Predictive simulation of mechanical behavior from 3D laser scans of violin
plates
Mirco PEZZOLI(1) , Riccardo R. DE LUCIA(2) , Fabio ANTONACCI(3) , Augusto SARTI(4)(1) Politecnico di Milano, Italy, mirco.pezzoli@polimi.it(2) Politecnico di Milano, Italy, riccardoroberto.delucia@polimi.it(3) Politecnico di Milano, Italy, fabio.antonacci@polimi.it(4) Politecnico di Milano, Italy, augusto.sarti@polimi.it

Abstract
In this paper we present a methodology for the predictive simulation of the vibrometric behaviour of a violin
plate. The 3D outer shape of the plate is acquired by means of a 3D laser scanner and then smoothed in orderto remove artefacts and details that are unnecessary for the acoustics simulation. The thickness of the plate isincorporated into the model through a technique that receives as input the thickness sampled at some points ofthe plate and interpolates it over the entire surface. We validate this 3D reconstruction technique by comparing
the vibrometric behaviour of the 3D model with data measured on the reference plate, and with simulations on
a model with uniform thickness.
Keywords: violin, vibration, simulation

1

INTRODUCTION

The acoustics of bowed strings musical instruments is greatly influenced by vibrations of the top and backplates. The dimensions and arching of the plates currently in use has been inherited from reference masters,
among which two of the most representative are Stradivarius and Guarneri del Gesú. Some of the instruments bythese masters are preserved in museums. On these instruments only non invasive analysis can be conducted, forclear reasons of preservation of the masterpiece. Among the geometric measurements that can be conducted,
a relevant one is the thickness of the plates [1, 2], which can be measured either by specialized thickness
gauges or through high resolution Computer Tomography (CT). The first method is not expensive but presents
the disadvantage that measurements in a limited number of points, if permitted, can be taken. The use of CT
is, instead, very expensive but presents the advantage of being per-sè non-invasive. It is however worth noticingthat this measurement requires to move the instrument to specialized centers, which could be an insuperable
obstacle in many cases. The knowledge of the thickness of the plate is, however, fundamental for determining
the vibrational behavior of the instrument, i.e. the modal shapes and the eigenfrequencies [3, 4, 5].
In this work we develop a methodology for the modeling of the vibrational behavior of the plates of a violinfrom the scanning of the outer surface of the plates and the knowledge of the thickness at some reference
points, measured by a thickness gauge or by former knowledge available in the literature. The reconstructionof the 3D of the plate from the outer surface proceeds through modeling the inner surface of the plate as thecomposition of two separate regions. The region of the inner surface close to the edge is flat, and therefore it
determines a plane on which the whole plate can lie. Conversely, the central part of the inner surface exhibitsa non uniform thickness. We assume that the thickness is known at some reference points, and we propose
here a methodology for the interpolation of the thickness on a regular grid that covers the whole central area.
The combined knowledge of the outer and inner surfaces makes it possible to reconstruct the three-dimensional
geometry of the plate.
The 3D model is then used for the simulation of the mechanical behavior of the plate through Finite Element
analysis (FEM) [6]. The proposed methodology is validated by means of simulations: a back plate is scanned
through a 3D laser scanner over the entire surface (inner and outer surfaces). The outer surface is then used

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Figure 1. The violin back plate acquired by the 3D laser scanner.
for reconstructing the 3D structure through the proposed methodology. The eigenfrequencies, modal shapes of
the 3D model are compared with those obtained on the original 3D structure and on a 3D model with uniform
thickness. Results demonstrate that the 3D model obtained through the proposed methodology yields moreaccurate results with respect to the model with uniform thickness.
It is well known in the literature [7] that the eigenfrequencies and modal shapes of the free plates are notrelated to those of the assembled violin body. Nonetheless, the use of free plates gives us the possibility tovalidate the reconstruction methodology in a controlled scenario. Moreover, it is important to notice that the
eigenfrequencies of the free plates are regarded by violin makers as an important clue for determining the
behavior of the assembled violin. Consequently, during the construction of the instrument, the final arching andthickness is decided based on acoustic or vibrational tests, aimed at tuning the eigenfrequency of some modes
in predetermined ranges.
The rest of the paper is organized as follows: Sec. 2 describes the method adopted for the reconstruction of the
3D structure from the outer surface. Sec. 3 discusses the technique adopted for the mechanical simulation of
the reconstructed surface. Finally, Sec. 4 draws some conclusions.

