The Influence of Wood Modification on Transfer Function of a Violin Bridge Utjecaj modifikacije drva na prijenosnu funkciju mosta za violinu

• The violin bridge is an important component of a violin since it transmits the excitation forces from the string to the violin body. Depending on its structure, at a certain frequency spectrum, the bridge acts as a damper or ampli ﬁ er of excitation forces, which depends on its transfer function. In the study, transfer functions in the range from 400 Hz to 7000 Hz in vertical directions of 3 bridges were measured. The bridges were made from maple wood and supplied by different manufacturers. The bridges were then thermally modi ﬁ ed, and the transfer functions were measured again. To determine the in ﬂ uence of thermal modi ﬁ cation on material properties, a sample of maple wood was also modi ﬁ ed together with the bridges, and the modulus of elasticity and shear modulus before and after the modi ﬁ cation were measured. Using Ansys software, a bridge was modelled by the ﬁ nite element method, by which natural frequencies and transfer functions before and after the modi ﬁ cation were calculated. It can be con ﬁ rmed from the research that wood modi ﬁ cation in ﬂ uences the bridge transfer function and that the ﬁ nite element method can be used to determine the dynamic properties of the bridge by knowing the wood material properties and, therefore, to predetermine the transfer function of the violin bridge before its production.

The purpose of the research was thus to study the infl uence of thermal modifi cation of wood on the dynamic properties of a violin bridge and to determine their transfer function.

MATERIJALI I METODE
Three bridges made from maple wood were taken ( Figure 1). Bridges No. 1 and No. 2 were made by Aubert and bridge No. 3 was made by the Teller company. Bridge No. 2 was partially designed and adapted for use on a specifi c violin.
The violin bridges were clamped on a Kistler dynamometer type 9272 ( Figure 2) and the transfer functions of the bridges were measured in the vertical direction. The bridges were excited on the upper side of the bridge with a harmonic force in the frequency range from 400 to 7000 Hz, in 3 Hz steps by Tira TV51120 exciter. The excitation was gradual from minimum to maximum frequency, whereby each excitation frequency was paused for 150 ms for a steady state response to be established. The excitation force was measured with a Bruel & Kjaser type 8200 piezo force transducer, which was inserted between the bridge and the exciter, and the response of the transmitted force was measured with a Kistler dynamometer. Both force signals were captured with a sampling frequency of 100 kHz using a NI USB 6361 measuring card and LA-bVIEW software, which was also used for the transfer function calculation. The transfer function of the experimental system was also measured, so that the Kistler dynamometer was directly excited, and the transfer function calculated from the measured excitation and response forces.
Bridge No. 1 was then modelled with Solidworks software, and a modal analysis was performed using fi nite element methods with Ansys software, whereby the natural frequencies of the bridge were calculated. The modulus of elasticity, shear modulus and Poisson ratios were taken from the literature (Kollmann, 1975) and adjusted relatively so that the calculated natural frequency of the main mode was in agreement with the natural frequency obtained from the measured transfer function. Material density and damping ratios, however, were determined from measurements of the mass and volume of the bridge, and from the measured transfer function, respectively, using the bandwidth method (Maia and Silva, 1998)

