Modal radiation patterns of baffled circular plates and membranes

Abstract

The far field velocity potential and radiation pattern of baffled circular plates and membranes are found analytically using the full set of modal velocity profiles derived from the corresponding equation of motion. The derivation is valid for a plate or membrane subjected to an external excitation force, which is used as a sound receiver in any medium or as a sound transmitter in a gaseous medium. A general, concise expression is given for the radiation pattern of any mode of the membrane and the plate with arbitrary boundary conditions. Specific solutions are given for the four special cases of a plate with clamped, simply supported, and free edge boundary conditions as well as for the membrane. For all non-axisymmetric modes, the velocity potential along the axis of the radiator is found to be strictly zero. In the long wavelength limit, the radiation pattern of all axisymmetric modes approaches that of a monopole, while the non-axisymmetric modes exhibit multipole behavior. Numerical results are also given, demonstrating the implications of having non-axisymmetric excitation using both a point excitation with varying eccentricity and a homogeneous excitation acting on half of the circular radiator.

Robust active sound radiation control of a piezo-laminated composite circular plate of arbitrary thickness based on the exact 3D elasticity model

“…Subsequently, Figure 4(d) presents the normalized radiation pattern plots of the simply supported circular piezo-laminated plate for the (1,0) axisymmetric and the (1,1) nonaxisymmetric modes (see equations (17)) at both long and short wavelengths (kro=10-5,10). Excellent agreements are obtained with the results presented in Figure 2 by Christiansen et al. 57 (note that the plots have been normalized with respect to their peak values). Here, it should be noted that at the low-frequency selection (kro=10-5), radiation patterns of the symmetric and asymmetric modes are very similar to those of a monopole and a dipole radiator, respectively.…”

“…Also, it is noteworthy that, although the numerical results presented by Christiansen et al. 57 include the nonaxisymmetric loading situation, they are based on the simplifying far-field approximation . Lastly, we used our main code to compare the calculated nondimensional natural frequencies (i.e.…”

“…Here, it should be noted that at the low-frequency selection kr o ¼ 10 À5 À Á , radiation patterns of the symmetric and asymmetric modes are very similar to those of a monopole and a dipole radiator, respectively. Also, it is noteworthy that, although the numerical results presented by Christiansen et al 57 include the nonaxisymmetric loading situation, they are based on the simplifying far-field approximation. Lastly, we used our main code to compare the calculated nondimensional natural frequencies (i.e.…”

“…[56][57][58] Unfortunately, due to presence of singularities, 59,60 the Rayleigh integral (equation (16)) is not actually responsive to direct numerical calculation of the near-field pressure, especially at high frequencies. 39 This difficulty can be circumvented by following the mathematical treatment presented for example in the literatures, 59,61,62 where the traditional Hankel transform (cylindrical wave) representation of the singular term (spherical wave) is adopted; 63 i.e.…”

“…Good agreements are obtained with the results provided in Figure 1 by Zawieska et al 86 as shown in Figure 4(c) (note that the numerical data in Zawieska et al 86 are for axisymmetric loading, and based on the low-frequency assumption). Subsequently, Figure 4 et al 57 (note that the plots have been normalized with respect to their peak values). Here, it should be noted that at the low-frequency selection kr o ¼ 10 À5 À Á , radiation patterns of the symmetric and asymmetric modes are very similar to those of a monopole and a dipole radiator, respectively.…”