2

RECONSTRUCTION OF THE 3D STRUCTURE FROM THE OUTER SURFACEIn this section we propose a technique for obtaining a reconstruction of the 3D shape of a violin plate froma the 3D scan of its outer surface. The method receives as input the plate outer surface scan and using the
thickness measured at some points, it returns as output a reconstruction of the entire plate 3D geometry.
2.1 Violin plate scanning and mesh generation
We briefly introduce the measurement process of the 3D geometry of the plate from laser scanning. A violinback plate with definite geometrical properties (outline, arch and thickness) is employed (see Fig.1). The lengthof the body is 376[mm] and the upper and lower radii are approximately 83[mm] and 103[mm], respectively.
The 3D geometry of the violin plate is acquired using a Romer ABSOLUTE ARM laser scanner with 0.01[mm]resolution and PolyworksTM acquisition software by InnovMetric. It is worth noting that the acquired 3D geometry and its mesh representation present unavoidable imperfections and small holes that are removed in a
pre-processing stage before exporting the polygon data in an STL file. A scan of the entire plate has been
performed and considered as the groundtruth in the evaluation of our reconstruction technique.
2.2 Reconstruction algorithm
Our goal is to estimate the inner surface from the outer surface scan and then merge the two faces in one single
mesh obtaining the reconstruction of the entire plate geometry.

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Figure 2. (a) Thickness measurements sampling points (white) on the violin plate. (b) External boundary
estimated in step 1. (c) Curved region boundary (red) and flat region points (black) computed in step 3. (d)Inner surface point cloud given by step 4. (e) Inner surface mesh build in step 5. (f) Reconstructed plate meshgenerated in step 6
In order to correctly reconstruct the structure of the violin plate, it is essential to model the inner surface
carefully. According to its curvature profile, the inner face can be subdivided in two main regions: the flat
one, which follows the plate border and a curved central area. The latter determines the non uniform thickness
profile of the plate.
Our method determines the thickness of the central region interpolating the inner surface point cloud from a setof measurements given by the user.
The reconstruction procedure is summarized in algorithm 1.
Algorithm 1 Violin Plate Surface Reconstruction
Input : Outer Surface Plate Point CloudThickness Samples
Steps:
1. External Boundary Estimation
2. Inner Surface Plane Estimation
3. Curved Region Boundary Estimation
4. Thickness Interpolation
5. Inner Surface Mesh Build
6. Outer and Inner Surfaces Union
Output: Reconstructed Violin Plate Mesh

1. External Boundary Estimation Given the point cloud of the outer surface mesh, the first step requires
the identification of the contour of the outer surface. This is accomplished by means of the well-known alphashapes algorithm [8]. The contour is used in step 3 to determine the curved region boundary.

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2. Inner Surface Plane Estimation In the global three dimensional Cartesian coordinate system the outersurface ideally lies on the x, y plane looking toward the z+ direction. Unfortunately, sampled plates may present
deformations along the three axis. As a result, the plate lies on a different plane with respect to the x, y plane.
In this step, we determine this plane employing a polynomial fitting of second degree in x and third degree
in y. The point cloud is then projected onto the lying plane, and rigidly translated to obtain the inner surface
plane with the required thickness along the edges. This step is essential to obtain the flat region along the edgeof the inner surface.
3. Curved Region Boundary Estimation In order to characterize the curved region of the inner surface, we
manually remove the spikes from the the contour (step 1) and we fit a smooth curve on the remaining points
using a spline kernel [9]. The identified curve is used as a boundary between the flat and curved regions andits location is determined by the flat area extension. The flat region width differs slightly from plate to plate,according to the choices of the violin maker. In general, the width profile follows the plate design and it iswider in correspondence of the spikes and on the upper and lower edges (see Fig.2(c)) Given the average, upperand lower flat region widths we can correctly identify the boundary of the curved surface area.
4. Thickness Interpolation Once the curved central region is identified in (step 3), we proceed to the interpolation of the thickness from the sampled points on a regular and dense grid. The measured values are used
to drive an interpolation algorithm [10] on the surface points. This operation computes the thickness so thatthe transition from the flat region to the curved area is smooth. The interpolated thickness map is used forcomputing the actual z coordinate of the inner surface point cloud from the outer surface z coordinates and thenormal directions, as shown in Fig. 2(d).
5. Inner Surface Mesh Build The output of step 4 consists in the final inner surface point cloud determinedby the thickness interpolation of the curved region. The data is used as a set of vertices and a triangulation
algorithm [11, 12] determines the mesh of the inner surface, as shown in Fig. 2(e). Note that after the triangulation of the vertices has been performed, a further cleaning of the output mesh may be required, based on themeshing process accuracy.
6.Outer and Inner Surfaces Union Finally, the mesh of the entire plate is obtained merging the outer andinner surfaces. This can be easily done with standard 3D computer graphics software, such as Blender[13]. The
selection of the mesh edge loops and the connection of the outer and inner edges can be accomplished withbuilt in functions. The unified output mesh of our procedure, consisting in the merged outer and inner surfaces(see Fig. 2(f)), represents a reconstruction of the entire violin plate geometry that can be employed for accurate
simulations and analysis, as shown in the next section.

