Tibetan bowls have been traditionally used for ceremonial and meditation purposes, but are also increasingly being used in contemporary music-making. They are handcrafted using alloys of several metals and produce different tones, depending on the alloy composition, their shape, size and weight. Most important is the sound producing technique used – either impacting or rubbing, or both simultaneously – as well as the excitation location, the hardness and friction characteristics of the exciting stick. Nevertheless, researchers are increasingly exploring the physical phenomenon behind such handcrafted materials. Several aspects of these instruments do not appear to be clear and a detailed understanding is needed.
Tibetan Singing Bowl
Tibetan singing bowl considered for current analysis is made of brass as shown in Figure 1. The singing bowl has been modelled in FEM using MSC NASTRAN- a commercially available tool widely used for dynamic analysis of structures. The thickness of the singing bowl varies from bottom to top from around 3-3.4 mm. Shell element is used to model the bowl with varying thickness to simulate the actual conditions. The structure is assumed to be made out of brass and its corresponding material properties have been used. The first 4 modes of the bowl are shown in Figure 2.
FEA Mode shapes of Tibetan singing bowl
Figure 2 shows coupled mode shape of Tibetan singing bowl due to symmetric nature of the bowl geometry.
The vibration data are captured through an accelerometer (Dytran, 10 mV/g). The data acquisition is done through Heim DIC-24 (refer APPENDIX-I), a 32-bit system. The data is analyzed and processed in ME Scope 5.0 (Vibrant Technology, USA). The structure is excited through a miniature impulse hammer (PCB, 22.5 mV/N). Using the vibration data, the Frequency Response Functions are built in the modal analysis software and processed to obtain its characteristic resonant frequencies, damping and mode shapes. To capture the sound signatures of the singing bowl, a microphone (GRAS, 12.6 mV/Pa) has been used. The test was conducted using impulse hammer. The modes of singing bowl obtained experimentally are shown in Figure 3.
FRF of Tibetan singing bowl
Mode shapes of Tibetan singing bowl obtained experimentally are shown in Figure 4.
Experimental mode shapes of Tibetan singing bowl
The results are summarized in Table 1
|Mode||FEA Frequency (Hz)||Exp. Frequency (Hz)||Damping (%)|
The experimental mode definitions are very much in line with mode shapes obtained through FEA with identical boundary conditions. Despite good manufacturing quality and symmetry of the structure imperfections exist and thus perfect symmetry is nearly impossible. This leads to the existence of two orthogonal modes which are very close to each other. FEA shows same frequency due to perfect symmetry in an ideal scenario even though the mode shapes are different. It is clear that when the structure is excited through an impulse excitation, the closely spaced modes interact to produce a beat and thus is heard as periodic rise and fall of sound. The microphone response of singing bowl is shown in Figure 5.
Microphone response of Tibetan singing bowl
The auto spectrum response clearly shows the two closely spaced modes (1& 2), when the excitation is done through puja stick. The microphone spectral response has been converted to octave band and is shown in Figure 6.
Octave band showing sound intensity of singing bowl
From Figure 6 it is clearly seen that the sound intensity is maximum (90 dB) near the first two modes of the singing bowl.
Vibration Experiments have been performed on a Tibetan singing bowl to understand its physics. It has been clearly observed that the structure possesses closely spaced modes due its near perfect symmetry. The loud periodic rise and fall of sound is clearly because of beat phenomena resulting due to interaction of closely spaced modes. The frequency spectrum data and octave band show high sound intensity around the vicinity of first two modes, basically responsible for the production of beats.