## Abstract

Even though string musical instruments made of synthetic materials such as carbon fiber reinforced polymer (CFRP) have respected acoustic performance, but a short manufacturing cycle and low product cost, they do not become an alternative to replace high-quality string instruments made of sound woods. For CFRP violins to approach high acoustic performance wood violins, they must exhibit approximately the same bending stiffness. The CFRP is denser, stiffer, and isotropic compared to the orthotropy of wood. In this work, the acoustic behavior of CFRP violins with the same geometry as high-quality wood violins was compared. A numerical modal study was developed by finite element simulations, comparing two violin top plates, one in CFRP and the other in Picea abies (PA) wood. The simulations were developed in the ansys mechanical software, using the Block Lanczos method with a mesh of 38,216 finite volumes, finding modal patterns for both the CFRP model and the PA model. Mathematical models based on solid state physics such as effective masses and maximum vibration amplitude between models were outlined. Both models were validated against experimental studies developed by other authors. It is concluded that for instruments with the same geometry, a sonorous superiority of the wood over the CFRP was evidenced, which leads to further reinforce the unique, enigmatic, and mythical behavior of violins made of sonorous woods such as the Stradivarius violins.

## 1 Introduction

There is a wide list of references related to the acoustic of musical instruments, and these references are mainly focused on the study and characterization of instruments of high acoustic performance [1]; in this research, this kind of instruments will be called “exceptional” or “high-quality.”

The behavior of acoustic waves changes as the properties of the medium change, and the vibrational properties of wood depend on the properties of the wood species, moisture content, and the direction of the nodal lines in relation to the structure of the fibers [2].

The direction of the wood fibers affects the acoustic impedance. When the acoustic signal is emitted parallel to the fibers in wood, the acoustic impedance is comparable to that of metals, while for the acoustic signal measured perpendicular to the fibers, it is similar to that of plastics and water [2]. That is why they are studied nowadays: stringed instruments with synthetic material are studied, like, for example, the violins made with carbon fiber reinforced polymer (CFRP) [3] (see Fig. 1).

The stringed instruments that have been most sought to replicate are the ones made by the Italian luthier Antonio Stradivari (1644–1737). This is due to the warmth, singularity, balance, and acoustic power of its sound in key frequencies. This is not completely clear how Stradivari could give these features to his instruments [4,5]. It is believed that he made it by trial-and-error method, modifying the extensive features (design), as well as the intrinsic features of woods using thermo-chemical treatments. It is also believed that unintentionally, the biological action of fungus was involved, improving in this way the acoustic performance of the instruments [6]. This interrelation between physical, chemical, and biological factors transformed these instruments into fine jewelry that until today are not being able to be replicated [7].

Since Stradivari’s time, luthiers have used sonorous woods with very special micro-structural characteristics. In the specific case of violins, for the acoustic box, as well as for the bass bar and the sound post, they are made with *Picea abies* (PA), also known as Norwegian Picea. For the rest of the structure, *Acer pseudoplatanus* (AP1) (European maple), the *Acer platanoide* (AP2) (Bosnian maple), and the *Caesalpinia echinata* (Brazilian Pernambuco) are often used.

In the efforts to replicate high-quality wooden instruments, one of the most promising materials is the CFRP, since it has a very similar acoustic performance when used in violins [3], replacing the traditional sonorous woods without high-cost and time-consuming manufacturing, as it is found in the traditional luthier workshops. This synthetic is a compound of 50% resin (polymer or epoxy) and 50% carbon fiber (CF) layers. It is manufactured by superposing three to six flat layers of CF, separated by resin interfaces, which induce homogeneous isotropic properties to the compound [8]. In order to give the instrument its shape, the CFRP is molded in an autoclave, with a viscoelastic behavior of a high strength–volume ratio [1].

