Resonant Frequency and Vibration Mode on U-Type Vibrator

Abstract

This paper deals with an analysis of a U-type vibrator restrained by arbitrary supporting stiffness at the center of the circular part of the vibrator, and presents some diagrams of the frequency constants and vibration modes, which are very convenient for design. As symmetric vibration on the U-type resonator is treated in this paper, it is possible to analyze by only a half section of the vibrator, and the center of the circular part is analogized for a roll-end binding with arbitrary stiffness s_s (Fig. 1). From eqs. (3), (4), equations of motion on a circular bar, a characteristic equation (7) is derived and its solutions are obtained as eqs. (8)〜(16). The conditions of connection are given as eqs. (17)〜(26), so the characteristic equations on the U-type vibrator are obtained as eqs. (28)〜(45) by using those mentioned above. The calculated results of the frequency constants α in relation to the dimension ratio l_a/l and some experimental results are shown in Fig. 2, and each curve demonstrates in order (a), (b), (c) the values on the first mode in the case of a clamped end (s_s=∞), roll end (s_s=0) and those of the second mode in the case of a clamped end (s_s=∞). Fig. 3 illustrates deviations of the frequency constants of the vibrator in the case of a roll end in relation to those of the clamped end on the first mode. It is evident that the values obtained in the case of roll end approach those of the clamped end in the region of practical use, about l_a/l≧0. 7, so that either value can be use in designing the vibrator. Fig. 4 and 5 illustrate the characteristics of frequency constant α to normalized supporting stiffness ζ as the parameter of the dimension ratio l_a/l, the diagrams demonstrate coupling resonance occurred with vibration of elastic symmetrical mode and rigid-vibration at a certain point of ζ. This indicates that spurious vibrations occur in the neighborhood of the coupling resonance, so that in designing, a suitable value of supporting stiffness must be chosen except the coupling resonance in accordance with the figures given above. The vibration modes at points numbered by the ○ mark in Fig. 4 are shown in Fig. 6 (a)〜(c). These figures indicate shifting aspects of modes from the rigid-vibration to the first mode of the clamped vibrator and from the vibration of the roll-end to the second mode of the clamped one. These results are convenient for use in the design of the U-type vibrator to the supporting system.