PROCEEDINGS of the23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Influence of orthotropic properties on vibration of violin top plates
Masao YOKOYAMA1; Riccardo Roberto DE LUCIA2; Fabio ANTONACCI2; Augusto SARTI2
1 Meisei University, Japan
2 Politecnico di Milano, Italy

ABSTRACTIn this paper the mode vibrations of violin plates are investigated by numerical simulations. The plategeometry is scanned by means of a 3D laser scanner. Numerical simulations are performed using Finite
Element Methods, in order to retrieve eigenmodes and mode shapes. The orthotropic properties of wood suchas maple and spruce are introduced in the numerical simulation. We show that the shapes of the main modes
calculated by numerical simulations are in good agreement with those of a real violin. We discuss how
material properties influence the eigenmode frequencies and shapes.
Keywords: Violin, Mode shape, Orthotropic material, Numerical Simulation

1. INTRODUCTION
Many studies about mode vibration of violin and sound radiation have been proposed during the years, seefor example [1-3], but it is very difficult to obtain results with a neat interpretation from a quantitative andeven qualitative standpoint. The shape and the dimension of current violin were settled by A. Stradivari(1644-1737) et al. before about 300 years ago. The mechanical properties of these instruments cannot be
studied in detail, due to the requirement of performing completely noninvasive analyses. It is therefore
impossible to infer the mechanical properties of the wood of the historical instruments and therefore, even
if the same geometry of the reference instrument is used, there is no way of obtaining a perfect acousticcopy.
As a consequence, numerical simulations based on Finite Element Method (FEM) and Boundary Element
Method (BEM) have gained importance in the last few years for the analysis of the vibration of violin
plates [4,5]. In this paper we focus on the problem of assessing how the Young’s modulus and density ofthe material impact on the eigenfrequency of the mode 5 (also known as “ring mode”), which is deemed asimportant to determine the resonances of the violin body. At knowledge of the authors, this study has notbeen conducted in the literature, yet. Moreover, in this paper we conceive the numerical analysis as acoupled problem between mechanical vibration and the sound field produced by the vibration. Also, thecoupling has received minor attention in the literature.
More specifically, in our study we conduct a numerical simulation (through FEM and BEM) of vibrationalmodes of the back plate of a violin whose geometry has been obtained by a precise 3D laser scanner, andwe investigate the influence of the mechanical properties of the wood (maple in the case of back plates ofthe violin) on mode vibration for a range of possible values.

126438masao.yokoyama@meisei-u.ac.jp
name.surname@polimi.it2. Method
2.1

Scanning violin surface and generating mesh for mechanical simulation

In this section we shortly describe the measurement of the 3D geometry of the plate from laser scanning.
We used a plate with a finished outline, arch and thicknesses. The body length is 352 mm and the weight is
123 g. The mean density estimated from the volume and weight is 0.637 g/cm3. A single violin has beenused for this study.
Figure 1 shows the snapshots of scanning thesurface of violin, revising the geometry andimporting the mesh used fornumericalsimulation. The resolution of laser scanner,
Romer ABSOLUTE ARM, is 0.01mm at theminimum. We took both the external surface
and internal surface of back plate of violin andcombined them using the Polyworks™ software
by InnovMetric. A stage aimed at removing all
the small holes and unavoidable imperfections
generated during the scanning has beenaccomplished before the creation of polygon
data in the STL file. The polygon data wasimported in the COMSOL Multiphysics™
software, and the three-dimensional geometryby tetrahedron mesh was generated by theautomatic mesh function offered by COMSOL

Fig.1 Snapshots scanning the surface of violin backplate (upper row, left), revising the geometry (upperrow, right) and importing the mesh data toCOMSOL Multiphysics (lower row).

Multiphysics. The number of vertices used isapproximately 234k, and the eigenfrequency and mode shapes were calculated by the finite element
method.