## 2 Theoretical Background

### 2.1 Orthotropic Nature of Wood.

Wood can be described as an orthotropic material due to its particular and independent mechanical properties, in its three mutually perpendicular planes: longitudinal, radial, and tangential. The longitudinal plane (*X*) which is parallel to the fiber or grain; the radial plane (*R*) which is normal to the growth rings, perpendicular to the grain in the radial direction; and the tangential plane (*T*), perpendicular to the grain and tangent to the growth rings (Fig. 2) [9].

### 2.2 Elastic Properties of Wood.

*E*, three modulus of shear

*G*, and six relations of Poisson. The elasticity modulus and the Poisson ratio are bound by the following expression [9]:

### 2.3 Modulus of Elasticity of Wood.

In wood, the three modulus of elasticity *E*_{L}, *E*_{R}, and *E*_{T} are defined along the longitudinal, radial, and tangential planes, respectively. Although in a same piece of wood, it can be found that sectors with quantities do not remain constant, like elasticity, humidity, and density [9].

### 2.4 Poisson’s Ratio of Wood.

In wood, Poisson’s ratios are defined as transversal deformations along the axial directions, described by $\u03d1LR$, $\u03d1RL$, $\u03d1LT$, $\u03d1TL$, $\u03d1RT$, and $\u03d1TR$. The first index refers to the direction of the strength or load and the second index to the direction of the lateral deformation. Thus, for example, $\u03d1LR$ is Poisson’s ratio related with the deformation of the radial plane, caused by the stress along the longitudinal plane of the piece of wood [9].

### 2.5 Shear Modulus of Wood.

The shear modulus measures the resistance of a given material when deformed by cutting forces. In the case of wood, the three shear modulus are *G*_{LR}, *G*_{LT}, and *G*_{RT}, defined on the planes *LR*, *LT*_{,} and *RT* [9]. All the mechanic properties of wood change intra and inter species and are affected by its moisture and density.

### 2.6 Simulations and Validations of Numerical Models.

Due to its orthotropic nature, wood is described by 12 Poisson’s ratios $\u03d1ij$ and by 12 shear modulus *G*_{ij}, *i* ≠ *j*. Therefore, it is not met for any of the mechanical tensorial quantities that *T*_{ij} = ± *T*_{ji}.

*k*

_{ij}=

*k*

_{ji}. Because of this symmetry, it is needed only six out of 24 tensorial quantities to run the simulations: three Poisson’s ratios $\u03d1xy$, $\u03d1yz$, and $\u03d1xz$ and three shear modulus

*G*

_{xy},

*G*

_{yz}, and

*G*

_{xz}.

*k*

_{xy}=

*k*

_{yx}and

*k*

_{xz}=

*k*

_{zx}

*y k*

_{yz}=

*k*

_{zy}.

#### 2.6.1 Free and Forced Vibrations.

*m*), the spring (

*k*), and the damping (

*c*), as shown in Fig. 3. When an external force is applied to the structure

*f*

_{(t)}, the movement is described by the following differential equation:

If such a system receives an external perturbation, it experiences free vibrations. The amplitude of the vibration weakens with time, depending on the damping constant of the system. The frequency of those vibrations is defined by the stiffness and mass. When the damping constant is zero (*c* = 0), the system oscillates with its natural frequency [10].

#### 2.6.2 Effective Mass.

The effective mass is the inertial effective mass tensor generally defined from the second derivative of the dispersion relation *E* versus *k*, although the first derivative is more directly relevant to measurable phenomena such as resonance [11].

Traditionally, there have been two main roles which the effective masses represent: first, they define mode shapes of structures “tied” to massless or virtual foundations, which are oscillatory accelerated. Second, they verify and update finite elements models of vibrant structures.

The effective mass indicates the sensibility of each vibrational mode in relation with the excitation that comes from the base. It also measures the spatial coupling of each mode and the excitation applied to the system. It can be described as a group of systems with parallel degrees-of-freedom joined to a rigid, oscillatory massless base (Fig. 4). Each mode of the distributed structure is represented by one oscillator. The mass of the oscillator is defined by the effective mass of the mode. This effective mass is not identical to the usual inertial mass of each *m*_{i}, because there are inertial bonds among the masses by way of “damping,” which tie them to the base. The intensity of the damping is determined by the data that represent the material of the structure, like its density, its mechanical modulus, and/or coefficients.