UVOD
A violin is a musical instrument that has been known for centuries. The fi rst violins were developed in Italy in the 16 th and 17 th centuries (Rossing, 2010) and reached a peak in the eighteen century in the hands of masters such as Antonio Stradivarius and Giuseppe Guarneri del Gesu of Cremona. Since then, many researchers have been involved in violin research. Some have explored the functioning of violins as a whole (Bissinger, 2008;Matsutani, 2018;Tai et al., 2018;Woodhouse, 2014), while others have studied the infl uence of various materials from which a violin is made and their surface treatment (Aditanoyo et al., 2017;Setragno et al., 2017), as well as the infl uence of damping and strings on the composition of the sound (Bissinger, 2004;Ravina, 2017).
A violin consists of several different components, whereby the bridge is defi nitely one of the more important parts. Its job is to transfer the excitation forces from the string to the top of the violin. When a violin player excites a string with a bow, it starts to oscillate at a frequency that depends on the string and can range from a few hundred to several thousand Hz. In addition to the basic frequency, the string also oscillates with higher frequencies, called harmonic frequencies, which are integer multiples of the basic frequency. Depending on the structure, the violin bridge can thus act as a fi lter, by which at a certain frequency range the excitation forces can be transferred to the body of the violin, in some regions even amplifi ed, while suppressed in another frequency range. Both frequency ranges can be identifi ed by a transfer function, which can be measured from the bridge or calculated using various approaches. The importance of the bridge and its dynamic properties is also evidenced by numerous studies that have been done by various researchers, who have investigated both its shape and the material from which the bridge is made (Bissinger, 2006   where r 2 and r 1 represent the ratios between the excitation and natural frequencies at the amplitude values of the transfer function which are by lower than the maximum value for a given vibration mode, as shown in Figure 3. Using the fi nite element method, a harmonic analysis was also performed, by which the transfer function was calculated. The clamped bridge was excited with a force of 1 N at different frequencies from 400 to 7000 Hz, in steps of 10 Hz, on 1/4 length of the bridge, so that asymmetric modes were also excited ( Figure 4).
In order to determine the effect of thermal modifi cation on the material properties, a maple board with no visible defects and uniform growth was taken. From the board, 4 specimens with dimensions 300 mm × 80 mm × 8 mm and 160 mm × 25 mm × 8 mm in L × R × T and R × L × T directions, respectively, were cut and modifi ed by the same procedure as the bridges.
Prior to the modifi cation, the elastic modulus in longitudinal and radial directions and the shear modulus were measured, the samples being freely supported at distances of 0.22 L and 0.77 L, and excited with a hammer ( Figure 5) to vibrate freely at their natural frequencies.
The vibration of the samples was measured using a Bruel & Kjaer Typ 4939 microphone, NI USB 6361 measuring card and LabVIEW software, with a sampling rate of 100 kHz. An FFT (Fast Fourier Transform) was made from the time signal and the natural frequencies f n for the various vibration modes were determined from the frequency spectrum. Modulus of elasticity E and shear modulus G were calculated using linear regression of the equation (Brancheriau and Bailleres, 2002) (2) where K is a constant that depends on the geometry of the cross-section (for rectangular, K = 5/6), I is the moment of inertia, A is the cross-sectional area of the beam, f n is the n-th natural frequency. Parameters m, F 1 (m) and F 2 (m) are calculated on the basis of index n for the n-th natural frequency.  creased by 1.2 % and 17 %. The latter value is relatively large compared to other values and this may be due to measurement error. In contrast to the elastic properties, the density decreased by 3 %, and the damping ratios in longitudinal and radial directions by 15 %. Table 2 shows the properties of the material that were used in the modelling of the bridge by the fi nite element method, together with the correction factors obtained from Table 1. To calculate the change in all three values of the shear modulus, a single factor of 1.012 obtained from a comparison of the shear modulus of G LT was used, since the factor of 1.169 obtained from parallel samples seemed to be somewhat large and unrealistic when compared to the G RT module. Table 2 also shows the material density for bridge No. 1 and a damping factor of 0.029 and 0.016 for the unmodifi ed and modifi ed bridge, respectively, which are significantly higher than the damping values obtained from the maple samples, where the damping factors in longitudinal samples for unmodifi ed and modifi ed wood amounted to 0.0081 and 0.0069, respectively, and for radial samples to 0.013 and 0.011, respectively.
The measured transfer functions of the violin bridges are shown in Figures 7 to 9. The fi gures clearly From the time recordings of specimen vibration, the material damping ratio  was also determined by means of logarithmic decrement (Maia and Silva, 1998) where X i and X i+n are the amplitudes at locations i and i+n, respectively, as shown in Figure 6.
The bridges and maple specimens were then thermally modifi ed. Thermal modifi cation was performed according to the Silvapro® commercial procedure (Rep et al., 2012). Samples were thermally modifi ed at 195 °C. Prior to modifi cation, samples were wrapped in Al foil to limit oxidation processes of wood. Al foil considerably slows down oxygen diffusion and consequently prevents unwanted polymer degradation. This solution proved to be effective for thermal modifi cation of smaller specimens. Results are comparable to modifi cation in an anoxic atmosphere. The time of modifi cation at the target temperature was 3 hours and mass loss of the samples after modifi cation was determined gravimetrically. After modifi cation, samples were stored in the laboratory for a week (23 °C; 65 %). After conditioning, both the transfer functions of the bridges and the material properties of the maple specimens were determined again. From the data, the infl uence of the thermal modifi cation, in terms of the modifi cation coeffi cient of wood properties, was calculated for the modulus of elasticity, shear modulus, density and damping ratios. Taking into account the changes due to the modifi cation, modal and harmonic fi nite element analysis with the fi nite element method was performed again, and a comparison between the measured and calculated transfer functions of the bridges was made. Table 1 shows the average values of the measured properties of maple specimens before and after modifi cation. The modulus of elasticity in the longitudinal and radial directions increased by an average of 3.5 % and 5 %, respectively, while shear modules in- Figure 6 Determination of parameters needed for calculation of a logarithmic decrement of an underdamped system from a transient response Slika 6. Određivanje parametara potrebnih za proračun logaritamske dekretacije prigušenog sustava iz prolaznog odziva In addition to the increase in the resonant frequencies, the images also show an increase in the amplitude of the modifi ed bridges, which coincides with a decrease in the damping. In addition to an amplitude increase at the mentioned frequencies, the amplitudes of the transfer functions are also increased at frequencies from 500 to 1500 Hz, which are due to the transfer function of the experimental system, and therefore should not be taken into account. The images show that the experimental system has the fi rst resonant frequency with the highest vibration amplitude at 1180 Hz, and additional resonant frequencies at higher values but with signifi cantly smaller amplitudes.

