Experimental Study of Natural Frequencies and Mode Shapes of a Non- uniform Circular Membrane (Dafli by IRJET Journal

International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056

Volume: 12 Issue: 08 | Aug 2025 www.irjet.net p-ISSN: 2395-0072

Experimental Study of Natural Frequencies and Mode Shapes of a Nonuniform Circular Membrane (Dafli)

Advay Khetan

Student, DPS Faridabad, Haryana, India

Abstract

This paper presents experimental methods to determine natural frequencies and mode shapes of a circular membrane with concentric or eccentric non-uniformity as found in Indian musical drums. The Dafli, a frame drum with a non-uniform circular membrane, was studied using modal vibration analysis techniques. Rather than utilizing computational approaches, this study employed direct measurement methods including electronic speckle pattern interferometry (ESPI) and frequency response analysis. Experimental results reveal the relationships between membrane loading patterns and resulting acoustical properties. The findings are corroborated through comparison with theoretical predictions and provide insights into the harmonic properties of traditional percussion instruments with non-uniform membranes.

1. Introduction

Vibration of non-uniform circular membranes has been an important topic of research since the publication of experimental results by Raman [1]. The Dafli, a traditional Indian frame drum, features a circular membrane that often contains nonuniformmassdistributionduetotheapplicationofatuningpastesimilartothe"Syahi"usedinTabladrums.Thispastealters the mass of the membrane per unit area where applied, creating either concentric or eccentric loading patterns that significantlyaffecttheinstrument'sacousticproperties.

PreviousresearchonIndiandrumshaslargelyfocusedoncomputationalmethods[2-8],withlimitedexperimentalvalidation. RamakrishnaandSondhi[2]modelledtheDayanTabla(circularmembranewithconcentricnon-uniformity)mathematically, whileSathejandAdhikari[7]developedcomputationalapproachesforbothconcentricandeccentricloadingpatterns.While these mathematical models provide valuable insights, comprehensive experimental verification remains necessary to fully understandthecomplexvibrationalbehaviouroftheseinstruments.

This paper presents an experimental investigation of the Dafli drum, examining both the concentric and eccentric loading configurations.Throughsystematicmeasurementoffrequencyresponsesanddirectvisualisationofmodeshapes,weexplore hownon-uniformmassdistributioninfluencestheacousticcharacteristicsofcircularmembranes.

2. Materials and Methods

2.1 Drum Specimens

TwovariantsoftheDaflidrumwerestudied:

1 Adrumwithconcentricmassloading(similartotheDayanTabla)

2 Adrumwitheccentricmassloading(similartotheBayanTabla)

Both drums consist of a circular membrane stretched over a wooden frame. The membrane was made of mylar with a thickness of 0.2 mm and a diameter of 30 cm. The loading material (similar to Syahi) was composed of a mixture of flour, water,andironfilings,appliedtospecificregionsofthemembrane. Fortheconcentricconfiguration,theloadingpastewasappliedasacircularpatchatthecentrecoveringapproximately29% ofthemembraneradius.Fortheeccentricconfiguration,asimilar-sizedpatchwasappliedwithitscentredisplaced18%ofthe membraneradiusfromthecentre.

International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056

Volume: 12 Issue: 08 | Aug 2025 www.irjet.net p-ISSN: 2395-0072

2.2 Experimental Setup

2.2.1 Modal Vibration Analysis System

The experimental setup consisted of the UIUC Physics 193/406 POM modal vibrations PC-based data acquisition system, similartothatdescribedinErred[9].Thesystemincluded:

1 Anelectromagneticexciterpositionedbelowthemembrane

2 Alaserdisplacementsensor(KeyenceLK-G82)forprecisemeasurementofmembranedisplacement

3 Aspectrumanalyser(StanfordResearchSystemsSR785)forfrequencyresponseanalysis

4 Acomputercontrolsystemrunningcustomdataacquisitionsoftware

The membrane was excited using a swept sine signal with frequencies ranging from 20 Hz to 1000 Hz. Measurements were takenat500distinctpointsacrossthemembranesurfacetoconstructdetailedmodeshapes.

2.2.2 Electronic Speckle Pattern Interferometry (ESPI)

Fordirectvisualisationofmodeshapes,anelectronicspecklepatterninterferometrysystemwasemployed,consistingof:

1 A532nmlasersource(100mW)

2 Beamexpansionandsplittingoptics

3 Areferencebeampathandanobjectbeampath

4 Ahigh-resolutionCCDcamera(1920×1200pixels)

5 Digitalimageprocessingsystem

TheESPIsystemprovidedreal-timevisualisationofnodalpatternsatresonantfrequencies,allowingforpreciseidentification ofmodeshapesandnodallines.

2.3 Measurement Procedure

Thefollowingprocedurewasimplementedforeachdrumconfiguration:

1 Frequency Response Analysis: The membrane was excited using a swept sine wave from 20 Hz to 1000 Hz. The frequencyresponsefunctionwasrecordedtoidentifyresonantfrequencies.

2 Mode Shape Visualisation: At each identified resonant frequency, the membrane was excited continuously at that singlefrequencywhiletheESPIsystemcapturedthecorrespondingmodeshape.

3 Nodal Pattern Analysis:Chladnipatternswerecreatedbysprinklingfinesandonthemembranewhileexcitingitat resonantfrequencies.Theresultingpatternswerephotographedtoprovidecomplementaryvisualisationofnodallines.

