AnaIysis of L-Ltype Resonator / By means of Coupled-Vibration Theory and Finite Element Method

A longitudinal-longitudinal vibration energy concentrator/divider, which has been devised by Ito et al. and is called "L-L type resonator", has such a construction that two half-wave-resonant (longitudinal mode) bars are crossed at their nodal portion of vidrational displace-ment. It is an epoch-makingly useful devise in the field of ultrasonic power applications because of its capability to concentrate the vibrational energy from plural sources into a single load or vice versa. Among several variations of those resonators (such as R-Ltype or L-L-L type etc. ), the authors took up an elemental L-L type resonator in this paper and analysed it, first, by means of coupled-vibration-theory to obtain its qualitative characteristics, and then analysed by means of finite-element-method in order to check the assumptions employed in the former analysis and to obtain quantitative values. From these studies, the analysis based on the coupled-vibration-theory is shown to be valid in qualitative sense, and the design charts of the resonators point-symmetrical with respect to its center are given. At first, the authors assumed that the vibrational displacement (u, v) of the arbitrary point (x, y) in the resonator can be expressed by eq. (1). The generalized equation of motion is given by eq. (2), which is a two dimensional eigenvalue equation in generalized form, and s_&ltij&gt and m_&ltij&gt are the coefficients given in eq. (3). Coupling factor k is defined as eq. (4). These s_&ltij&gt and m_&ltij&gt are given as eq. (7) and (8) under the assumption of two-dimensional plane-stress condition. Eq. (2) can be written as eq. (9) which is the standard expression of two-dimensional eigenvalue equation. Resonant condition of the system is given by eq. (10). The resonant frequencies f_&lt0S&gt are given by eq. (12) or eq. (13), or found in Fig. 3. Amplitude ratios for both arm ends are given by eq. (15) or (16). The product of the two respective ratios of the first and the second mode is given by eq. (17), which is always negative. By examining the sinse of the double-sign contained in eq. (16), it is found that the amplitude ratio of the first mode is negative and the second mode positive. Both ratios can be expressed as eq. (19) and are shown in Fig. 5 in normalized from. Form this analysis, coupling factor is given by eq. (14), which suggests that it is proportional to Poisson's ratio of the material employed. This relation is confirmed by the results of finite-element-analyses. Analysis described here, based on the coupled-vibration-theory, has a deficiency in that the vibrational displacement is assumed as eq. (1), which means the neglecting of the shearing stresses. The authors examined this assumption by finite-element-method, and clarified that the neglection of the shearing stresses can be permitted for the first mode, but it shouldn't be made for the second mode. For the finite-element analysis, a quarter of the resonator, as shown in Fig. 7, was analysed as a two-dimensional plane-stress eigenvalue problem by the program "FEM23". This program uses triangular-elements with three nodal points for an element and first order polynomial function as trial function. Main results are shown in Figs. 8 to 11. Comparisons with the experimental values are given in Table 1. Charts for the design of resonators point-symmetrical with respect to its center are shown in Figs. 12 and 13. Fig. 14 shows a designed example.