Reconstruction of an early viola da gamba informed by physical modeling

The first textbook on the viola da gamba appeared in 1542 in Venice by Silvestro Ganassi, followed by a second volume one year later.¹ In addition to the practical playing instructions, the pictures included in both bands are of great interest, since Ganassi describes himself as someone who is very interested in pictorial representations and, hence, values a correct and elaborate realization of the graphics.

In search of a construction framework for an early viola da gamba, the detailed—though unusual—outline of Ganassi seems to correspond with early string instrument representations.² The outline alone, however, does not give concise indication of the design of the individual parts of the instrument, especially the top plate, which has a major effect on the sound characteristics. This uncertainty also involves the existence or not of a bass bar and a soundpost, since these were not always present in renaissance viols.³ In fact, according to Ref. 4, in order to enhance the sound quality, some internal asymmetry was introduced initially by a gradually thickening top plate and subsequently by a bass bar and soundpost. The possibility of moving the bridge for tuning purposes¹ also constitutes the placement of a soundpost unlikely. Therefore, following research on written and iconographic sources, an evidence-based reconstruction of a viol after Ganassi has been postulated in Ref. 5, where an asymmetric top plate is used to compensate for the lack of a soundpost and a bass bar.

The present manuscript studies the vibrational characteristics of such an asymmetric design. This can be achieved by the use of physical modeling and experimental measurements. Physical modeling may be used to increase our understanding of how various constructional details of string instruments influence sound radiation.⁶ Experimental measurements can be used to visualize the vibrational behavior of the instruments.^7,8 Of particular interest is whether a top plate of variable thickness has a significant effect on the acoustic efficiency of the instrument. To this end, a finite element model of an early viola da gamba is formulated, as outlined in Sec. II. The model is compared in terms of its vibrational characteristics with a reconstructed instrument of the same design. In Sec. III, it is used to compare an asymmetric design with a symmetric one, without the need to reconstruct the latter. The results show that the instrument with an asymmetric top plate may radiate sound more efficiently. Finally, Sec. IV discusses the findings of this work in the context of music acoustics research.

The most important factor to determine the quality of bowed-string instruments is the vibrational behavior of the instrument body.⁹ In particular, the top plate is responsible for the largest part of the radiated sound. The thickness graduation of the top plate for the viola da gamba presented in Ref. 5 is visualized in Fig. 1. This is in contrast to traditional designs, for instance the cello top plate by Stradivari shown in Ref. 10, the thickness of which follows the body geometry.

Bowed-string instruments radiate sound mainly through the vibrations of the instrument body. These are imposed by the oscillations of the strings, which couple to the instrument body at the bridge. The forces applied by the motion of the strings on the bridge are largely parallel to the plate of the instrument. The most common method to characterize the response of the instrument body is an input admittance measurement,¹¹ where the mobility of the bridge is measured by exciting (e.g., using an impulse hammer) on one side of the bridge (usually the treble side) and measuring the motion of the bridge on the opposite side. The bridge admittance is then defined as the bridge velocity divided by the excitation force. In this study, such measurements are carried out on the reconstructed instruments and replicated in a numerical simulation.

In fact, two different types of simulations are performed using the finite element method (FEM). A structural analysis that only considers the vibrations of the instrument body and an acoustic-structure interaction simulation that couples the instrument's vibrations to the surrounding air. Besides the bridge oscillations excited by the motion of the strings, another factor that needs to be taken into account is the static load applied on the bridge due the tension of the strings. This static load is applied on the bridge before the dynamic vibrations of the instrument are computed, as it may have a significant influence on the stiffness of the instrument body. The total tension of all six gut strings (tuned at 78, 104, 139, 175, 233, and 311 Hz) amounts to T = 97.7 N. They form an approximate angle $θu=81°$ with the bridge towards the neck and $θd=71°$ towards the tailpiece. The resulting static forces on the bridge may be calculated as

where F_x is the force acting along the length of the instrument (vertical to the bridge) and F_y pushes the bridge downwards. Figure 2 shows the deformation of the viol body due to the static load presented by the tension of the strings.

