159 Modal Analysis of Plates With Partial Constrained-Layer Damping Treatments

8/13/2019 159 Modal Analysis of Plates With Partial Constrained-Layer Damping Treatments

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MODAL ANALYSIS OF PLATES WITH PARTIALCONSTRAINED-LAYER DAMPING TREATMENTS

Karl K. Stevens, ProfessorDepartment of Mechanical EngineeringFlorida Atlantic UniversityBoca Raton, Florida USA

of modal analysis techniques in theof the modal parameters and modeof an edge-fixed rectangular plate with ayer viscoelastic damping treatmenting over a portion of the surface is des-The test specimen and test procedures areand experimentally-determined valuessystem natural frequencies and loss factorsarying degrees of damping treatment are pre;and compared with predicted values. Infor-o is presented on the influence of thetreatment on the flexural mode shapes of

- natural frequency (in Hz)

--

--

quality factorkinetic energy factordecay timeenergy dissipated per cyclemaximum stored strain energylogarithmic decremetifraction of critical dampingsystem loss factorcircular fnequencysystem natural circular frequency

structures include plates and panels, whichto flexural vibration. Controloften is important in reducingfailures and noise problems, and frequentlyhed by the use of viscoelastic dampinglastic damping materials are used either as atreatment consisting of a single damp-applied to the structure or in a con-yer treatment wherein hhe(idampling layerwith a sti ffe r elastic constrainingIn both configurations, the various layersd together and energy is dissipated in the

tic material as a result of the cyc lic

Rajendra A. Bhat, Graduate StudentDepartment of Ocean EngineeringFlorida Atlantic UniversityBoca Raton, Florida USA

deformations induced by the flexural vibration ofthe underlying structure. The advantages anddisadvantages of both types of damping treatmentare well established and are discussed in the ex-tensive literature which now exists on this subject.A corn rehensivein(l, i ) review of this literature is givenand a good discussion of the current state-of-the-art may be found in (3). Su ffice it here tosay that free-layer treatments are effec tive forthin plates and panels, while constrained-layertreatments are more effect ive for thicker members.Only constrained-layer treatments are considered inthis paper.Cost-effective design of damping-layer treatmentsfor plates and panels frequently requires thatmaterial costs and added weight be held to a mini-mum. This requirement can be met only by judicioustradeoffs between the amount of damping treatmentused and the amount of damping achieved. It isknown intuitively, and from experience, that thedamping achieved by complete coverage of a platewith a viscoelastic damping treatment usually isnot significantly greater than that achieved by apartial coverage extending over some lesser frac-tion of the surface area. However, the designerneeds information that is more specific if thenecessary tradeoffs are to be made intelligently.This information is most useful when presented inthe form of plots, nomographs, or similar designaids.Generation of design aids for plates with con-strained-layer damping treatments is made diff i-cult by the number of variables involved and by theranges of parameter values which must be considered.Because of the large number of different caseswhich must be investigated, use of finite elementmethods is not economically feasible. Accordin l y,an approximate analyt ical method was developed( Bfor predicting the effec tiveness of partial andcomplete constrained-layer damping treatments incontrolling the lower flexural modes of vibrationof rectangular plates of finite extent. Thispaper describes the use of modal analysis tech-niques to obtain experimental verificat ion of theprocedure.A series of experiments was conducted on an edge-fixed rectangular plate with a constrained-layerviscoelastic damping treatment applied over a