This points out that the displacement of the masses is equivalent to the displacements of the centers of gravity associated to the effective modes, which at the same time are associated to the effective masses. These effective masses are determined by the matrix algebra of eigenvalues and eigenvectors.

To this effect, beginning with the defined geometry, the effective masses of the natural vibrational modes, and the quantities of those modes by unit of frequency, it is possible to outline, in a reasonable way, the acoustic performance of an oscillating structure.

## 3 Materials and Methods

### 3.1 Materials

#### 3.1.1 Technical Data.

For the technical data of PA, experimental measures published by Bucur [12] were taken. For the model formulation and methods it was considered the studies [13–18] for violins in PA.

Two coordinate systems of reference, one local and one global (Fig. 5), were added to the model, a top plate of a violin. In order to introduce the orthrotopy or the quasi-orthrotopy of the materials, (*L*, *R*, *T*) = (*X*, *Y*, *Z*) was made in the local coordinate system (LCS).

The density of the material is the only quantity that remains invariant, when the LCS is translated and/or rotated. In the presence of translations of the LCS, only the following parameters remain constant: Young’s modulus, Poisson’s ratio, and the shear modulus.

In wood, the discrepancies of the mechanical properties are relatively small when defined on different tensorial planes. Therefore, it is guaranteed that by assuming *ij* = *ji*, the mechanical performance of the PA remains almost the same. It is resumed in Table 1 the values of density (*D*), Young’s modulus (*E*), and the shear modulus (*G*) when averaged.

In Table 1, $GIJ\xaf=Gij$, as well as for its conjugated *G*_{ji}.

It was not possible to import to the simulations, all the values $\u03d1ijGij$. It is due to ansys^{®} software, which makes calculations only on three orthogonal planes, related with the three pairs of symmetric tensorial indexes *TR*, *TL*, *RL.* Then, to make the strain–stress matrix positive, the following three coefficients were selected: $\u03d1TR=0.25$, $\u03d1TL=0.013$, and $\u03d1RL=0.028$.

### 3.2 Methods

#### 3.2.1 Simulations.

Part of the structure of a violin is studied: the front plate with the bass bar (Fig. 7). These two pieces have an important role in the instrument performance [20]. A bond-type contact was selected between these instrument parts. By means of a modal analysis in finite elements, it is studied the acoustic performance of a CFRP violin front plate with the same thickness found for the wooden violin front plates.

A modal analysis was performed through finite elements method in ansys^{®} using a mesh of 38,216 finite volumes (Fig. 8). Natural patterns of vibration were found in the interval of 150–6000 Hz. The imported technical data did not consider damping. Nodal-boundary conditions were established, and the equations were numerically solved by means of the Block Lanzos method. These simulations were run using mechanical data of sonorous woods and carbon fiber reinforced polymer. The way some intrinsic parameters and the plate thickness determine the acoustic performance of the model was found.

#### 3.2.2 Validation of the Model.

The validation of both numerical models—the one in PA and the one in CFRP—was made against three types of experimental studies of PA violins: 1. modal analysis [21,22]; 2. interferometry studies [23,24]; and 3. holographic studies [1,8,25]. In this way, reliable analysis and inferences were made for all models.

## 4 Results and Discussion

Modal simulation

The modal simulation in PA shows a maximum of 150 mode shapes in the interval of 150–6000 Hz. Due to the strong orthotropy of the wood, the mode shapes are very symmetric, especially for low frequencies, as observed in Figs. 9(a) and 9(b). For frequencies higher than 1000 Hz, patterns become more complex in shape, with deformation amplitudes diminishing (Figs. 10(a) and 10(b)).

