Experimental and Numerical Modal Analysis of The Turkish Tradi- tional Instrument ‘Bendir’ Uğur Tatlıer 1 İstanbul Technical University İTÜ Gümüşsuyu Kampüsü, Makina Fakültesi, 34437, Beyoğlu - İstanbul / Türkiye Sinem Öztürk 2 İstanbul Technical University İTÜ Gümüşsuyu Kampüsü, Makina Fakültesi, 34437, Beyoğlu - İstanbul / Türkiye Nurşah Öner 3 İstanbul Technical University İTÜ Gümüşsuyu Kampüsü, Makina Fakültesi, 34437, Beyoğlu - İstanbul / Türkiye Sevinc Aycan Yetim 4 İstanbul Technical University İTÜ Gümüşsuyu Kampüsü, Makina Fakültesi, 34437, Beyoğlu - İstanbul / Türkiye

ABSTRACT This study carried out an experimental and numerical modal analysis of the Turkish traditional in- strument ‘Bendir’ by an acoustical excitation source. In experimental modal analysis, Bendir was excited acoustically for specified frequencies and elastic deformations of different areas of Bendir due to the excitation were observed with a laser vibrometer. Using the Fourier series handled gained data, modal analysis was done. The first six natural frequencies obtained experimentally and numer- ically were compared. In addition, other hits in Bendir, symbolizing three fundamental voices such as ‘dum’, ‘te’, ‘ke’ were used to excite the instrument. As a result, time-dependent deformation & velocity data were determined and different responses of Bendir were compared.

1. INTRODUCTION

Vibrating membranes are the fundamental sound source, such as percussion instruments. Cultures around the world use vibrating membranes for different reasons. Even though percussion instruments were just used for rituals in early times, they are used to make musical beats today. These vibration- acoustic sources, used for traditional percussion instruments, also it is being a primary problem for studies of both acoustic and vibration.

In studies on membrane vibrations, [1] Gonçalves et al. and [2] Chakravarty made model verifi- cation by theoretically comparing results created by the nonlinear circular vibrating membranes model and the finite element method. [3] Namdari ve Dehghan examined 2D rectangular and circular membrane vibrations and used numerical analysis to determine mode shapes. [4] Jenkins & Korde

1 tatlier15@itu.edu.tr 2 ozturksi@itu.edu.tr 3 onernursah@itu.edu.tr 4 yetim17@itu.edu.tr

worm 2022

have investigated the acoustical excited membrane's vibration of the instrument whose membrane is created using Mylar, by using laser vibrometer, also determined the first two natural frequency and acoustical maps. [5] Tronchin analyzed the kettledrum vibrations, which are excited acoustic radia- tion, did a modal analysis, compared the conclusions in terms experimentally and theoretically, and created acoustic maps whose mode shapes. [6] Bilbao obtained 3D acoustic maps whose drum by determining the states of linear-nonlinearities by excitation of a double-membrane drum. [7] Ono et al. created the traditional Japanese Drum Wadaiko using two different materials and determined their vibration responses in different frequencies.

In this study, the Turkish traditional percussion instrument Bendir was investigated numerically and experimentally. In experiment studies, a modal analysis was done by acoustic and force excita- tion. An accelerometer and a laser vibrometer were used for acquiring the data. As a result, natural frequencies and mode shapes were obtained. Also, these data were obtained by the finite element method and cross-checked to get accurate results.

2. THEORY

Theoretically, mode shapes can be obtained using Bessel's first and second-order functions.

e-

φሺu, θሻ= ൫C 1 𝐽 𝑚 ሺ𝛽𝑢ሻ+ 𝐷 1 𝑌 𝑚 ሺ𝛽𝑢ሻ൯ሺ𝐶 2 cosሺ𝑚𝜃ሻ

(1)

+ 𝐷 2 sinሺ𝑚𝜃ሻሻ

Receptance is calculated mathematically by the gaining acceleration data were obtained from ac- celerometer and force amplitudes data were collected from the modal hammer, is used in order to get natural frequencies.

𝐻ሺ𝜔ሻ= 𝛼ሺ𝜔ሻ= 𝑋 ሺ 𝜔 ሻ

𝐹ሺ𝜔ሻ

(2)

𝑀ሺ𝜔ሻ= 𝑉 ሺ 𝜔 ሻ

𝐹ሺ𝜔ሻ = 𝑋 ሺ 𝜔 ሻ

𝑖𝜔𝐹ሺ𝜔ሻ

(3)

𝐼ሺ𝜔ሻ= 𝐴 ሺ 𝜔 ሻ

𝐹ሺ𝜔ሻ = 𝑋 ሺ 𝜔 ሻ −𝜔 2 𝐹ሺ𝜔ሻ

(4)

| \ iy

3. EXPERIMENTAL MODEL

Firstly, to string up ‘Bendir’, special apparatus was designed and 3D printed. The designed apparatus and experiment set-up was shown in Figure 1.