Formulating a physical model that can predict the sound radiated by the instrument allows the comparison of different designs without having to construct the actual instruments. This virtual prototyping has therefore been used during the phase of designing the instrument geometry in order to optimize the opening area of the sound holes and to ensure that the top plate is strong enough to support the static load from the strings. In order to formulate a reliable model for such a complex instrument body it is necessary to possess a detailed geometrical description, as well as accurate values for the material parameters. The former has been obtained after an extensive review of historic material, as reported in Ref. 5, involving a weakly arched top plate and a back plate with strengthening back-plate cross bars (see sketch in Fig. 3). The latter is in accordance to the wood varieties used by the instrument makers, corresponding to a carved-plate configuration (i.e., the axes run through the arched plates in fixed absolute directions). The top plate is made out of Sitka spruce, and the ribs (2.5 mm thick), the bridge, and the back plate (3.5 mm thick) are made out of red maple. Material parameters are given in Table I, considering the orthotropic nature of wood.^12,13 E is the Young's modulus of elasticity (in GPa), G the shear modulus (in GPa), ν the Poisson ratio, ρ the density (in kg/m³), and η the damping factor; r indicates the radial direction, l the longitudinal direction, and τ the tangential direction; in the case of double subscripts, the first index refers to the stress direction and the second to the strain direction.

In order to derive the displacement u of the structure due to a given load, stress-strain relationships need to be established. For linear elastic materials, these may be written in matrix form as

where E is the elasticity compliance matrix, ε the strain vector, and σ the stress vector. In the case of orthotropic materials, such as wood, special attention needs to be taken regarding the orientation of the wood grain. Depending on the way the wooden pieces are cut and placed in order to form the top plate, ribs, and back plate of the instrument, the elasticity matrix can be generated that relates stresses and strains in the viol body. Defining x as the direction across the instrument body, y vertical to it and parallel to the top and bottom plates, and z vertical to the top plate this matrix has the following form:¹⁴ 

The equation of motion of the system may then be written as

F_V being the total load per unit volume applied on the body. This may stem from the tension of the strings, the surrounding acoustic pressure, and any excitation given on the body (such as the harmonic excitation for the admittance calculations in Sec. III).

An approximate solution to Eq. (4) may be calculated using the finite element method (see, e.g., Ref. 15). The instrument is discretized into 33 016 tetrahedral elements (see Fig. 4), 12 380 of which are required for the discretization of the top plate. Note that the neck and the tailpiece of the instrument are omitted from the simulations since their acoustic effect is of minor importance and their geometry too complex to be included in the simulations with sufficient accuracy. Instead, fixed boundary conditions are used for the edges where the top plate meets the neck and the tailpiece. Before considering how the vibrations of the instrument body are coupled to the surrounding air, the reliability of the simulated structural vibrations needs to be validated. This is carried out using two different approaches as follows.

In order to assess the accuracy of the numerically predicted oscillations of the instrument body, comparisons are carried out between numerical and experimental results. Assuming a linear behavior, which should hold for low-amplitude vibrations, the equations describing the body vibrations form a system of linear equations and may be cast in matrix form,¹⁵ which can be used to perform an eigenfrequency analysis, calculating vibrating modes, and the corresponding resonance frequencies. Such an analysis is carried out on the assembled body of the viola da gamba as well as on the isolated top plate.

The vibrational behavior of the top plate, before the assembly of the final instrument, can be experimentally visualized with the aid of Chladni patterns.⁹ This method allows the identification of nodal lines on a vibrating surface while exciting at a resonance frequency. These nodal lines are also present in an eigenfrequency analysis, assuming a harmonic displacement field, that may be used to calculate the modal shapes of the top plate. A comparison between the modes of the reconstructed top plate (according to the asymmetric design discussed above) and those simulated using FEM is depicted in Fig. 5. It can be observed that the modal shapes are similar and occur at frequencies that lie close to each other. This agreement not only points towards the accuracy of the finite element model but also validates the material parameters used in this study (though one should still keep in mind that wood properties may vary even within a single object).

In the case of the full body structural simulation, an eigenfrequency analysis is carried out after the static load due to the tension of the strings is applied on the bridge. This is necessary, since such a load acts as a prestress on the instrument body that has an effect on its modal shapes and the corresponding eigenfrequencies. As such, the vibrating modes may be derived in accordance to the final reconstructed instrument, including the placement of the strings. In order to verify the calculation of the modal shapes and their resonance frequencies, the modes of the reconstructed instrument have been visualized using electronic speckle pattern interferometry (ESPI).¹⁶ 

ESPI is a nondestructive optical technique generally applied in a full-field surface deformation analysis. It uses properties of coherent light reflected from a rough surface and captured by a camera with a superimposed reference beam created by the same source. The formed image is called a subjective speckle pattern.¹⁷ The experimental setup in this work consists of a continuous wave laser (532 nm wavelength) that is separated by a beam splitter in two arms; one is reflected from the vibrating object (in this case the viol top) towards a camera; the second one is directly sent to the camera, where the two laser beams interfere with each other.¹⁸ By subtracting subsequent images from each other a contour map (containing interferometric fringes) can be generated that yields a qualitative description of the instrument's vibrations. The instrument is excited by a shaker (Brüel & Kjær, type 4810) at one side of the bridge and images are captured at resonance frequencies, where modal shapes can be observed. Figure 6 shows the first four modes of the instrument calculated using FEM and compared with those measured using ESPI.