REZULTATI I RASPRAVA
To get the representative vibration modes, only bridge No. 1 was modelled with the fi nite element method, using the data from Table 2. Figure 10 shows the vibration modes of bridge No. 1 together with the associated natural frequencies. The most important vibration modes for a violin are #3, known as the "rocking" mode, and vibration mode #6, known as the "bouncing" mode, where the calculated natural frequency for unmodifi ed and modifi ed bridges amounted to 4705 Hz and 4960 Hz, respectively. In addition to the modal analysis, a harmonic analysis was performed, by which a transfer function was calculated for bridge No. 1, whose values are shown in Figure 7. Very good agreement between the measured and calculated transfer functions can be seen between 3000 Hz and 7000 Hz for both unmodifi ed and modifi ed bridges, which deteriorates at frequencies below 3000 Hz, since the measured transfer function has higher values due to the infl uence of the experimental system. If the measured transfer function of the bridge were to be correct-  ed considering the experimental system transfer function, the agreement would be better at lower frequencies, too. The calculated transfer function of the bridge, in addition to a pronounced increase in amplitudes at 4720 Hz and 4950 Hz, also has a slight in-crease in amplitudes at frequencies of 2060 Hz and 2200 Hz, which are the natural frequencies of the rocking oscillation mode, and, due to its low vibration amplitude, are not evident in the measured transfer functions.

ZAKLJUČAK
From the analysis of the violin bridges, whereby the transfer functions were measured and calculated, it can be confi rmed that thermal modifi cation of wood affects the transfer function of violin bridges. Since the modulus of elasticity and shear modulus increase and the density decreases with modifi cation, the natural frequencies increase accordingly, which means that the frequency range of the forces transferred by the bridge from the strings to the violin body increases. In addition, the modifi cation also reduces the damping factors of the material, which helps to increase the amplitudes at the resonant frequencies of the bridges. The research thus confi rmed that thermal modifi cation is reasonable when the transmitted frequency range is to be increased and bridge damping is to be minimized. It is also reasonable to use the fi nite element method to calculate the transfer function and the natural frequencies of the bridge, since the study showed very good agreement between the calculated and measured values.