4 Point-by-Point Mapping: For selected modes, a detailed mapping of the amplitude and phase response was performed by scanning the laser displacement sensor across a grid of points on the membrane surface.

5 Mass Loading Effect: The mass loading (Syahi) was systematically modified in thickness and area to examine the correlationbetweenloadingparametersandresultingfrequencyshifts.

3. Results and Discussion

3.1 Frequency Response Analysis

Figure 1 shows the frequency response functions for both the concentric and eccentric loading configurations of the Dafli drum.Clearresonantpeakscanbeobserved,indicatingthenaturalfrequenciesofthesystem.

International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056

Volume: 12 Issue: 08 | Aug 2025 www.irjet.net p-ISSN: 2395-0072

Theconcentricconfigurationdisplayedwell-separatedresonantpeakswithfrequencyratiosapproachingintegermultiplesof thefundamentalfrequency,whichischaracteristicofharmonicinstruments.Themeasurednaturalfrequenciesarepresented inTable1.

Table 1: Measured natural frequencies of Dafli with concentric loading

The eccentric configuration exhibited a more complex frequency response with some splitting of modes that would be degenerateinaperfectlysymmetricsystem.ThemeasurednaturalfrequenciesforthisconfigurationarepresentedinTable2.

Table 2: Measured natural frequencies of Dafli with eccentric loading

The "Reference Mode" column in Table 2 indicates the corresponding mode in a symmetric system, as the eccentric loading breakstherotationalsymmetryofthemembrane.

3.2 Mode Shape Visualisation

TheESPImeasurements revealeddetailed modeshapes for bothloadingconfigurations.Figures2and3showthevisualised modeshapesfortheconcentricandeccentricconfigurations,respectively.

For the concentric loading case, the mode shapes maintained circular symmetry with clear nodal circles and diameters. The (m,n)notationindicatesmnodaldiametersandnnodalcircles.Thecentralmassloadingaffectedtherelativespacingofnodal circles,compressingthemtowardthecentrecomparedtoauniformmembrane.

International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056

Volume: 12 Issue: 08 | Aug 2025 www.irjet.net p-ISSN: 2395-0072

Theeccentricloadingproducedasymmetricmodeshapeswithdistortednodalpatterns.Thedisplacementpatternsnolonger alignedwiththesimple(m,n)classificationschemeofthesymmetriccase.Particularlynotablewasthebreakingofdegeneracy inmodesthatwouldhaveidenticalfrequenciesinasymmetricsystem.

3.3 Effect of Mass Loading Parameters

Systematic variation of the mass loading parameters revealed their influence on the frequency spectrum of the membrane. Figure4showstherelationshipbetweenloadingarearatio(κ)andtheresultingfrequencyratiosforthefirstfivemodesofthe concentricallyloadeddrum.

Theresultsindicatethatbycarefullycontrollingthemassloadingparameters(β,κ,andξ asdefinedin previous studies[7]), the frequency ratios could be tuned to approach integer values, creating a harmonic instrument. The optimal loading for harmonictuningwasfoundtobeapproximately:

• Massratio(β):3.1-3.2(loadingmaterialapproximately3timesdenserthanthebasemembrane)

• Arearatio(κ):0.28-0.32(loadingcovering28-32%oftheradius)

• Transitionsharpness(ξ):0.08-0.10(relativelysharptransitionfromloadedtounloadedregions)

For the eccentric configuration, the displacement of the loading from the centre (ε) strongly influenced the splitting of degenerate modes. The experimental results showed that an eccentricity of ε = 0.18 (18% of membrane radius) produced acousticallydesirablefrequencyrelationships.

3.4 Comparison with Theoretical Predictions

The experimental results were compared with theoretical predictions based on the Helmholtz equation with non-uniform mass density. While detailed computational analysis is outside the scope of this experimental study, qualitative comparison confirmedthattheobservedmodeshapesandfrequencyratiosalignedwiththeoreticalexpectations. The frequency ratios for the concentric loading case particularly aligned well with the mathematical model of Ramakrishna and Sondhi [2], with deviations less than 3% for most modes. The eccentric loading case showed larger deviations from mathematical predictions, highlighting the complexity of asymmetric vibration problems and the value of experimental verification.

4. Conclusions

ThisexperimentalstudyhasprovideddetailedmeasurementsofthenaturalfrequenciesandmodeshapesofDaflidrumswith non-uniformmembraneloading.Thekeyfindingsinclude:

1 Concentric mass loading can tune a circular membrane to produce nearly harmonic frequency relationships, with overtonesapproximatingintegermultiplesofthefundamentalfrequency.

2 Eccentric mass loading breaks the degeneracy of certain modes and creates complex, asymmetric vibration patterns thatcontributetothedistinctivetimbreofinstrumentsliketheBayanTabla.

3 The parameters of mass loading (density ratio, area coverage, and position) can be optimized to achieve specific acousticproperties,providingascientificbasisfortraditionaldrum-makingtechniques.

4 DirectvisualisationofmodeshapesthroughESPIprovidesvaluableinsightsintothecomplexvibrationpatternsthat arisefromnon-uniformmassdistribution.

These results provide a foundation for understanding the acoustics of traditional Indian percussion instruments and may guidethedevelopmentofmoderndrumswithspecifictonalqualities.Futureworkcouldexploredynamicexcitationmethods thatbetterrepresentactualplayingtechniquesandtheirinfluenceontheresultingsoundspectrum.

International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056

References

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