Again, there is a qualitative agreement of the modal shapes, while the resonance frequencies lie close to each other (now considering the complex geometry of the assembled viol body and the fact that two types of wood are present with different material properties). Furthermore, the measured (respectively, simulated) resonance frequencies coincide with the peaks of the measured (respectively, simulated) peaks of the bridge admittance presented in Sec. III A, Fig. 12 (note that the reconstructed instrument labelled “Schuerch” was used for the ESPI measurements). Since at the low-frequency range the vibrating modes are well separated from each other,¹⁹ and assuming a linear behavior, the function of the instrument may be simulated in the frequency domain. Hence, the agreement between measured and calculated modes serves as a tool towards the reliability of the physical model design, including the choice of material parameters.

After the physical model of the instrument has been formulated and its structural behavior validated against experimental measurements, the objective becomes to compare the sound radiation properties of the postulated asymmetric design to that of an instrument with a top plate of constant thickness (4 mm). To this end, a physical model is formulated where the vibrating body radiates sound towards a spherical boundary. Frequency-domain simulations are carried out with the instrument being excited at the side of the bridge, assuming a harmonic excitation of the form $F=|F| cos(ωt+ϕ)$⁠, where F is the force per unit volume, ω the angular frequency, and ϕ the phase of the excitation. The tension from the strings is still taken into account in the form of a static load applied on the bridge prior to the harmonic excitation.

The interaction between the vibrating viol body and the surrounding air is bidirectional; the vibrating surfaces of the body act as moving boundary conditions, imposing an acceleration of the air particles (a_n) normal to the surface of the instrument that is equal to the surface acceleration at the interface between the solid and the air, i.e.,

with n being the unit vector normal to the surface. At the same time, the acoustic pressure p surrounding the instrument presents an external load F_p that acts normal to the instrument's surfaces, that is

Both these effects also occur at the boundaries in the air cavity inside the instrument. The surrounding air domain (including the air inside the instrument) is discretized using an increasing number of tetrahedral elements depending on the simulation frequency; it is thus ensured that at least ten elements per wavelength are used to resolve the wave propagation.

The radiated pressure field can be modeled using the wave equation²⁰ 

where c is the speed of sound, and the acoustic pressure p depends on both position r and frequency ω [i.e., $p=p(r,ω)$] and is generated by the surface vibrations. By applying a Fourier transform, this can be converted into the frequency domain as

where k = ω/c is the wavenumber and $λ=2πc/ω$ is the wavelength of the sound pressure. A highly absorptive layer surrounds the spherical air domain at a distance of 1 m from the instrument to ensure anechoic conditions.²¹ This is necessary in order to design a computationally efficient model, and results in a rapid pressure decrease due to the fact that sound radiation is heavily damped. This is illustrated in Fig. 7 and the accompanying animation.²² Discretization and numerical solution are carried out after the weak form of the underlying equations is derived.¹⁵ Figure 7 shows the radiated sound field for a single frequency of excitation (at 120 Hz). An animation showing sound radiation for a frequency range between 50 and 700 Hz, in relation to the simulated bridge admittance, is also provided.²² 

In order to assess the acoustic efficiency of the instrument, the sound pressure is calculated 0.5 m in front of the instrument. At the same time the input admittance of the instrument is stored by calculating the bridge velocity (since a unity force is used to excite the instrument). Both input admittance and acoustic efficiency are plotted in Fig. 8 for the asymmetric and the symmetric design. It can be observed that the mobility of the bridge is similar for both geometries—and in fact slightly higher in the symmetric case. However, the sound radiation is clearly higher for the asymmetric design over the whole simulated range. This is achieved by distorting the symmetry of the instrument,⁸ which is usually enforced by the presence of a bass bar and a soundpost, but in the presented design it is only due to the variable thickness of the top plate. The asymmetry of the top plate allows the bridge motion to couple to the symmetric fundamental mode of the top plate. Note, however, that this coupling is rather weak for this particular instrument design. The symmetric mode hardly appears in the simulated admittance curve and is very weakly present in the measured curves (see Fig. 12). Nevertheless, when exciting with a shaker, it has been possible to visualize this mode using ESPI by exciting with a force vertical to the bridge, as illustrated in Fig. 9.

