Finite Element Analysis of Ultrasonic Exponential Solid Horn

Abstract

Many papers have reported on the one-dimensional analysis of an ultrasonic solid horn in half-wave resonance. However, there is a considerable difference in behavior between the horn analyzed by the one-dimensional theory and the horn produced practically by the one-dimensional analysis. As many ultrasonic solid horns are axisymmetric, their vibration is analyzed two-dimentionally and axisymmetrically by means of the finite element method. In this paper, an exponential solid horn is treated and its free vibration is analyzed for the following two conditions:(1) When both ends are free;(2) When a concentrated mass is attached at the small end. Then, the horn is composed of an assembly of triangular ring elements, and the shape of horn is defined by two parameters d_1/l and d_2/d_1 (Fig. 1, 2, 3). As a result of this analysis by the finite element method, many important vibrational properties of the horn which cannot be obtained by the one-dimensional analysis are found, and the examples of these properties are discribed in this paper. The precision of the values obtained by this analysis is very reliable in the first mode (Fig. 5). As the value of d_1/l increases, the value of αl decreases and the higher modes are drawn near respectively (Fig. 6). The discrepancy between the first resonant frequency obtained by this analysis and that obtained by the one-dimensional analysis is less than 0. 2 percent for d_1≦0. 2 (Fig. 7). The variation of αl as a function of Poisson's ratio ν is approximately linear (Fig. 8). As any inner displacement of the horn is a vector displacement that consists of the axial component and the radial component, the models of the horn vibration are very complex (Figs. 10, 11). The amplification factor obtained by this analysis is equal to that obtained by the one-dimensional theory for d_1/l≦0. 5 (Fig. 9). Then the inner stress of the horn becomes larger, and the flatness of the large end becomes worse in vibration (Figs. 12, 13). The nodal surface, where the axial displacement is zero, is not a plane but a curved surface, and the radial displacement exists even in the surface nodal points (Figs. 14. 15. 16). For d_1/l&gt=0. 2, as the value of d_2/d_1 decreases, the effect of the supporting point on αl decreases (Fig. 17). As for the horn with a concentrated mass, its properties obtained by this analysis have a same tendency as those obtained by the one-dimensional analysis (Figs. 18, 19, 20, 21). As mentioned above, the results obtainsed by this analysis can serve as a reference for the designing of an exponential solid horn.