23I SVC23rd International Congress on Sound & Vibration

10-14 July 2016Athens, Greece

EXPERIMENTAL MODAL ANALYSIS AND FINITE ELEMENTMODELLING OF A CONTEMPORARY VIOLIN

Roberto Corradi, Alessandro Liberatore, Stefano MiccoliPolitecnico di Milano - Department of Mechanical Engineering, Via La Masa 1, I-20156 Milano, Italyemail: roberto.corradi@polimi.it

This paper illustrates the present state of a research project aimed at developing and validatinga vibroacoustic numerical model of a violin. The attention is focused on the evolution of thesoundboard and the back plate mechanical characteristics during instrument construction. Exper-imental modal analysis was performed on a contemporary violin at several stages of the musicalinstrument manufacturing process and finite element models, corresponding to the same testedconfigurations, were developed. A preliminary comparison between experimental results andsimulations is presented.

1. Introduction

During the past decades several researches were performed on musical instruments in order tostudy their mechanical behaviour and enhance the sound quality [1, 2]. The design of the violin, aswell as any other musical instrument, has been modified during the centuries and the modern instru-ment is the result of this natural and empirically driven evolution carried out by luthiers. Severalstudies have been conducted on the vibroacoustic behaviour of violins [3, 4, 5], nevertheless a com-plete understanding of the influence of the manufacturing details on the acoustical performance hasnot yet been reached.

This paper illustrates the present state of a research project aimed at developing and validatinga vibroacoustic numerical model of a violin that is based on accurate structural modelling. The at-tention is focused on the evolution of the soundboard and the back plate mechanical characteristicsduring instrument construction. Experimental modal analysis was performed on a contemporary vio-lin at several stages of the manufacturing process. In parallel to the experimental tests, finite elementmodels corresponding to the same tested configurations were developed, starting from the geometryobtained from 3D laser scanner measurements.

Some experimental modal analysis results of the soundboard and the back plate are presented. Apreliminary comparison with finite element simulation, in terms of both natural frequencies and modeshapes, is presented. The final objective is the validation of the finite element models which will bethe basis for the subsequent sound radiation simulation.

2. Experimental modal analysis

Experimental modal analysis tests were performed on the soundboard and the back plate of acontemporary luthiery instrument at nine steps of its construction. The main advantage of followinga step-by-step analysis method is the possibility of investigating the influence of the manufacturingprocess on the dynamic behaviour of the sound radiating components of the violin [6, 7].

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The 23rd International Congress of Sound and Vibration

The instrument was suspended by nylon cables (Fig. 1) so as to reproduce a free boundary con-dition. The surfaces of the soundboard and the back plate were excited, on a 67 nodes grid, ac-cording to the roving hammer technique. A light impact hammer (mass 5 g, sensitivity 22.5mV/N)was used, with a rubber tip, and the output vibration was measured using three piezoelectric uniaxialaccelerometers (mass 0.5 g, sensitivity 1mV/(m/s2), range 9000m/s2, bandwidth 10 kHz). At leastfive repetitions were performed for each excitation position to minimize the influence of uncorrelatedinput disturbances.

Figure 1: Experimental modal analysis set up: finished violin (with damped strings) suspended bynylon cables.

The H1 estimator was adopted to derive the Frequency Response Function (FRF):

Hoi(ω) =Goi(ω)

Gii(ω), (1)

where Goi(ω) is the crosspectral density function between the input force i and the output vibra-tion o, whereas Gii(ω) is the autospectral density function of the input force [8]. The coherencefunction γ2oi(ω), calculated as

γ2oi(ω) =|Goi(ω)|2

Goo(ω)Gii(ω), (2)

indicates the degree of correlation between input and output and defines the frequency range overwhich the FRF data can be considered reliable. Fig. 2 shows, as an example, three mobility FRFs ofthe soundboard of the finished violin.

The mobility FRFs (Fig. 2) show that up to about 1 kHz the coherence is close to one and themodes are clearly visible, so that it is possible to efficiently apply an identification algorithm. Theadopted modal parameters identification procedure is based on a two-step approach.

First, the Least Squares Complex Exponential (LSCE) algorithm [9, 10] is applied to estimate thesystem poles and the corresponding natural frequencies and damping ratios. The LSCE is a time-domain algorithm which requires in input the Impulse Response Functions (IRFs), the latter beingobtained via inverse Fourier transform of the experimental FRFs, for all the considered input-outputpairs.

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Figure 2: Example of experimental mobility FRFs of the soundboard of the finished violin, for threedifferent input-output location pairs.

The second step consists in the classical FRF based curve fitting method (least squares minimiza-tion): according to the idea of modal superposition, the measured FRFs are fitted with an analyti-cal model consisting of a series of second order systems. Least squares minimization operates in auser-defined frequency band which includes a certain number of vibration modes. To reach suitablecurve-fit, high and low frequency residuals are added to the resonant modes [8], so as to account forthe contribution of the modes not included in the selected frequency band.

By following this procedure it was possible to identify the natural frequencies, damping ratiosand mode shapes up to at least 1 kHz for each manufacturing stages. Figure 3 shows two examples ofidentified mobility FRFs superimposed to the corresponding experimental data (free soundboard andfree back plate).

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Figure 3: Comparison between experimental (blue) and identified (dashed red) mobility FRFs: freesoundboard (left) and free back plate (right).

