COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan,China
©2006 Tsinghua University Press & Springer
Sensibility Analysis of Violin Plates
M. Razeto*, C. Staforelli, G. Barrientos
Department of Engineering Mechanics, University of Concepcion, Concepcion, Chile
Email: [email protected]
Abstract The study of dynamic and vibratory behavior of violins considers the effect of the different parts that
compose it. The top and back plates have great importance in the transmission of vibrations and the characteristic tone
of the violin; therefore, the study is centered in their vibratory behavior, since the ideal form of the modeshapes and
frequency of violins of greater prestige is known, according to bibliography. A numerical model of the top plate is
made by means of the finite elements method, to determine the influence of the different mechanical and geometric
properties on its main modes of vibrations and natural frequencies; with the objective of obtaining the expected
acoustic response, making precise changes in their structure. In this work a sensibility analysis is applied on a finite
elements model of the top plate, to determine the influence of the thickness in different zones on its natural frequencies.
Key words: violin, vibratory behavior, sensibility analysis, dynamic, modeshapes
INTRODUCTION
Famous luthiers have studied the importance of geometric properties of wood in the acoustic qualities of violins [1, 2].
Later investigations developed by scientists have given varied bibliographical information on the acoustic
characteristics of the great violins [3-5], centering attention in the mechanical and acoustic properties of their main
components (top and back plates) [6-9]. In the present investigation a model was made by means of finite elements to
analyze the dynamic behavior of the plates and the influence of thickness.
Construction of the violin A violin is made up of 35 different components wich are assembled in such a way that by
passing the arch trough its cords, their vibrations are transmitted to the structure and the air contained in it (Fig. 1).
Figure 1: Construction of the violin
A violin has four strings, of steel or gut, tuned to G3, D4, A4 and E5 (196, 294, 440 and 660 Hz). Strings of small
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diameter displace very little air as they vibrate; therefore, they radiate very little sound. The vibrating strings, however,
set the body of the violin into vibration, wich results in radiated sound of considerable strength [4, 5].
The resonance box consists mainly of the top plate, with two openings, called f-holes cut to both sides on the top, the
back plate, the ribs, the bass bar and the sound post. The top plate is usually made of spruce, the ribs and back plate of
curly maple. The bass bar is glued to the top plate under the foot of the bridge and the sound post is extended from the
top to the back plate, near the other foot of the bridge.
The top and back plates are carved from blocks of wood, where its boards are sawn parallel to the axis of the tree,
containing one symmetric axis of the material (called “quarter-cut”) [5].
The sound post and bass bar help to distribute the loads and the vibrations produced by the cords inside the box, they
are two important structural members that also serve important acoustic functions [3, 5, 8, 10].
The bridge transforms the motion of the vibrating strings into periodic driving forces applied to the top plate of the
intrument and transmits it to the acoustic box, producing the movement of large areas (the plates). These are the main
actors of the intense sound of the instrument [5].
The study of the plates is then a basic step to understand why some instruments are superior to others [11]. The
vibrational characteristics of the wood plates depend not only on the size, shape, thickness and arching of the wood but
also on the density, stiffness and internal damping of the wood [9]. Fig. 2 is a photograph of the modeshapes of a violin
top and back plate using hologram interferometrytes [5, 12]. These nodal patterns are similar among the plates of
famous violins. The experience has demonstrated that the modes 1, 2 and 5 are the more important modes to define the
tuning of the plates, called main modeshapes of the top and back plates [7, 11, 13].
Figure 2: Violin plates modeshapes and frequencies respective (Hz)
ANALYSIS ON THE NUMERIC MODEL OF A SPRUCE TOP PLATE
With the finite element software Samcef, a finite elements model of the violin top plate is used to simulate the effect of
geometric and mechanical modifications of the dynamic and acoustic properties. The initial geometry for the model is
obtained from a real sized Stradivarius.
The FE model of the top plate uses the orthotropic mechanical properties determined experimentally by means of a
vibratory analysis [1]. The modifications that were studied are: curvature, thickness, acoustic holes and the bass bar
glued to the plate [6]. In this work we show the influency of the thickness.