Figure 1: Modal analysis experimental set-up.

To determine the first six natural frequencies, the modal analysis of ‘Bendir’ was carried out. 24 points were defined on the Bendir’s face. An accelerometer was fixed at point 23 then a force trans- ducer was used to excite the system at different points of the system. Every frequency response func- tion (FRF) was collected in-store in terms of acceleration and force data. The acquisition interval for frequency was 0 to 500 Hz and was divided into 1600 parts. As can be seen in Figure 2, all the FRFs were plotted in the same graphic to determine where the natural frequencies exist.

Figure 2: All FRF (dB) vs frequency. From the graph shown in Figure 2, 7 natural frequencies were detected in the 0-300 Hz frequency range.

Figure 3: Single FRF (dB) vs frequency respectively accelerance, mobility and receptance.

By calculating the averages of natural frequencies from the peaks of FRF, the first seven natural frequencies data given in the table below were obtained.

Table 1: Experimental natural frequencies.

Mode Number Natural Frequency [Hz]

1 0

2 77.4027

3 145.6471

4 204.4334

5 218.2251

6 260.7816

7 281.6286 To validate the experimental model, a solid model was created to use in finite element method (FEM) software. Figure 4 shows the original Bendir model and the finite element model.

=>

Figure 4: ‘Bendir’ and its Solid Model

For the FEM model of the Bendir, it was significant to define the correct material properties. Jen- kins & Korde [4] used Mylar as instrument membrane material in their study. It was not known which material was used to manufacture the Bendir. Mylar or Polyetilen materials are the general percussion instrument membranes. But beech wood material for the wooden part and real goat leather for the face of the Bendir were the most suitable materials. Material properties and dimensions used in the FEM model were given in Table 2.

Table 2: Material properties & dimensions.

Tambour Body Leather

Dimensions

Inner Diameter 469.6 mm Inner Diameter 469.6 mm

Outer Diameter 449 mm Outer Diameter 449 mm

Depth 65.2 mm Depth 65.2 mm

Material Properties

Density 500 kg/m 3 Density 500 kg/m 3

Elasticity Modulus 1.19×10 10 Pa Elasticity Modulus 1.19×10 10 Pa

Poisson’s Ratio 0.42 Poisson’s Ratio 0.42

The first six natural frequencies and mode shapes were obtained by numerical modal analysis. From Table 3, it can be noted that the natural frequencies were close to the values obtained experi- mentally.

Table 3: Numerical Natural Frequencies.

Mode Number Natural Frequency [Hz]

1 0

2 78.824

3 150.99

4 227.14

5 254.67

6 282.84

The mode shapes obtained numerically were depicted in Figure 5.

Figure 5: First six mode shapes.

The sound source was placed behind the center of ‘Bendir. Then a sine wave in different frequen- cies such as 300, 500, and 800 Hz was created. To get deformations in out-plane vibration, a Polytec PSV-400 Scanning vibrometer was used. The entire surface of ‘Bendir’ was scanned for different nodes. Figure 6 shows the experimental setup with the laser vibrometer.

Figure 6: The experimental setup with the laser vibrometer.

Deflection shapes of the Bendir due to the excitation with 300 Hz, 500 Hz and 800 Hz sine waves were shown in Figure 7, Figure 8 and Figure 9, respectively.

Figure 7: Deflection shapes due to the excitation with 300 Hz wave.

Figure 8: Deflection shapes due to the excitation with 500 Hz wave.

1 | iia

Figure 9: Deflection shapes due to the excitation with 800 Hz wave.

The mode shapes obtained by the laser vibrometer at the natural frequencies of 80 Hz, 150 Hz, 200 Hz, 220 Hz, 260 Hz and 280 Hz can be seen in Figure 10.

Figure 10: Acoustic maps due to the excitation with different natural frequencies

80 Hz – 150 Hz – 200 Hz – 220 Hz – 260 Hz – 280 Hz.

The experimental and FEM results were compared as depicted in Figure 11. It can be easily said that there is a close correlation between those results.

Figure 11: Numerical and Experimental Comparison of Natural Frequency. 4. CONCLUSIONS

In this study, the natural frequencies and mode shapes of the Turkish traditional percussion instrument Bendir were determined by using both modal analysis and finite element methods. In modal analysis, two different excitation and data acquisition methods were applied. In the first method, the Bendir was excited acoustically for different frequencies and the data were collected by Polytec PSV-400 Scanning vibrometer. In the second method, the Bendir was excited by a force transducer the data were collected by an accelerometer. In conclusion, the natural frequencies and mode shapes that were obtained in two different experimental methods and the FEM method were compared. The results were found to be compatible with each other. 6. REFERENCES

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