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3. Finite element model

A finite element model of the violin structure was developed in parallel to the experimentaltests, starting from the geometry acquired with a 3D laser scanner (6-axis, arm length 2m, toler-ance 0.03mm). By simulating the violin at the same manufacturing stages of the experimental tests,it is possible to perform a step-by-step validation of the structural model.

A solid model, suitable for FE simulation, was obtained by a reverse engineering process from thecaptured external geometry of the instrument. In Fig. 4 the model of the finished soundboard, withthe “f-holes” and the bass bar, is presented. About 92500 quadratic tetrahedral elements, with averageedge dimension approximately 3mm, where used for this model. Such a large number of elements is aconsequence of the reverse engineering process of the complex shape of the soundboard. Consideringthe reasonable computational time of the eigenvalues/eigenmodes extraction, it was decided not topursue the definition of a FE model with a coarser mesh.

The soundboard and the back plate under study are made respectively of red spruce and maple.Both woods were modelled as linear elastic orthotropic materials with nine elastic constants [11].Comparing the numerical natural frequencies and mode shapes, (obtained from eigenvalue analysis),with the modal parameters identified from the experimental data, it is possible to verify the accuracyof the model. Then it is possible to tune the material parameters, in order to reach the maximumreliability of the structural model of the complete violin.

Figure 4: Finite element model of the finished soundboard.

4. Numerical-experimental comparison

Fig. 5 shows some experimental and numerical results for the finished soundboard. It is possibleto note the good agreement in terms of both natural frequencies and mode shapes [3, 2].

The FE model allows also to investigate the mechanical behaviour of the violin in a higher fre-quency range with respect to that covered by experimental modal analysis. For example, in Fig. 6three modes of the soundboard, at 2039.3Hz, 2975.9Hz and 4097.3Hz, are shown. Note that, at highfrequency, the bass bar strongly affects the lobes distribution, increasing the characteristic structuralasymmetry of the instrument.

The last step of the experimental modal analysis focused on the soundboard and the back plate ofthe finished instrument. Fig. 7 shows some of the well known mode shapes of the violin described inthe literature [2, 5] that were identified on the specific tested instrument.

The finite element model of the finished violin is currently under development.

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(a) f1 = 90.6Hz (b) f2 = 162.6Hz (c) f5 = 359.1Hz (d) f10 = 574.3Hz

(e) f1 = 88.3Hz (f) f2 = 156.2Hz (g) f5 = 353.9Hz (h) f10 = 542.2Hz

Figure 5: 1st, 2nd, 5th and 10th mode of the free soundboard identified from experimental data (top)and simulated with the FE model (bottom).

(a) f40 = 2039.3Hz (b) f62 = 2975.9Hz (c) f85 = 4097.3Hz

Figure 6: High frequency modes of the free soundboard (from FE modelling).

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(a) f1 = 180.6Hz (“B1 mode”) (b) f6 = 282.7Hz (“A0 mode”)

(c) f11 = 448.8Hz (“C1 mode”) (d) f14 = 562.0Hz (“C3 mode”)

Figure 7: Experimental modes of the soundboard and the back plate of the finished violin (labelledaccording to [2]).

5. Conclusions

The research project presented in this paper is focused on the investigation of the evolution ofthe soundboard and the back plate of a contemporary luthiery violin during instrument construction.Experimental modal analysis was performed at different manufacturing stages in order to validate afinite element structural model of the violin to be used for predicting the dynamic response of thesoundboard. In a next step, the structural model will be linked to a sound radiation model of theviolin, which will be used for tone and sound power prediction.

6. Acknowledgements

This project was carried out in the Musical Acoustics Lab of Politecnico di Milano, located in thepremises of the Violin Museum of Cremona. The authors are grateful to luthiers Elena Bardella andSebastiano Ferrari for providing the violin analysed in this research project and to Sara Gonizzi forhelping us with the reverse engineering process.

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REFERENCES

1. Fletcher, N. H. and Rossing, T. D. The physics of musical instruments, Springer, New York, 2nd ed., (1998).

2. Rossing, T. D. The Science of String Instruments, Springer, New York, (2010).

3. Hutchins, C. M. The Acoustics of Violin Plates, Scientific American, 245 (4), 126–135, (1981).

4. Bissinger, G. Some mechanical and acoustical consequences of the violin soundpost, The Journal of theAcoustical Society of America, 97 (5), 3154–3164, (1995).

5. Woodhouse, J. The acoustics of the violin: a review, Reports on Progress in Physics, 77 (11), 1–42, (2014).

6. Corradi, R., Fazioli, P., Marforio, S., Paluello, A., Ripamonti, F. and Squicciarini, G. Modal analysis ofa grand piano soundboard, Proceedings of ISMA 2010 - International Conference on Noise and VibrationEngineering, Leuven, Belgium, 20–22 September, (2010).

7. Corradi, R., Fazioli, P., Miccoli, S. and Squicciarini, G. Modal analysis and finite element modellingof a grand piano soundboard, Proceedings of the 8th International Conference on Structural Dynamics,EURODYN 2011, Leuven, Belgium, 4–6 July, (2011).

8. Ewins, D. J. Modal Testing: Theory, Practice and Application, Wiley, New York, 2nd ed., (2001).

9. Maia, N. M. and Silva, J. M. Theoretical and Experimental Modal Analysis, Wiley, New York, (1997).

10. Heylen, W., Lammens, S. and Sas, P. Modal Analysis Theory and Testing, K. U. Leuven, (1998).

11. Bucur, V. Acoustics of wood, Springer, New York, 2nd ed., (2006).

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