細い棒をつけた片持梁の振動 : 角のついた特殊鐘及び棒をつけた和鐘のモデル : 固有振動数の連続変化の研究第2報
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概要
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As a model of the special bell with horns and of the Japanese bell attached with a bar, which permits mathematical analysis, a canti-lever attached with a thin bar is considered(Fig. 1). The kinetic and potential energies of the system are represented by the coordinate system shown in Fig. 2 (Eqs. 3, 3', 4, 4'). Expanding the deformation y_I, y_<II> of the system by the eigen-functions of canti-levers (Eq. 9), the kinetic and potential energies are represented by the coefficients q_<Ir>, q_<IIr> of the expansions. (Eqs. 12. 12', 13, 13'). Thus, Lagrange's Eqs. of motion are established and Eqs. (17, 18) which determine the vibration of the system, are obtained. These are linear differential Eqs. with constant coef. of ∞ degree of freedom. The characteristic frequencies of the system could be obtained in principle, by the ordinary method. The procedures are shown briefly from Eqs(19) to(35). The difficulty arises from a practical point of view due to the extent of the degree of freedom. However considered physically, the higher mode of the expansion may be neglected, but to solve the Eq. (33) rigorously is not easy. An effort is made to get out of the difficulty by the use of the perturbation theory. But, since these results do not agree with the experimental results, preference is given to solve the Eq. (33) by the computer (IBM 1130) using Jacobi's method. The order of symmetric matrix, the elements of which are defined by Eqs. (32, 32', 32"), is limited to 10. The results of the calculations are shown in Table 4 and Fig. 6. The agreement with experimental results is valid up to 1%. Because the above calculations are too formal, it is next shown that the nature of the Eqs. (17, 18) has the characteristic feature of the continuous changes of frequencies of the special bell and the Jap. bell attached with a bar, shown in the Fig. 3, 4, of the previous article. Firstly, if the terms containing μ'_<II> in the Eq. (17) is neglected, and μ'_<II> L_<II> in Eq. (18) is deleted, Eqs. (37, 37') are obtained. The Eq. (37) shows that q_<Ir> vibrates with angular frequency ω_<Ir> and by inserting the results of Eq. (37) in (37'), the Eq. representing the typical forced vibration is obtained. The Eqs. shows that the system has the characteristic frequencies ω_<Ir> and ω_<IIs> (L_<II>) approximately (Fig. 3), which are the characteristic frequencies of the bare canti-lever and those of the thin bar as a canti-lever. The curves which represent the characteristic frequencies of the system as a function of the length of the attached bar should pass close to these frequency curves. These characteristics of the Eqs. (17, 18) may be foreseen as those of the special bell and the Jap. bell attached with a bar. Next, the perturbation treatment is mentioned briefly in Eq. (33), which shows the behavior of the continuous changes of the frequencies where the curves approach each other (Fig. 4, 5). Finally, the calculated results by the computer and the experimental results are shown. At the end of the article, the beat phenomena observed at the part where the curves approach each other are touched on and their relation to the past studies of the author is explained.
- 社団法人日本音響学会の論文
- 1972-12-01
著者
関連論文
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