For the same interval of frequencies, the modal simulation in CFRP shows only 50 mode shapes, Figs. 11(a) and 11(b), which are also very symmetric as the ones found in PA. As in PA, in mid to high frequencies, the patterns become more complex, and the deformation amplitudes become more tenuous, although for these frequencies, symmetries weaken less than PA (Figs. 12(a) and 12(b)).

For both models (PA and CFRP), it can be observed that in general for low and mid frequencies, the region with more acoustic work (deformation amplitude) is around the central zone of the plate, close to the locus where the violin bridge is placed.

The curves in Fig. 13 show that for the frequencies range in this study, the modal version in PA has more density of modes than in CFRP, especially for low to mid frequencies. On frequencies below 882 Hz, the CFRP model totally lacks resonance modes. For the PA model, the first mode shape presents at 373 Hz, and for the model in CFRP, the first mode shape presents after 882 Hz.

It was found that no one of the 50 mode shapes in CFRP is identical to any of the 150 mode shapes in PA. Modal patterns in PA are more complex in shape than the ones in CFRP. The obtained effective mass participation is sufficient to characterize most of the acoustic behavior of CFRP in violins (Table 3).

As it is shown in Table 3, the total CFRP effective masses

*E** are nearly twice the total PA effective masses*E*, thus*E** >*E*. The opposite happens for the relative effective masses (ratios*I**,*J**,*I*,*J*):*I** <*I*and*J** <*J*.The ratio of

*effective mass versus numbers of modes*is 6.5 times higher in CFRP compared with the same ratio for the PA version. Additionally, it is found that the ratio of*total effective mass versus total model mass*in CFRP is 50% higher compared with the one in PA version.Figures 14 and 15 show the effective masses of each normal mode for all of the frequencies studied. The higher participation of those masses is present in the transversal direction (axis

*z*of the LCS).Figures 14 and 15 show that for low and mid frequencies (150–800 Hz), the CFRP model presents just one normal mode and has a considerably lower contribution to the effective masses compared with the PA model. Due to the importance of the Bridge Hill effect on the acoustic performance of violins, it is interesting to study the participation of modal effective masses in frequencies where this effect is found.

In the interval of this study, 150–6000 Hz, the model in CFRP presents a greater average participation of effective mass than the model in wood. However, for the Bridge Hill frequencies, the model in PA is more active in relation to these masses (compare Fig. 16 with Fig. 17). The participation of the effective mass of the CFRP model in the Bridge Hill zone do not change significantly in relation to the output this model has for the rest of the frequencies.

It is convenient to know the dynamics of finite elements of the CFRP model. The acceleration of its maximum deformations can be observed in Fig. 18.

It can be observed that for low to mid frequencies, in average the CFRP model significantly improves its dynamic performance. This improvement does not happen for the Bridge Hill zone.

Numerical models validation

In Fig. 19, a close similarity between the simulated mode shapes of this study and the experimental mode shapes in Refs. [23,24] can be observed. Real models are shown in gray scale (Figs. 19(a) and 19(c)). Valleys and crests are separated by nodal curves. The simulated model also shows its valleys and crests separated by nodal curves (Figs. 19(b) and 19(d)), but they are thinner and less visible than those seen in the real models.

Since ansys

^{®}do not admit damping in modal simulations, it is expected to find discrepancy between the modal frequencies of real models and the modal frequencies of simulated models. Thus, a discrepancy between the two frequencies associated to each pair of matching mode shapes, like*mode b*versus*mode*1 (Figs. 19(a) and 19(b)),*mode d*versus*mode*5 (Figs. 19(c) and 19(d)),*mode d*versus*mode*13 (Fig. 20), and*mode a*versus*mode*5 (Fig. 21), is found.Damping does exist when experimental modal studies are made. This is why the modal frequencies given by Refs. [23,24] are lower. These discrepancies on frequencies between damped and undamped systems do not weaken the validation since the shape of the natural mode of vibrations are determined, on the first place, by the elastic (mechanical) parameters and the geometry of the models.