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concentric rectangular region of varying size onone side of the member. Impulse testing techniqueswere used to determine the sys tem natural frequen-cies, loss factors, and mode shapes for the firstfive flexural modes of the plate. The test speci-men and test procedures are described in detail,and results are presented and compared with predic-ted values for two of the modes. The mode shapedata obtained confirm the validity of the assump-tion that addition of the damping treatment doesnot alter significantly the flexural mode shapesof the plate. As this assumption is commonly usedin the analysis of plates with damping layers, thisresult should be of particular interest.ANALYTICAL APPROACHThe analytical approach is described in detailin (4). Here, we give only a brief summary asbackground for the discussion to follow.The analysis is based upon an energy balancefor the sys tem which, for sinusoidal oscillationswith frequency 6.1, yields the result

w* = Us + i(l/*s)UD (11TIn this expression, is the maximum systemkinetic energy, US is the maximum stored strainenergy, and UD is the energy dissipated per cyc le.This result is the counterpart of the Rayleighquotient for elastic sys tems. Since the Rayleighquotient has a stationary val ve in the neighbor-hood of the syssem eigenfunctions(5), accurateestimates for w can be obtained from Eq. (1) ifreasonable estimates of the eigenfunctions areavailable. The natural frequency, wn, and lossfactor, n, of the damped sys tem are obtained fromEq. (1) using the relationsrum 11w =n 1 1+-and 11Drl(wn) = -**us

2)

The frequency dependence of the loss factor stemsfrom the frequency dependence of the mechanicalproperties of the viscoelastic damping material.If desired, the system damping can also be ex-pressed in terms of other damping parameters, suchas the logarithmic decrement, A, the fraction ofcritical damping, 5, the amplification at reso-nance, Q, or the decay time At. These parametersare related to the loss factor via the expressions

(4)where f, is the natural frequency in Hz.It was assumed that addition of the damping treat-ment does not alter significantly the mode shapesof the undamped plate and that the latter can beused as an approximation to the flexural modeshapes of the system. The validity of this assump-tion has been demonstrate flayer damping treatments t6aqj pt~,e~a:i~~a?efoundto be valid for the plate specimen with constrainedlayer treatment used in the experiments described

herein. This same assumption is commonly used inthe analysis of constrained-layer damping treat-ments for beams and plates; e.g., see (839).TEST SPECIMENThe test specimen was a 1.59mm (1/16th in.) thickrectangular aluminum plate with overall dimensionsof 0.46m (18 in.) by 0.51m (20 in.). When in-stalled in the support frame, the plate had aworking area of .30m (12 in.) by .34m (13.4 in.),which gives an aspect ratio of 1.11. This aspectratio was chosen to provide reasonable separationbetween the resonant frequencies for the modes ofinterest, thereby minimizing problems with modaloverlap in the data reduction. The working areaof the plate was covered on one side with a con-strained-layer damping treatment consisting of a0.254mm (10 mil) thick layer of #468 AdhesiveTransfer Tape manufactured by 3M Corporation and a0.381mm (15 mil) thick constraining layer ofdead-soft aluminum. The viscoelastic adhesivebonds the various layers together and provides thedamping. The plate specimen and the mechanicalproperties of the damping layer as functions offrequency and temperature were provided by AnatrolCorp. , Cincinnati, Ohio. Space does not permit in-clusion of these properties here; they are givenin (4, loI. The specif ic weight of the damping ma-terial was 10.2kN/m3 (64.81b/ft3), and it wasassumed to be incompressible. In separate tests ,the elastic modulus and Poissons ratio of theplate and constrainpg layer were found to be67.6GN/m2 (9.8 x 10 lb/in ) and 8.33, respectively;the specif ic weight was 26.3kN/m (1671b/ft3).A test frame used in a previous study of s uareplates with free-layer damping treatments ? 7) kasmodified to accommodate the rectangular platespecimen. The frame was made of 25.4mm (lin.)thick by 76.2mm (3in.) wide steel bar stock andconsisted of two parts - an upper and lower half.Each part was fabricated from four lengthsof barmachined smooth on their mating edges and boltedtogether to form a rigid rectangular unit. Themating surfaces of the two halves of the frame weremachined fla t to provide good surface contact withthe plate. Some of the rounded edges of the rolledbar stock were not completely removed in thisprocess, however, and did contribute to a veryslight variation of the boundary along the extremeedges of the plate. Clamping of the specimen wasaccomplished by sandwiching the plate between theupper and lower halves of the frame and boltingthe two halves together with sixteen bolts aroundthe periphery of the plate. To provide uniformclamping, all bolts were tightened to the sametorque using a torque wrench. The plate specimenexhibited some initial curvature so the testspecimen actually was a slightly curved plate con-strained to a flat edge condition by the steelframe. A photograph of the test frame, withspecimen installed, is shown in Figure 1.TEST PROCEDUREImpulse testing techniques (11-14) and a Hewlett-Packard 5423A Structural Dynamics Analyzer wereused to determine the mode shapes and modal para-meters for the first five flexural modes of theplate specimen. The plate was excited by impact-

