Unsymmetrical Vibrasion of Circular Diaphragm with Concentric Rigid Part at th Center

Abstract

A piston diaphragm is an ideal diaphragm for electoroacoustic transducers such as telephone transmitter and receiver. In order to realize the ideal diaphragm, the concentric part at the center of a circular diaphragm is bent into the form of a cone or a dome. As a result, the concentric part, which is rigid compared with the circumferential part of the diaphragm, performs a piston vibration. So a circular diaphragm with a concentric rigid at the center is adopted as the model of vibration analysis. Steep peak and dip of the frequency response curve are often observed in the neighbourhood of resonace frequency in measuring the displacement of diaphragm. This is explained by the fact that the resonance frequency of symmetrical mode, with which the rigid part at the center vibrates in translational motion and that of unsymmetrical mode with which the rigid part vibrates in rotational motion around the diameter of the rigid part are close to each other. The equation of motion and the boundary condition of the diaphragm of Fig. 1 are given by Eqs. 1 and 2 respectively. A general solution of Eq. 1 is represented by Eq. 3 and the displacement along the circle of radius b is given by Eq. 4. While the displacement along the same circle of Fig. 2 is given by Eq. 5. Comparison of Eqs. 4 and 5 shows that the displacement of translational motion is represented by Eq. 6 and that of rotational motion by Eqs. 7 and 8. The force acting on the boundary element of the rigid part is a uniform transverse force in the case of translational motion and a periodic transverse force with respect to angle θ in the case of rotational motion. Accordingly the boundary conditions along radius b are given by Eq. 13 and Eq. 14. On the basis of the displacements and the boundary conditions, the frequency equations 16 and 17 with respect to symmetrical mode and unsymmetrical mode respectively are obtained. Roots α_&ltm, n&gt (m=0, 1) called normal constants are shown in Fig. 3 as a function of β or γ. The relation between normal constants and angular resonance frequency is given by Eq. 18. If 4κ^2/a^2, the ratio of γ or β, is nearly equal to 1, where a is the radius of the diaphragm and κ the radius of gyration of the rigid part, then two steep peaks appear in about the same frequency of the frequency response curve. The diaphrams used in the measurement are simple diaphragms shown in Fig. 4 and that bent into the form of a cone or a dome. The curve(a) of Fig. 5 is the relative displacement of the center point of the rigid part and curve(b) is that of the circumferential point of the rigid part. Comparison of curves (a) and (b) shows that the first resonance corresponds to the resonance of the translational motion of the rigid part and the second resonance to that of the rotational motion. The relative displacement of the diaphragm used in the telephone receiver R-60 is shown in Fig. 9. These measured values are in good agreenment with the computed values.