2.2

Setting of orthotropic property of wood

In order to setup accurate simulations, it is of utmost importance to set the mechanical properties of thematerial, which in this case is wood. Wood is an orthotropic material [6, Chap. 1], as it exhibits different
mechanical properties along three main axes, which are determined by the grain, radial and tangentialdirections. Spruce and maple blocks for violinmaking are usually cut as shown in Fig.2.
The nominal mechanical properties of the material that are used in the simulations are shown in Table 1 [7].
Here, the values of ET and ER are expressed in terms of the ratio with the Young’s modulus along thelongitudinal direction, EL. It is important to notice that different samples of maple exhibit values of EL thatcan greatly differ from the nominal value reported in Table 1. Moreover, it is difficult to measure theaccurate Young's modulus of a plate when its arch has been already sketched.

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Y (Radial)

Table 1 Values for Orthotropic properties forsetting in numerical simulation

Z (Tangential)
X(Longitude)

Property

Value

Young’s module EL

12.6(GPa)

ER / EL

0.132

ET / EL

0.065

Rigidity modulus GLR / ELFig.2 Orthotropic property of violin plate. Because
wood is orthotropic material, we set the typical (orreal) value of orthotropic properties of the wood tothe parameters of COMSOL Multiphysics.

0.111

GRT / EL

0.021 *

GLT / EL

0.063

Poisson’s ratio μLR

0.424

μRT

0.774

μLT

0.476

* GRT is assumed by the average of other hardwoods because the value is not showed in thearticle[7].

3. Simulation Result
3.1

Comparison of mode shape between virtual thickness and real thickness

The thickness of plate of violin is not uniform along its extension. In particular, the central part of the plateis the thickest and the thickness gradually become thinner toward the peripheral part. More specifically, thethickness at the edge is in the range 4.2 - 4.5 mm, whereas 3 cm inside from the edge of lower bouts andupper bouts it is in the range 2.0 - 2.5 mm. The exact values of the thickness depend, of course, on thechoices of the violin maker.
Thus, the thickness has a very relevant impact on the vibrational behavior of the plate. As a first study, wecompared eigenfrequencies and modal shapes obtained from the actual 3D geometry, with those of a platewith a uniform thickness. We analyzed three modes, which are deemed as important by violinmakers,
which are namely mode 1, mode 2 (also known as “cross-mode”) and mode 5.
The eigenfrequencies were different for all the modes under analysis. Moreover, the nodal lines (i.e. theregions of the modal shapes with zero displacement) of mode 1 and mode 2 are similar, whereas the nodal
lines of mode 5 are different, as the nodal line exhibits a rounder pattern when the real thickness is used.
This rounded contour matches with the experimental results obtained with the Chladni analysis on the sameback plate (Fig. 4).

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ConstantThicknessd = 3.5mm

Mode 1Mode 2Mode 5

RealThickness

Fig.3 Mode shapes of back-plate of violin, (a) mode 2 (x-mode), (b) mode 5 (ring mode).

mode 2mode 5

Fig.4 Mode shapes (mode 2, 171 Hz, and mode 5, 352
Hz) of back-plate of violin by Chladni Method.

This experimental validation proves the importance of using the real thickness in modeling the vibrational
behavior of the plates, and it is easy to guess how it is important also for the violin body with assembledplates. It is important to notice that the measurement of the thickness is not always possible for historicalinstruments.
In the process of violin making, violin makers adjust and finish the thickness and arch of violin plate so thatthe eigenfrequencies match target ranges. In particular, the eigenfrequency of mode 5 is known as “tap tone”
which many violin makers hear from old days. It is generally adjusted to 340 - 350Hz in a back plate.
When the nominal value EL = 12.6 GPa of the Young’s modulus is used, the eigenfrequency of mode 2 was150 Hz, that of mode 5 was 340Hz. On the other hand, by the Chladni method, the eigenfrequency of mode
2 was 171 Hz and that of mode 5 was 352 Hz. We interpreted this difference as a consequence of thedeviation of the mechanical properties of the wood under analysis from the nominal ones. If we ascribe thedifferent behavior to the Young’s modulus only and we keep the remaining parameters to the nominalvalues, we need to increase EL to 13.5 GPa, approximately 7.2% upper than the representative value, in6441