The high-Q peaks in the sound radiation correspond to eigenmodes of the air oscillations inside the cavity of the instrument, the first one being the Helmholtz resonance. They mainly assist in reinforcing sound radiation in frequency regions where the bridge admittance is relatively low, especially in the lower frequency range. Figure 10 shows the first two air modes at 123 and 296 Hz. At the lower one, the air inside the cavity oscillates in phase, whereas at the second one, the top and bottom portions of the enclosed air oscillate with opposite phase.

Finally, in an attempt to assess the playability of the reconstructed instruments, the input admittance of two of them was measured inside an anechoic chamber, in accordance with the perfectly matched layers of the simulation. The treble side of the bridge was excited using an impulse hammer (PCB 086C03) that was mounted on a pendulum controlled by a foot pedal (see Fig. 11). The acceleration was measured on the other side of the bridge using a ceramic miniature accelerometer (PCB 352C23). Both excitation and motion signals were recorded and post-processed in order to obtain the input admittance curve. Five repeated measurements were carried out and the results were averaged in order to reduce any effects of measurement noise. The strings of the instrument were damped using a piece of foam, so that energy does not flow from the body oscillations towards the vibrating strings (at the strings' resonance frequencies).

The obtained results are shown in Fig. 12 where the instruments are labelled by the manufacturers' names and the simulated admittance is plotted for reference. Measured and simulated signals show an overall similarity, although it can be observed that several vibrating modes are only present in the measured data. This is a result of the complexity of the instrument geometry and the material properties of wood; neither of which may be easily captured by a physical model and, indeed, differences can be observed between the two reconstructed instruments although, in principle, the same design and material have been used. Nevertheless, both measurements and simulations show that the energy is adequately distributed over a large frequency range, with no isolated peaks in the admittance curve. The playability of both reconstructed instruments has also been confirmed by musicians during a live performance.⁵ 

A final observation regards the amplitude of the simulated admittance in comparison to the measured ones. It can be observed that the measured admittances are decreasing in amplitude with increasing frequency. This does not hold for the simulated curve. This discrepancy stems from the fact that the damping factor used in the simulations is frequency independent. Material damping is difficult to estimate, especially for wooden, complex shapes; therefore, no attempt is carried out to vary the damping factor with the frequency of the simulations. Determining material parameters for wooden structures is out of the scope of this study. Nevertheless, the observations regarding frequency-dependent damping may be of interest to the acoustics community.

The attempt to virtually reconstruct an early viola da gamba after Silverstro Ganassi is reported, following a design that includes no soundpost or bass bar. Thus, the asymmetry of the instrument body is only ensured via the variable thickness of its top plate. The structural and acoustic characteristics of the instrument are analyzed using a finite element model. In order to validate the ability of the model to accurately reconstruct the vibrations of the instrument, comparisons are carried out with measurements on the top plate of the viol (using Chladni patterns) and on the full complete instrument body (using laser interferometry). The agreement between measurement and simulations confirms the accuracy of the model geometry and of the values used for the material parameters.

The interaction of the vibrating body of the viol with the surrounding air has been modeled in the frequency domain in order to compare the postulated asymmetric design with an equivalent symmetric one. It has been shown that, while the bridge mobility is larger for the symmetric case, the asymmetric design has better radiation properties. Furthermore, the simulated bridge admittance is qualitatively comparable to the one measured for two reconstructed instruments. Both instruments have the ability to radiate sound efficiently over a large frequency range. However, several modes that appear in the measured impedance curves are not recovered by the simulations. These modes may be attributed to the parts of the instrument that have not been included in the simulated geometry (such as the neck and the tailpiece) as well as the effects caused by loose connections, geometric imperfections, complex boundary conditions, and other factors not captured by the physical model. Nevertheless, the presented results are valuable to instrument makers that aim to reconstruct early bowed-string instruments, as well as to researchers studying the evolution of such instruments.

The author would like to thank Thilo Hirsch for the conception of the instrument design and instrument makers Stephan Schürch and Günter Mark for the reconstruction of the instruments, the latter also for providing the Chladni patterns.