The Journal of the Acoustical Society of Korea (한국음향학회지)

The Acoustical Society of Korea (한국음향학회)
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Transmission Line Based Plucked String Model

전송선로 기반 탄현 모델

  • Lee, Jingeol (Department of Electronic Engineering, Paichai University) ;
  • French, Mark (Department of Mechanical Engineering Technology, Purdue University)
  • Received : 2013.02.18
  • Accepted : 2013.03.26
  • Published : 2013.07.31
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Abstract

As one way to describe the behavior of a vibrating string, analogies to a transmission line have been made based on the fact that they have oppositely travelling waves on each of them. In such analogies, a rigid end to the string has been represented as an open circuit, and the displacement of the string as the current on the transmission line. However it turns out that the rigid end corresponds to a short circuit, the displacement to the voltage by the theory of the transmission line, and it is confirmed by experiments with circuit simulations. Based on these discoveries, a transmission line based plucked string model comprising a transmission line, two piecewise linear current sources, and switches is proposed. The proposed model is validated by showing that the voltage at the arbitrarily chosen location, and the voltage calculated over an infinitesimal portion at the end of the transmission line are consistent with the displacement at the corresponding location and the force on the rigid end of the string from the well known difference form of a wave equation governing the behavior of the string with its fundamental frequency tuned to that for the proposed model, respectively. Moreover, the applicability of the proposed model to modeling string and wind instruments is presented.

진동하는 현의 성질을 나타내는 방법으로 반대 방향으로 진행하는 파가 현과 전송선로에 존재한다는 사실에 기초하여 현은 전송선로에 비유되어왔다. 이러한 비유에서 현의 강역(rigid end)과 변위는 각각 전송선로의 개방회로와 전류로 나타내어졌다. 그러나, 본 연구에서 강역과 변위는 각각 단락회로와 전압에 해당됨이 전송선로의 이론으로부터 밝혀졌고 이를 회로시뮬레이션으로 확인하였다. 이러한 발견에 기초하여 전송선로, 구분적 선형 전류원, 스위치들로 구성된 전송선로 기반 탄현 모델을 제안하였다. 임의로 선택된 지점에서의 전압과 전송선로 끝 극소 부분 양단에서 계산된 전압이 현의 성질을 지배하는 파동방정식의 차분형식(difference form)으로 구한 해당 지점에서 변위와 강역에서의 힘과 일치함을 보임으로서 제안한 모델이 정당함을 증명하였다. 또한, 제안된 모델의 현악기 및 관악기 모델링의 적용성을 제시하였다.

Keywords

References

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  2. Transmission line based struck string model vol.111, 2016, https://doi.org/10.1016/j.apacoust.2016.04.002