order to obtain the same eigenfrequencies of the sample violin.
3.2

Impact of the properties of wood on the eigenfrequency

Figure 5 plots the eigenfrequency of the modes as a function of the thickness of the central part of the plate,expressed in millimeters. The thickness of the rest of the plate was scaled proportionally. The actual
thickness of the central part of the plate under analysis was 4.0 mm. The range of thicknesses in ournumerical simulation was set from 3.5 mm to 4.2 mm, here the range is general thickness when violinmakers make a violin.
In order to modify the thickness of the plate, we developed an ad-hoc algorithm that modifies thez-coordinate of the outer mesh (the mesh of inside arch is fixed), once the representative value of thethickness at the center of the plate is given. It is possible to notice from Fig. 5 that for both modes 2 and 5,
in the considered range there is a linear relationship between eigenfrequency and thickness.

thickness of test violin

Fig.5 Change of eigenfrequency by the change in thickness of back plate. The thickness
in horizontal axis is the thickest point in the center of back plate. 4.0 mm is the thicknessof test violin used in this study.

On the other hand, we could be interested in assessing the impact of a modification of the Young’s modulus
on the eigenfrequency. Figure 6 and Figure 7 show the variation of the eigenfrequency when the Young'smodulus and the density span in the range from -20% to +20% of the nominal value. This range has beensuggested by violinmakers. We notice that if the elasticity in the grain direction, EL, increases, theeigenfrequencies of mode 2 and mode 5 become higher. Conversely, if the density increases theeigenfrequencies decrease.

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EL of test violin

Fig.6 Change of eigenfrequencies by the difference in EL ofback-plate.

Density of test violin

Fig.7 Change of eigenfrequencies by the difference in density ofback-plate.

In practical situations, one could be interested in determining the thickness of a plate when the Young’s
modulus differs from the nominal value so that the eigenfrequencies match with the target values. For thispurpose, Figure 8 shows the relation between the density, the Young’s modulus, the thickness at the centerof the plate and the eigenfrequency, under the assumption that the three variables are independent eachother. As mentioned above, violin makers check the eigenfrequency of the mode 5 listening to the tap tone
and, based on this, trim the arch and the thickness of the plate so that the mode frequency falls in the range
from 340 to 350Hz. Figure 8 quantifies the modification of the thickness of the plate so that theeigenfrequency increases or decreases when the elasticity and density of wood change.
As an example, it is possible to notice from Fig. 8(a) that when the density increases by 10%, the frequency
decreases approximately by 16Hz. To restore the frequency of 352Hz of the mode 5, the thickness must be
increased by approximately 0.3 mm. Moreover, from the inspection of Fig. 8(b), we can notice that when
EL decreases by 10%, the frequency decreases approximately by 18Hz. To keep the eigenfrequency of352Hz, a violin maker should increase the thickness of the plate approximately 0.3 mm.

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However, it is important to stress that the Young's modulus and the density are not independent variables,
and therefore we cannot directly apply the rule of thumb proposed above simply. Nonetheless, this analysis
gives us some ideas, at least qualitative, of the trend.

(a) density v.s. thickness

Fig.8(b) EL v.s. thickness

Change of mode 5 eigenfrequency and relation between thickness d(center), density and elasticity.

4. CONCLUSIONSIn this study we performed a numerical simulation of the vibration of back plate of violin by the finite
element method using COMSOL Multiphysics and analyzed the tendency of the change of the vibration
mode by the change of the physical characteristics of maple. The analysis of these relationships by thechange of physical characteristics helps violin production, and it is thought that these analyses contributethe analysis of the sound of historical Italian violin in future. Focusing on the acoustic radiation from aviolin body, now we are preparing the geometry of not only back plate of a violin but also whole body andparts, and we will simulate numerically the coupled problem of structure and acoustic in near future.

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Ravina, E. “Violins characterization through vibro-acoustic experiments” In Acoustics 2012 (2012).

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