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with an impulse hammer with an attached PCBModel 3338 force transducer to senseforce imparted to the plate. The plate re-was determined using a PCB 303A miniaturizedter. Signals from both the force trans-and accelerometer were amplified by PCBamplifiers and fed into the two channels ofHP 5423A analyzer to determine the systemfrom which the modal parametersdetermined. A photograph of the overall testis shown in Figure 2.

define the point of excitation andn of the accelerometer, a 6x6 mesh ofzed elements was drawn on the side o f thewithout damping treatment (Figure lb). Thefive mesh nodal points not located on theboundary were used as the points of excita-and accelerometer locations during the ex-For identification, the mesh pointsnumbered as indicated in Figure 3. The xy-coordinates of these nodal points were usedcoordinate table to define the plate configur-in the analyzer.strip of damping treatment was removedall four edges of the fully-coated plate toa narrow margin between the dampingand test frame. The purpose of thiswas to prevent the edges of the dampingfrom rubbing against the frame duringthereby distorting the measured valuesstem damping. This resulted in a dampingcovering 97.3% of the surface of the

plate was then excited at a selected point byhammer and the response at a differentby the accelerometer. For eachof excitation and response points, data fromrepetitions of the impact were averaged totransfer function. To obtain theshapes, the tests were repeated with thelocated at each of the mesh nodalnot on the plate boundary, with the pointkept the same. Cross-correlation ofta was checked by repeating the experimentdifferent points of excitation and the samelocation. A coherence of 0.99, orwas required for data acceptance.

study (15) revealed that, of the severalavailable on the analyzer for determin-the modal parameters, the X-band procedure ismore reliable. Accordingly, this method wasin the present investigation. In this pro-the modal parameters are determined byanalyzer from a curve fit to the transferover a frequency interval spanning thent frequency for the mode of interest. Tose the reliability of the data, experimentsdetermining the frequencies and damping factorsrepeated four times each and experiments tothe mode shapes were repeated twice.was done for each of the f ive modes investi-

second and third modes of the plate specimenfound to be relative ly close, with theirfrequencies approximately 15 Hz apart.nt modes are accounted for in the curve-fitof the analyzer, However, the interaction