The frequencies of the Bridge Hill zone [26] are highlighted by the dotted line in Fig. 22. An accumulation or active zone in “small peaks” of admittance values is shown. In this region, the majority of violins, especially the ones with outstanding acoustic performance, show a better admittance performance in relation to the rest of the frequencies.

## 5 Conclusions

For wooden violins and CFRP violins whose thickness has not been made thinner, none of the 50 mode shapes of the CFRP is identical to any of the 150 in *Picea abies*, although there is a strong similarity between them.

For the same frequency, an unaltered-thickness CFRP violin has a different mode shape with respect to a high-quality PA violin; thus, if it is played the same musical note using both violins at the same time, two different waveforms are obtained. Then, it is possible to notice using electronic equipment, and possibly just by ears, the difference between the sound of an unaltered-thickness CFRP violin and the sound of a high-quality PA violin.

The energy of the longitudinal pressure waves (the sound) that an oscillating structure causes is proportional to the effective mass participation of the structure. On average, along the interval of frequencies studied, the unaltered-thickness CFRP model has more effective mass participation than the PA model. In this way, on average, an unaltered-thickness CFRP violin sounds louder than a high-quality PA violin.

Regarding efficiency, the opposite takes place. On average, the PA model has a higher ratio of *effective mass* versus *total mass*, compared with the unaltered-thickness CFRP model. It is easier (energy-saving) for a violinist to deliver sound using a violin in *Picea abies* compared with its correlation in CFRP.

Even if for some few frequencies the unaltered-thickness CFRP violin sounds louder than a PA violin, the PA violin is “easier to play,” since it is more responsive: it has a better *effective mass* versus *total mass* ratio and much more normal modes of vibration, well-distributed along the frequencies.

The resonance (the harmonic performance) of a musical instrument is strongly related to the quantity of its normal modes of vibration. The fact that for low frequencies the unaltered-thickness CFRP model shows an absolute lack of normal modes implies that for such frequencies this model has low volume, low intensity of sound, and low harmonic performance (it is a “deaf violin” on that frequencies, on violinist terms). The CFRP is stiffer than the wood used for violins plates; it causes a low resonance capacity on low frequencies when it is used in violins with standard thickness. An unaltered-thickness CFRP violin will ask the violinist for more stamina and technique to play on those frequencies.

A high-quality PA violin, compared with an unaltered-thickness CFRP violin, has a more complete set of “sound colors” (waveforms) because it has more resonances and natural reverberations. The fact that the PA has many natural modes of vibration provides the violinist with greater aesthetic possibilities, since the musician takes advantage of these acoustic features to generate effects that for physiological, and psychological motives, result agreeable to the human ear.

By the results obtained in this research, it is understandable that the violin makers adjust the thickness of the CFRP violins to have them thinner than the wooden ones. In violinists’ jargon, it can be said that an unaltered-thickness CFRP violin is a low-cost instrument that can be labeled as a “good advanced-student violin.”

Regarding professional violinists, an exceptional high-cost high-quality wooden violin is required for them because it assures a frequency-well-distributed sound power and normal modes, and a special admittance performance on the Bridge Hill frequencies.

## Acknowledgment

The authors thank the Postgraduate Program in Mechanical Engineering at the University of the Andes, at the Salesiana Polytechnic University, at Physics Research Group of the School of Physics and Mathematics of the Higher Polytechnic School of Chimborazo, and the Doctoral Program in Mechanical Engineering and Materials of the Federal University of Technology of Paraná.

## Conflict of Interest

The authors have no conflicts of interest to declare that are relevant to the content of this article. The data that support the findings of this study are available upon reasonable request from the authors.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

## References

*Department of Paper and Bioprocess Engineering*. N.C. Brown Center for Ultrastructure Studies, https://www.esf.edu/pbe/ncb/