3

MECHANICAL SIMULATION OF THE RECONSTRUCTED SURFACEIn this section we discuss the results of a mechanical simulation of the reconstructed violin plate. Our reconstruction procedure is compared to the results of a vanilla reconstruction method, consisting in a plate withuniform thickness.
We reconstruct the entire back plate surface of Fig.1 using its outer surface. We follow the methodology
described in 2.2, where the thickness of the curved region has been sampled in 20 points (see Fig. 2(a)). The
average thickness of the flat region is set to 3[mm] and as for the width we measured 30,4[mm] and 16.9[mm]for the upper and lower edges, respectively. The average width for the remaining part of the contour is set to6[mm]. The plate with uniform thickness has been obtained by rigidly translating the outer surface by 3[mm]along the z direction.

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(a)

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Figure 3. (a) Scanned plate mode 1. (b) Scanned plate mode 2. (c) Scanned plate mode 3. (d) Uniform thickness reconstruction mode 1. (e) Uniform thickness reconstruction mode 2. (f) Uniform thickness reconstruction
mode 5. (g) Proposed reconstruction mode 1. (h) Proposed reconstruction mode 2. (i) Proposed reconstruction
mode 5.
We analyze the mechanical behavior of the three plates (reference, vanilla and proposed reconstructions) withfree boundary conditions through an FEM simulation performed with Comsol Multiphysics R [14] software. For
each plate, we import the polygon mesh data and a tetrahedron mesh is automatically generated by the software.
In order to accurately simulate the vibration behavior of the plate, the material properties of the object must be
carefully set. In this case, the back plate is made of spruce wood, whose elastic properties along the three axes
are shown in Table 1 [15].
Table 1. Values of the orthotropic properties of the simulated material
Young’s modulus
Ex = 12.6[GPa]Ey /Ex = 0.132
Ez /Ex = 0.065

Rigidity modulus
Gxy /Ex = 0.111
Gyz /Ex = 0.021
Gxz /Ex = 0.063

Poisson’s ratioµxy = 0.424µyz = 0.774µxz = 0.476

As a first study, we compare the eigenfrequencies of three modes that are considered important by violinmakers, which are namely, mode 1, mode 2 (the so called “cross-mode") and mode 5 (also known as “ringmode"). In Fig. 3 the mode shapes of the eigenfrequencies in analysis are depicted for the scanned surface (Fig. 3(a),(b),(c)), the uniform thickness reconstruction (Fig. 3(d),(e),(f)) and for the surface reconstructed7546

through the proposed methodology (Fig. 3(g),(h),(i)).
In order to evaluate the effectiveness of the proposed reconstruction technique, we consider the absolute error
in Hz computed as the absolute value of the difference between the eigenfrequencies obtained through themechanical simulation of the scanned back plate and the ones given by the reconstructed plates.
The eigenfrequencies of the three simulations are reported in Table 2, along with the relative and percentage
error with respect to the reference plate. From the values reported in columns Eu and E p of Table 2, it ispossible to notice that the plate reconstructed with the proposed process is characterized by eigenfrequencies
that are closer to the reference values with respect to a plate reconstructed with uniform thickness. In particular,
the error obtained with our reconstruction technique E p is reduced with respect to the error given by a plate of
uniform thickness Eu for all the eigenfrequencies considered.
This simulation proves the importance of an accurate thickness reconstruction in the context of eigenfrequency
analysis. More precisely, our methodology is able to better approximate the eigenfrequencies of an actual violin
plate, improving significantly the simulation effectiveness.
Table 2. Eigenfrequency values and the relative error Eu of the uniform thickness plate and the proposed
methodology E p
Mode 1 [Hz]Mode 2 [Hz]Mode 5 [Hz]4

Scanned143.51224.08484.43

Uniform
128.94207.99480.68Proposed
140.40222.43480.84Eu [Hz]14.57 (10.15%)
16.09 (7.18%)
3.75 (0.77%)