between these two modes was minimized in the testsfor the modal parameters by locating the point ofimpact or response measurement on an anti-nodalline for the second mode and on a nodal line forthe third mode, or vise versa.After all testing was completed for the fully-treated plate, a strip of damping treatment 25.4mm(1 in.) wide was removed f rom around the outeredge of the plate, leaving a concentric rectangu-lar partial damping treatment centered at themiddle of the plate, as illustrated in Figure la.Modal parameters and mode shapes were then deter-mined for the partially-treated plate. The processwas repeated until a bare plate condition wasreached. The percentages of the plate surfacecovered by damping treatment during the tests were97% (ful l coverage), 71%, 47%, 27%, 13%, 4% and0% (bare plate).Temperatures in the laboratory ranged from 22C(7ZF)to26C (78F) during the course of theexperiments, with most tests conducted at 24C(75Y). Potential problems arising f rom thedifferent coefficients of thermal expansion ofthe steel frame and aluminum plate were alleviatedby releasing the plate and reclamping it to thesame bolt torques before each set of experiments.Careful investigation revealed that release andreclamping of the plate had negligible ef fect onreproducability of the data.Measurement tables were set up on the analyzer foreach of the fiv e modes of vibration with each ofthe different degrees of damping treatment. Atypical table is shown in Figure 4. These tableslist the natural frequency, percent critical damp-ing, test temperature, impact and accelerometerlocation, and degree of damping treatment. ThePoint No. listings refer to points on the plategrid in Figure 3 and the residues of the transferfunct ion are proportional to the vibrational am-plitude at these points. The shape listings arenormalized modal displacements, which define themode shapes.MODE SHAPESFigures 5 and 6 give overall views of the ex-perimentally-determined mode shapes for the secondand fourth flexural modes of the plate with com-plete (97%) damping treatment coverage. Figures7-10 show these same mode shapes along traverses inthe x and y-directions through an anti-nodal pointfor three different degrees of damping treatment:full treatment (97%), 47% coverage and bare plate.For comparison purposes, the theoretical modeshapes also are shown. In these figures, thenumbers listed in the captions correspond to plategrid points (Figure 3) included in the traverse.These data, which are representat ive of the resultsobtained for the other three modes investigated,indicate clearly that the theoretical mode shapesfor the undamped plate are reasonable approxima-tions to the mode shapes of the damped member, asassumed in the analysis. The zero slope conditionassociated with a clamped edge is not evident inFigures 5-10, but could be made more apparent bytaking more data points in the vic inity of theplate boundary.

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MODAL PARAMETERSFigures 11-14 give a comparison of the predictedand measured values of natural frequencies andloss factors for the first and second flexuralmodes of the plate test specimen. Agreement be-tween theory and experiment was at least as goodfor the other three modes, for which results arenot shown.The properties of the damping material are ratherdependent upon frequency and temperature in theranges involved in the experiments. For each mode,the values of these properties used in the compu-tations were determined at the natural frequency ofthe undamped plate. Since the natural frequencydid not va ry significantly with the percentage ofdamping treatment, the properties of the dampingmaterial were assumed constant for each mode ofvibrat ion. Predicted results based on the dampingmaterial properties at three dif ferent tempera-tures over the range encountered in the experi-ments are presented. The resulting set of threecurves for temperatures of 72F, 75F and 78Fgives some indication of the scatter that could beexpected in the experimental values due to temp-erature variations encountered during the ex-periments.The loss fac tors for the bare plate ranged fromabout 0.005 to 0.010 for the modes considered.These values include radiation losses and the in-herent damping in the plate and within the suppor-ting f rame. Since the loss factor for the bareplate is small , it can be added direc tly to theloss factor resulting from the dam ing treatment toobtain the total sys tem damping 67. Accordingly,the loss factor measured for the bare plate wasadded to the predicted values in Figures 12 and14 in order to obtain a more realistic comparisonbetween theory and experiment.It is difficult to achieve a ful ly clamped edgecondition in experiments, and this investigationproved to be no exception. It is evident fromFigures 11 and 13 that the measured natural fre-quencies of the bare plate were consistentlybelow the theoretical values, which is indicativeof flexib ility in the supports. To compensate forthis ef fec t, and to obtain a more realistic com-parison between theory and experiment, the pre-dicted results were adjusted, as follows. In thecomputations, the maximum stored energy, US, wasreduced by an amount AUcomplete edge fix ity . ? to account for the lack ofhe quantity AUS was chosento bring the predicted and measured natural fre-quencies for the bare plate into agreement, andwas assumed to be constant for all percentages ofdamping treatment. This adjustment yields theresults indicated in Figures 11-14, and broughtthe predicted and measured frequencies and lossfactors into substantial agreement for all fivemodes of vibration. The fac t that this adjustmentyields good agreement for all percentages of damp-ing treatment in strong evidence that the initialdifferences between the theoretical and experimen-tal results were associated with test systemflexibi lity and not with some aspect o f the damp-ing treatment.

