Measurement and Visualization System of Eigenmode Shapes Considering Orthogonal Modes on Circular Membrane

Abstract

This work proposes a visualization system of eigenmode shapes of a tapped circular membrane of membranophones. Under tapped membrane vibration, it is not enough to divide modes in frequency domain, because of the existence of eignmodes whose frequencies are close to each other including orthogonal modes. Therefore, we structured this visualization system of eigenmode shapes, which could divide the orthogonal modes appearing at slightly different frequencies. This system firstly measures the sound pressures emitted from the vibration on tapped circular membrane of membranophone with a circular microphone array placed in proximity to the membrane. Secondly, it extracts the circumferential shape of each eigenfrequency. In this system, dominant frequencies of estimated power spectrum density obtained by Yule-Walker method are considered as approximate eigenfrequences of this vibration. Multiple eigenmodes whose azimuthal mode order is different, are divided by Fourier series expansion of the circumferential shapes of each eigenfrequency. In addition, orthogonal modes, which have the same mode order and slightly different frequencies are divided by considering the orthogonality of mode shapes, sine and cosine functions. Lastly, the system estimates the eigenmode shape of the whole membrane using the Bessel function of the first kind correspond to each mode order as radial shape, and shows them to users. It is expected that user could recognize the unbalance of tension distribution intuitively using this system. It is also expected the system to be applied to the tuning assistive system of the tuning of circular membranophone.