差分法による不規則形状の音場解析

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An automobile compartment is considered as an irregularly shaped acoustic field enclosed by body panels. The purpose of this paper is to report on a numerical investigation of an acoustic field of general shape, while it is impossible to get the exact solution of normal frequencies or sound pressure distributions of such a field. The calculation method used in this paper is based on the usual finite difference method, but a special consideration is paid for an approximation of the boundary-condition equation (Eq. 5) in order to improve the accuracy of calculations. Eq. 2 or Eq. 3 is wave equation and it is approximated by the finite difference equation given by Eq. 4. A set of simultaneous equations on the sound pressure of interior grid points (Eq. 12 or Eq. 13) is generated by using the suitable relation described in Eq. 7〜Eq. 11 for the approximation of the boundary-condition equation in each boundary illustrated in Fig. 2, where, the quadratic in stead of linear distribution of the sound pressure around the boundary is assumed (Eq. 6). In this way much error can by avoided without involving a longer computation time and a larger computor memory. In case all the boundaries are rigid (V_n=0 in Eq. 5), the set of equations is homogeneous and it is reduced to the eigen value problem as expressed in Eq. 12. The eigen values (normal frequencies) and the eigen vectors (sound pressure modes) are obtained from this equation. The calculated normal frequencies of the lowest four modes for a rectangularly shaped room converge on the exact solution as the grid size vanishes (Table 1). For a trapezoidal shape model, both results obtained with coarse and fine grids quite precisely agree with experimental values as shown in Fig. 3 (normal frequencies) and Fig. 4 and Fig. 5 (sound pressure modes). When some parts of the boundary are given arbitrary vibration (V_n), forced vibration problem of the acoustic field must be studied. The sound pressure gradient which is equivalent to the vibration velocity of the boundary panel forms constant terms of Eq. 13. The sound pressure at each interior grid point is evaluated by solving this equation. The response of forced vibration (Fig. 7) gives the `transfer acoustic impedance density' that is the sound pressure at an arbitrary point in the field radiated from a panel of a unit area vibrating with unit velocity. This result shows that a vibration of the panel which faces an anti-nodal part of sound pressure modes has a higher sensitivity for the interior sound pressure than the one which faces a nodal part. Fig. 9 shows the calculated sound pressure of interior points for various vibration modes of boundary panels. The interior sound pressure can be predicted as the resultant vector of the sound pressure which is radiated from whole boundary panels. This indicates that the vibration prevention of boundary panels does not always reduce the interior sound pressure. The present paper lays emphasis on the calculation method and includes only a simple example. But the results for more complex shapes which simulate automobile compartments verify the method that is mentioned. The effective consideration for the vibration of body panels is also expected for the purpose of reducing the interior noise of an automobile. These are to be discussed at another opportunity.
1972-01-10

差分法による不規則形状の音場解析

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