Given the difficulties involved in obtaining iden-tical conditions between theory and experiment andthe scatter normally encountered in damping data,the agreement between theory and experiment in thepresent case is considered excellent.CONCLUDING REMARKSIt has been demonstrated that impulse testing andmodal analysis techniques, when used with properattention to detail, can provide values of sys temdamping with sufficient accuracy and consistencyto evaluate analyt ical damping models. The modeshape information obtained confirmed the assumptionthat addition of the damping treatment had littleeffect on the flexural mode shapes of the plate.While this result is for the particular platespecimen and modes of vibration investigated, it isbelieved to hold for other constrained-layer con-figurations for while the flexural rigidity of theconstraining layer is not significantly larger thanthat of the underlying structure, which usually isthe case.Some variation in damping values obtained fromdifferent sets of measurement points was observed.This variation often exceeded the expected scatterband, and seemed to indicate some path dependenceof the damping values. As might be expected, thiseffect was more noticeable for the smaller per-centages of damping treatment coverage. Time didnot permit a more detailed investigation of thispoint.

REFERENCES1.

2.

3.

4.

5.

Nakra, B.C., Vibration Control with Visco-elastic Materials, The Shock and Vibration---Digest, Vol. 8, No. 6, pp. 3-12, 1976.Nelson, F.C. , Techniques for the Design ofHighly Damped Structures, The Shock and---Vibration Digest, Vol. 9, No. 7, pp. 3-11,1977.Rogers, L., ed., Conference on AerospacePolymeric Viscoelastic Damping Technology forthe 1980s, Air Force Flight Dynamics Labora-tory,rt No. AFFDC-TM-78-78-FBA, July ,1978.Hsu, H.Y ., Vibration Analysis of RectangularPlates with Complete and Partial Constrained-Layer Damping Treatments, M.S. Thesis, De-partment of Ocean Engineering, Florida Atlan-tic University, 1983.Meirovitch, C., Analytical Methods in Vibra-tions, The MacMillan Co., Nm,T967.

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Stevens, K.K., Kung, C.H. and Dunn, S.E.,"Damping o f Plates by Partial ViscoelasticCoatings-Part I-Analysis", Proceedings Noise-Con. 81, Raleigh, N.C. , pp. 445-448, June,1981.Dunn, S.E., Kung, C.H. , Jaising, V. andStevens, K.K., "Damping o f Plates by PartialViscoelastic Coatings-Part II-Experimental,"Proceedings Noise-Con. 81, Raleigh, N.C., pp.449-452, June, 1981.

E.M. and McQuillan, R.J ., "Plate Damp-ing by a Constrained Viscoelastic Layer: Par-tial Coverage and Boundary Effects ", BoltBeranek and Newman Report No. 760, 1960.Johnson, C.D. and Kienholz, D.A., "FiniteElement Prediction of Damping in Structureswith Constrained Viscoelastic Layers", AIAA J.,Vol. 20, No. 9, pp. 1284-1290, 1982.Nashif, A.D., "Control of Noise and Vibrationwith Damping Materials", Sound and Vibration,--Vol. 17, No. 7, pp. 28-36, 1983."Dynamic Testing of Mechanical Systems UsingImpulse Testing Techniques", Hewlett PackardApplication Note 140-3, 1972.Ramsey, K.A., "Effective Measurements forStructural Dynamics Testing, Part l", Soundand Vibration, Vol. 9, pp. 24-35, 1975.-Ramsey, K.A., "Effec tive Measurements forStructural Dynamics Testing, Part 2", Soundand Vibration, Vol. 10, pp. 18-31, 1976.-Halvorsen, W.G. and Brown, D.L., "ImpulseTechnique for Structural Frequency ResponseTesting", Sound and Vibration, Vol. 11, pp.--8-21, 1977.Jaising, V.R., "Analysis of Damping LayerTreatments for Plates using ExperimentallyDetermined Mode Shapes", M.S. Thesis,Department of Ocean Engineering, FloridaAtlantic University, 1981.