Fig. 3 shows the first 10 modes FEM. The shape of modes 1, 2 and 5 are close to the ideal configuration illustred in the
bibliography. Frequencies are lower, but, the harmonicity remains according to the 1:2:4 rule [9, 11].
1. Thickness modifications The objective is to know the behavior of the natural frequencies of main modeshapes as a
function of a range of values attributed to the different thickness, this way to find a good distribution of thickness that
allows to obtain the ideal natural frequencies.
In this study the program Boss/Quattro is used to achieve the sensibility analyses of the frequencies in function of the
thickness, starting from a FE model in Samcef.
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Figure 3: First to 10th shape modes of the numeric model of the top plate. The darker lines of the
model are the nodal lines of the modeshapes
The model is divided into 45 domains to find a tendency in the influence of the thickness of diverse areas on the
frequencies of the main modes (Fig. 4).
Figure 4: 45 sub-domains of the model
Plots are made to visualize the variation of the frequency of modes 1, 2 and 5, modifying the thickness in ±2.5 mm , of
the 45 different areas.
It is necessary to consider the size of each domain, and their influence on the behavior of the variation in the frequency
of different modes. For this, the variation was obtained in frequency divided by the area (cm2) of the domain in study;
in such a way to visualize the influence that the thickness of each area has over the frequency, independent of the size
of the domain.
These data group is organized according to their value to obtain a tendency of the behavior, revealing the influence of
the area in question about the frequency in each vibrating mode (Fig. 5-7).
1) Model 1 For the first mode of vibration the conclusions are, from the central zone toward the ends, the influence of
the different thickness over the frequency spreads to decrease (Fig. 8). However, the biggest influence is low,
compared to the frequencies of the other modes. This can be explained considering that this area whose thickness has a
higher influence over the frequency of mode 1, corresponds to zones that suffers a torsion movement without
displacement, due to the modeshapes of the top plate (mode 1).
The mass increase in these areas tend to stiffen the top plate, so the stiffness increase has a higher importance over the
frequency than the mass increase.
It is remarkable that the extreme lateral areas influence negatively over the frequency (Fig. 8(d)), since when being an
area of great nodal displacement without bending, thickening those areas has more influence over the mass increase
than the stiffness.
2) Model 2 Figure 10 shows that the central and inferior zones have a higher influence (Fig. 9(a)), it is due to the
bending movement taken place in that area, son the stiffness of the plate increases with the thickness.
The superior zone has little influence, regarding the inferior zone (Fig. 9(b)).
The zones with low nodal displacement have little influence over the frequency (Fig. 9(c)).
It is remarkable that the lateral extreme sectors influence inversely over the frequency. So, when the mass is increased,
the stiffness growth on the top plate is less significant, diminishing its frequency (Fig. 9(d)).
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(a) variation of frequency in 4.9-7.8 Hz/cm2
(c) variation of frequency in 0.003-1.94 Hz/cm2
(b) variation of frequency in 2-4.5 Hz/cm2
(d) influence between frequency and thickness
Figure 5: Variation of frequency of mode 1, in function of the domains thickness; grouped,
according to frequency variation reached
(a) variation of frequency in 8-16 Hz/cm2
(b) variation of frequency in 3-7.5 Hz/cm2
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(c) variation of frequency in 0.1-2.6 Hz/cm2
(d) influence between frequency and thickness
Figure 6: Variation of frequency of mode 2, in function of the domains thickness; grouped,
according to frequency variation reached
(a) variation of frequency in 8-16 Hz/cm2
(b) variation of frequency in 3-7.5 Hz/cm2
(c) variation of frequency in 0.1-2.6 Hz/cm2
(d) influence between frequency and thickness
Figure 7: Variation of frequency of mode 5, in function of the domains thickness; grouped,
according to frequency variation reached
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(a) case a
(b) case b
(c) case c
(d) case d
Figure 8: Top plate with it marked section when is retiring ±2.5 mm of thickness, on the nominal thickness in the specified
zones; of each domain, reaching the following mode 1 frequencies variation
