著者

柳沢 猛 中村 喜十郎 白柳 伊佐雄

出版者

一般社団法人 日本音響学会

雑誌

日本音響学会誌 (ISSN:03694232)

巻号頁・発行日

vol.33, no.8, pp.412-416, 1977-08-01 (Released:2017-06-02)

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The longitudinal displacements of a point on a piano string changing with time immediately after hammering are calculated, together with the transverse displacements, using the finite element method. This method assumes that the string is elastic and completely flexible with small mass points and a large mass point of the sound board on it. From equations (1)〜(6) and Fig. 1, the values of transverse and longitudinal displacements changing with time at any point of the string immediately after hammering may be calculated (Figs. 2, 3, 4). It is found from the analysis that the longitudinal vibration generated in the string causes the sound board to produce an inharmonic tone of high frequency (Fig. 5).

著者

柳沢 猛 中村 喜十郎 白柳 伊佐雄

出版者

一般社団法人 日本音響学会

雑誌

日本音響学会誌 (ISSN:03694232)

巻号頁・発行日

vol.31, no.11, pp.661-666, 1975-11-01 (Released:2017-06-02)

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In this paper, the vibration system of a piano string and a sound board (Fig. 1) is analyzed by the finite element method (Fig. 2). The string is stretched with constant tension P between the upper bearing and the lower bearing, and is assumed to be completely flexible. The mass point m are distributed along the string at equal distances, and m_1 indicates the upper bearing, m_&lt103&gt the lower bearing, m_&lt97&gt the equlivalent mass of the sound board, and k the spring constant. It is also assumed that the mass-point of the hammer m_H collides with the point m_N on the string with an initial velocity x^^^. _H, that they repel each other according to Newton's law, and that m_N is decelerated by the tension P. Then it collides with m_H again, and this series of motions is repeated. These motions of all points m_i, m_H, m_N are expressed by Eqs. (1), (2), and (3). The flow diagram of the program is shown in Fig. 6. The calculated values by this simulation program and the measured values of an actual piano are presented in Figs. 7 and 8. Comparisons between them show good agreement.

著者

柳沢 猛 中村 喜十郎 白柳 伊佐雄

出版者

一般社団法人日本音響学会

雑誌

日本音響学会誌 (ISSN:03694232)

巻号頁・発行日

vol.31, no.11, pp.661-666, 1975-11-01

---

In this paper, the vibration system of a piano string and a sound board (Fig. 1) is analyzed by the finite element method (Fig. 2). The string is stretched with constant tension P between the upper bearing and the lower bearing, and is assumed to be completely flexible. The mass point m are distributed along the string at equal distances, and m_1 indicates the upper bearing, m_&lt103&gt the lower bearing, m_&lt97&gt the equlivalent mass of the sound board, and k the spring constant. It is also assumed that the mass-point of the hammer m_H collides with the point m_N on the string with an initial velocity x^^^. _H, that they repel each other according to Newton's law, and that m_N is decelerated by the tension P. Then it collides with m_H again, and this series of motions is repeated. These motions of all points m_i, m_H, m_N are expressed by Eqs. (1), (2), and (3). The flow diagram of the program is shown in Fig. 6. The calculated values by this simulation program and the measured values of an actual piano are presented in Figs. 7 and 8. Comparisons between them show good agreement.