E p [Hz]3.11 (2.17%)
1.65 (0.74%)
3.59 (0.74%)

CONCLUSIONSIn this paper we proposed a methodology for reconstructing the entire surface of a violin plate when only thescan of the outer surface and thickness measurements at a few sample points are available. The core of ourprocedure, described in Sec. 2, consists in the reconstruction of the inner surface from the outer surface and thesampled thickness. The inner surface is estimated considering its typical geometry and the characteristics of theanalyzed object. Therefore, the reconstructed plate geometry follows the actual non uniform thickness profile ofthe violin plate thanks to an interpolation of the measured thickness values.
The effectiveness of the reconstruction methodology is studied in Sec. 3, where a numerical simulation of
the mechanical behavior is proposed. We analyzed the three modes that are reckoned by violinmakers to be
the most important, which are namely the first, second and fifth modes of vibration of the plate. The obtained
eigenfrequencies are compared with the ones given by the simulation of the entirely scanned plate. Moreover, inorder to asses the improvements of our methodology we compared the eigenfrequency analysis of our plate witha plate with uniform thickness. The results suggest that our reconstruction succeed in improving the accuracyof the eigenfrequency analysis with respect to the uniform method. This can be deduced from the analysis of
the error, expressed as the distance between the actual eigenfrequency, of the simulation results. Therefore, theproposed procedure is introduced as a promising tool for performing more accurate analysis, especially when
there is no possibility of an entire geometry scan, as in the case of historical violins.

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ACKNOWLEDGEMENTS
The authors would like to thank Mr. Gioele Greco for the support in the reconstruction of the 3D geometry
from the outer surface.
Authors acknowledge the support of the Cultural District of Violin Making of the city of Cremona.

REFERENCES[1] M. Tinnsten and P. Carlsson, “Numerical optimization of violin top plates,” Acta Acustica united with Acustica, vol. 88, no. 2, pp. 278–285, 2002.
[2] N.-E. Molin, L.-E. Lindgren, and E. V. Jansson, “Parameters of violin plates and their influence on theplate modes,” The Journal of the Acoustical Society of America, vol. 83, no. 1, pp. 281–291, 1988.
[3] C. Gough, “The violin: Chladni patterns, plates, shells and sounds,” The European Physical Journal Special
Topics, vol. 145, no. 1, pp. 77–101, 2007.
[4] G. Knott, Y. Shin, and M. Chargin, “A modal analysis of the violin,” Finite Elements in Analysis andDesign, vol. 5, no. 3, pp. 269–279, 1989.
[5] J. Bretos, C. Santamarıa, and J. A. Moral, “Vibrational patterns and frequency responses of the free plates
and box of a violin obtained by finite element analysis,” The Journal of the Acoustical Society of America,
vol. 105, no. 3, pp. 1942–1950, 1999.
[6] M. Elejabarrieta, A. Ezcurra, and C. Santamaría, “Vibrational behaviour of the guitar soundboard analysedby the finite element method,” Acta Acustica united with Acustica, vol. 87, no. 1, pp. 128–136, 2001.
[7] N. H. Fletcher and T. D. Rossing, The physics of musical instruments. Springer Science & Business Media,
2012.
[8] F. Bernardini and C. L. Bajaj, “Sampling and reconstructing manifolds using alpha-shapes,” 1997.
[9] C. De Boor, C. De Boor, E.-U. Mathématicien, C. De Boor, and C. De Boor, A practical guide to splines,
vol. 27. springer-verlag New York, 1978.
[10] I. Amidror, “Scattered data interpolation methods for electronic imaging systems: a survey,” Journal ofelectronic imaging, vol. 11, no. ARTICLE, pp. 157–76, 2002.
[11] N. Amenta, “The crust algorithm for 3 d surface reconstruction,” in Symposium on Computational geometry,pp. 423–424, 1999.
[12] N. Amenta, M. W. Bern, M. Kamvysselis, et al., “A new voronoi-based surface reconstruction algorithm.,”in Siggraph, vol. 98, pp. 415–421, 1998.
[13] Blender Online Community, Blender - a 3D modelling and rendering package. Blender Foundation, Blender
Institute, Amsterdam, 2018.
[14] COMSOL R , COMSOL Multiphysics R v. 5.4. COMSOL R , COMSOL AB, Stockholm, Sweden, 2019.
[15] D. W. Green, J. E. Winandy, and D. E. Kretschmann, “Mechanical properties of wood. wood handbook:wood as an engineering material,” Forest Products Laboratory, 1999.

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