(a) variation of frequency in 4.9-7.8 Hz/cm2: (b) variation of frequency in 2-4.5 Hz/cm2:
(c) variation of frequency in 0.003-1.94 Hz/cm2: (d) inverse influence between the frequency and the areas thickness
(a) case a
(b) case b
(c) case c
(d) case d
Figure 9: Top plate with it marked section when is retiring ±2.5 mm of thickness, on the nominal thickness in the specified
zones; of each domain, reaching the following mode 2 frequencies variation
(a) variation of frequency in 8-16 Hz/cm2: (b) variation of frequency in 3-7.5 Hz/cm2:
(c) variation of frequency in 0.1-2.6 Hz/cm2: (d) inverse influence between the frequency and the areas thickness
(a) case a
(b) case b
(c) case c
(d) case d
Figure 10: Top plate with it marked section when is retiring ±2.5 mm of thickness, on the nominal thickness in the
specified zones; of each domain, reaching the following mode 5 frequencies variation
(a) variation of frequency in 8-16 Hz/cm2: (b) variation of frequency in 3-7.5 Hz/cm2:
(c) variation of frequency in 0.1-2.6 Hz/cm2: (d) inverse influence between the frequency and the areas thickness
3) Mode 5 The extreme lateral areas (Fig. 10(d)) have inverse influence over the frequencies of modes 1 and 2 (when
increasing their thickness, the frequency diminishes)
The thickness in central zones and inferior and superior lateral central zones (Fig. 10(a)) have great influence over the
frequency of mode 5. This marked influence reaches a variation in frequency of up to 16 Hz/cm2, it is due to the
bending movement with high displacement that takes place in those areas for the 5th mode.
Moving away from those domains, the neighboring sectors have smaller influence (Fig. 10(b), 10(c)).
It is difficult to lower the frequency of mode 5 without diminishing the frequency of mode 2. It is necessary to find
those areas that have a high effect over the frequency of each mode of vibration that does not coincide to each other.
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This way, it is possible to modify the required modeshapes without changing the frequency of other main mode of
vibration.
CONCLUSIONS
1) A study on the acoustic behavior of a violin was conducted, considering the effects of the different components from
the instrument. It is centered on the dynamic behavior of the plates, due to its importance in the characteristic sound of
the violin.
2) With the modal analysis method, the frequencies and nodal displacements for the 1st, 2nd and 5th mode were found
for the top plate with acoustic holes and a bass bar. This method validated the results from the numerical model.
3) The numerical method allowed a study of the different mechanical and geometric parameters of the main mode
shapes and natural frequencies. As result, a set of criteria was established to modify the components and to obtain the
desired dynamic behavior.
4) With the sensibility analysis, those areas whose thickness had a higher importance about the value of the frequency
of the main modes are found. After that, modifying the height in each point, the frequency variation of each mode is
obtained.
5) The developed analysis states that ther is no zone where wood can be extracted to increase the frequency of the main
modes.
6) In this paper, a methodology is presented for the vibrate-acoustic analysis of the violins that can be applied in future
studies of diverse musical instruments.
REFERENCES
1. McIntyre ME, Woodhouse J. On measuring the elastic and damping constants of orthotropic sheet materials.
Acta Metall, 1988; 36(6).
2. Wangaard FF. The Mechanical Properties of Wood.
3. Gough C. Science and the Stradivarius. 2001 Science Writing Award for Professionals in Acoustics from the
Acoustical Society of America.
4. McIntyre ME, Woodhouse J. The acoustics of stringed musical instruments. Interdisciplinary Science Reviews,
1978; 3(2).
5. Rossing TD. The Science of Sound. 2nd Edition.
6. Bissinger George. Experimental violin acoustics. Journal American Lutherie, 1986; 7.
7. Carruth Alan. Free Plate Tuning, Part 2: Violins.
8. Curtin Joseph, Alf Gregg. Set-Up and Adjustment. Joseph Curtin Studios, Concert Violins & Violas.
9. Hutchins CM. The Acoustics of Violin Plates. Scientific American, October 1981, p. 170.
10. Johannsson Hans. Construction of a violin. Violin Making.
11. Massmann Herbert, Ferrer Rodrigo. Instrumentos Musicales. Artesanía y Ciencia.
12. Hutchins CM, Stetson KA, Taylor PA. Clarification of "Free Plate Tap Tones" by Hologram Interferometry.
CAS Newsletter #16, Nov. 1971.
13. Curtin Joseph. Some Principles of Violin Setup. Journal of the Violin Society of America, 1995; 15(1).
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