinq obtained from various damping treatments.

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FIG. 9.

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Dampingfactor7

Frequency c/sDamping of steel-m- Aquaplas sandwich

.

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.. .

0 02

o=o IDamping

7O-00!

Calculatedt

\ t

30 50 loo 30 0 500 IO00 300 0Frequency c/s

Damping of a backed rubber Ia=

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FIG. 12.

I I I

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I I I

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14.IG.

t-t

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15.IG.

Permacell 45tape

/

Damping tape with flanged backing.

FIG. 17.

Spaced damping layer

Both diagrams are sections perpendicular to the planeof bending.

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FIG. 16.

-

,

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. . .-. -- - - - --

I I IA.R.C.C.P. No. 596. September, 1960. A.R.C. C.P. No. 5%. September, 1960.Parfitt, G. G. and Lambeth, D. - Imperial College Parfitt, G. G. and Lazibeth, D. - Imperial College

TI-E DAMPING OF STRUCK VlBIWl.!IONS TIE DAMPING OF STRUC- VIBRATIONSA mainly experimental study is described of the

damping of structural vibrations obtainable from softviscoelastic layers backed by metal strips, in both singlean1 multiple forms. These can give seater damping tha3available unbacked layers of equal weight. Making cutsacross and forming flanges along the backings can beadvantageous.

A mainly experimental study is described of the~ damping of structural vibrations obtainable from softviscoelastic layers b acked by metal strips, in both singleaa multiple forms. These can give greater damping thanavailable unbacked layers of equal weight. Making cutsacross and forming flanges along the backings can beadvantageous.

iA.R.C.C.P. No. 596. September, 1960.Parfitt, G. G. and Lambeth, D. - Imperial College

TIE-EDAMF'INGOF SmUCm VIBRATIONSA mainly experimental study is described of thedamping & structural vibrations obtainable from soft

viscoelastic layers backed by metal strips, in both singleaa multiple forms. These can give greater damping thanavailable unbacked layers of equal weight. Matig cutsacross ax d forming flanges along -the backings can beadvantageous.

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0 Crown copyright 1962Printed and published by

HER MAJESTYS STATIONERY OFFICETo be purchased fromYork House, Kingsway, London, w.c.2423 Oxford Stree4, London w.1

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