Figure 2. Experimental Set-Up.

Y

Figure 3. Plate Measurement Grid.

Figure 1. Plate Specimen with (a) PartialDamping Treatment and (b) Measurement Grid.

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Frequen cy: 124.597 Hz.Damping: 3.290%

Residueoint No.9 1.403 K10 2.881 K11 3.463 K12 2.896 K13 1.606 K16 2.843 K17 5.655 K10 6.653 K19 6.167 K

20 3.423 K23 3.373 K24 6.545 K25 8.851 K26 6.775 K27 3.871 K30 2.690 K31 5.779 K32 6.437 K33 6.160 K34 3.231 K37 1.212 K38 2.605 K39 2.937 K40 2.607 K41 1.305 K

Accelerome ter at 25Temperature: 74F

Shape

FIRST MODE (46.57% COVERAGE)

I374.034 m768.292 m923.414 m772.134 m428.162 m758.037 m

1.50791.77401.6442

912.685 m099.407 m

1.74512.36001 JO661 SO320

7 17.234 m1.54101.71641.6426

861.482 m323.291 m694.694 m783.150 m695.127 m348.096 m

Point of impac t was moved to different locations.

Figure 4. Measurement Table.

Figure 5. Plate Mode Shape (2nd Mode).

Figure 6. Plate Mode Shape (4th Mode).

97.3%CovEffACE 46.6%COVCRAGE DANEPIATE THEORETICAL-- _-. - - ___ .Dofsct on/Cantar Deflection

Da Dlatanc a from p atr sdpa In Inohae

SECOND MaoE-DIRECTION (8.9.10.11.12.3.14)Figure 7. Second Mode-X Traverse.

97.3xCOVEFADE CO%&E P& $ THEORETICAL-- --. - - _ _ _ _ .

I.5 Dcflsction/Daflection at 4.50 inch.

1

.s

0

-3

-I

-1.5 L0.M) 2.25 4.50 6.75 9.00 11.25 13.375

Distance from plats ed9* in inchesSEC OND MDDE. Y-DIRECTLON (4.11.18.25.32.39.46)

Figure 8. Second Mode-Y Traverse.

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97.3x 46.6XCovERA0E COVERAOE PEGi THEORETICAL-- --. - _ - - _ _ _

1.5 Defecton/Defect on at 4.0 inch.

2.00 4.00 6.00 6.00 10.00Distance from pIat* edga in Inchew

FOURTH MODE, X-DIREC TION (36.37.30.39.40.41.42)

12.00

Figure 9. Fourth Mode-X -Traverse.

97.3% THEORETICALCOERAOE C&%?OE P%i-- --- ____--_

Deflsction/Dafkction ot 9.0 inch.1.5

2.25 4.50 6.75 9.00 11.25Dlstoncs ram plate edge in lnchsa

FOUR, MODE. Y-DIREC TION (5.12.19.26.33.40.47)

13.375

Figure 10. Fourth Mode-Y Traverse.

9.9,I.199nr

19.19.14.

139II.

Lt.1.9

. 69 3. 4. 69 66 79 9. 9, I..

PER CENT DAMPING TREATMENT COVERAGE

I. :Adjusted9.67

*..9..95..a4..93

6.99..6,

9.996 1. 96 3. 4. si 9; 9; A

PER CENT DAMPING TREATMENT COVERAGEFigure 12. Loss Factor (1st Mode).

I*I 69 4; 5; 9: 9;

PER CENT DAMPING TREATMENT COVERAGEFigure 13. Natural Frequency (2nd Mode).

9.14

..,I ..

. ..I - I I I- I I I . .1 I . i d

9 1, 6, 3. 49 9. 66 79 96 9. 1.9PER CENT DAMPING TREATMENT COVERAGE

Figure 14. Loss Factor (2nd Mode).

Figure 11. Natural Frequency (1st Mode).

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159 Modal Analysis of Plates With Partial Constrained-Layer Damping Treatments