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Viscoelastic Internal Damping Model for Sound Production: Instrument Physical Properties

Explore the importance of internal damping in musical instruments with a finite-difference model. Learn about wood damping, Young's modulus correlation, and viscoelastic properties in modeling sound production.

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Viscoelastic Internal Damping Model for Sound Production: Instrument Physical Properties

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  1. Viscoelastic internal damping finite-difference model for musical instruments physical model sound production Rolf Bader Institute ofSystematicMusicology University of Hamburg

  2. Importanceof internal damping Physical Model of Steinway M piano Single string, with normal damping Normal soundboarddampingwithoneparameterfor all frequencies 2) Extreme soundboarddamping 3) Nosoundboarddamping

  3. Soundboard wooddamping From: I. Bremaud: What do we know on “ resonance wood ”” properties? Selective review and ongoing research. Proceedings of the Acoustics 2012 Nantes Conference.

  4. Wood dampingcorrelationwithYoung‘smodulus, not withdensity From: Iris Brémaud, Joseph Gril& Bernard Thibaut: Anisotropy of wood vibrational properties: dependence on grain angle and review of literature data. Wood SciTechnol (2011) 45:735–754. DOI 10.1007/s00226-010-0393-8

  5. Microfibergrain angle dependency Normal wood Compressionwood From: Iris Bremaud, Julien Ruelle, Anne Thibaut, Bernard Thibaut: Changes in viscoelastic vibrational properties between compression and normal wood : roles of microbril angle andoflignin. Holzforschung 67(1), 75-85, 2013.

  6. Leatherglasstransitiondamping From: SujeeviniJeyapalina. Studies on the hydro-thermal andviscoelasticpropertiesofleather. PhD, Leichester 2004.

  7. Models of internal damping • Thermal losses (Zener, Harris) • Molecularconformationalchanges (Glassstone, Laidler, Eyring) From: Alfred D. French and Glenn P. Johnson: Advanced conformational energy surfaces for cellobiose. Cellulose 11: 449–462, 2004.

  8. Thermal lossesproduce band gaps From: A.D. Pierce: Intrinsic damping, relaxation processes, and internal friction in vibrating systems. POMA 9, 2010

  9. Kinds ofdamping Onedampingparameter Frequency-dependentdampingspectrum Spectrawhenknocking on woodenplate 30x30cm (left) beforeand (right) after laquering

  10. Modellingthefrequency band gapofdamping:Maxwell / Kelvin-Voigt model • Maxwell model • Kelvin-Voigt model • ComplexYoung‘smodul (FEM, FDM) Maxwell Kelvin-Voigt

  11. Fractionalmodel, FEM calculation From: Sebastian Müller, Markus Kästner , Jörg Brummund& Volker Ulbricht: On the numerical handling of fractional viscoelastic material models in a FE analysis. ComputMech (2013) 51:999–1012. DOI 10.1007/s00466-012-0783-x

  12. ComplexYoung‘smodulus Complex stress-strainmodel: σ : stress ε: strain Phase-shiftbetween stress andstrain: EI, ER: imaginaryand real partsof E

  13. Model Complex stress-strainmodel: σ : stress ε: strain α: damping ω: frequency s: complexfrequency u: displacement A: Amplitude μ: dampingconstant Transfer (1) into time domain (multiplication in frequencydomain -> convolution in time domain) Inverse Laplace transform: γ: integrationconstantneededforconvergence, influencesdampingstrengthDs

  14. Model Membrane Differential Equation u: displacement T(x,y): tension μ: areadensity D: dampingcoefficient Viscoelasticmembrane differential equation:

  15. Discretemodel σt : stress at discrete time point t εt : strain at discrete time point t N: numberof time points r: sample rate hτ: dampingfunction at time pointτ Discretefrequency: hτneedtobe real, therefore:

  16. Model parameters Input parameter: • Inverse Laplace integration (damping) exponentγ • Dampingamplitude A = Re(Ek) • Lengthof h (numberofperiods) Output parameters: • Dampingexponentμ • Filter constant Q Contraints: • Avoid (orcontrol) overflow • Reducecomputationcost

  17. Computation time

  18. Result Target f = 4174 Hz • Lengthof h: • 10periodsof f • 25periodsof f • 35periodsof f • 50periodsof f • Eachcaseγ = n x [10: 1/(n 10) ; 25: 1/(25 n) ; 35: 1/(25 n); 50: 1/(50 n)] with n = 1, 2, 3, … 10 • Eachcase Re{Ek} = n x 0.0003 with n = 0, 2, 3, … 10

  19. Dampingof f = 4174 Hz Example: 50 periods Re{E(s)} = 1 / 0.0003 Partial decayμfromsimulation

  20. Masspoint, Spetra Non-exponentialdecayleadstosidebandswith high γ.

  21. 20 kHz 10 periods, Re{E(s)} = 0.0003 γ = .1 20 Hz Time 400 ms 0 ms 20 kHz 10 periods, Re{E(s)} = 0.0003 γ = .01 20 Hz 400 ms 0 ms Time 20 kHz 50 periods, Re{E(s)} = 0.0003 γ = .01 20 Hz

  22. 20 kHz 50 periods, Re{E(s)} = 0.0003 γ = .01 20 Hz Time 0 ms 400 ms 20 kHz 50 periods, Re{E(s)} = 0.0003 γ = .002 20 Hz 400 ms 0 ms Time

  23. 10 periods

  24. 25 periods

  25. 35 periods

  26. 50 periods

  27. Energysupplybyovershooting

  28. Examples Drum withlong fundamental: withviscosity noviscosity Piano soundboard (Steinway M-Model) knocking: noviscosity withviscosity 125 Hz damped 250 Hz damped 1000 Hz damped Piano soundboardonestringplayed: noviscosity withviscosity

  29. Summery Viscoelasticitydoes not leadto an exponentialdecayofsinglefrequencies. Threeparameterstunable: Inverse Laplace transform real value: γ ComplexYoung‘smodulus: Re{E(s)} Lengthof h (amountofperiods) • Maximum μindependentoflengthof h • Maximum μdependens on targetfrequency • Q isgettingsharperwithsmallerγ

  30. Future work • Fit modelto experimental data • Implement flexible filterfrequenciesusing multiple hτ • Add controlledenergysupplytomodelinteractionof multiple musicalinstrumentpartsby time-dependent h

  31. String instrumentdampingdependency on antivibrationparameterρ/c From: Shigeru Yoshikawa:Acoustical classification of woods for string instruments. JASA 122 (1), 568-573, 2012.

  32. Vacuumexperiment Thünen Institute, Hamburg 30x30cm plate, sine sweep Radiation (microphone) Energy on plate (piezo) Fig. 8: Amplitudes relative to 760 Torr of the piezo measurements on the wooden plate in dB. Contrary to the clear picture with the microphone measurements, the vibrational energy on the plate is much less consistent and does not follow simple rules. In the high frequency range from about 2 kHz the tendency is quite clear, the negative dB values point to an increase of energy on the plate with decreasing vacuum consistent with the expectation. Still in the low and middle range the behavior is much more complex and not yet understood.

  33. Model Exampleforonly a singlefrequencydamped: δ: Dirac delta E0: Absolute Young‘smodulus All non-viscoelasticfrequenciesaddto a Dirac delta at theτ=0. Onlythedampingfrequencyleadsto a time series in τ. Frequency-dependentdampingleadsto integral overspectrum

  34. f = 306 Hz

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