1971 年 27 巻 10 号 p. 515-523
The ever-increasing use of electromechanical resonators vibrating in flexural modes of tuning forks, free-free bars and cantilevers in various fields of communication and control calls for another look into the mounting techniques, which can yield a more compact overall size without impairing the stability and Q of the vibrators. The mounting technique presented in this paper utilizes the supporting flexural vibrators which are inserted between the main vibrator and the base. The length of the supporting vibrator is chosen to be a quarter or a half wavelength so that the free or fixed boundary condition can be realized at the junction between a supporting vibrator and the main vibrator. Man features of this technique are :(1) The coupling between a supporting vibrator and the base can be kept to a minimum. This is because the thickness of the supporting vibrator is so thin that a nearly perfect fixed condition is realized at the junction between the supporting vibrator and the base. This improves Q and suppresses variations in resonance frequency due to environmental changes. (2) The length of the supporting vibrators is proportional to its thickness and can therefore be made very short. This makes the miniaturization of the overall size possible. (3) Supporting points are not restricted to the nodal area of the main vibrator. Even support at the antinodes is possible. This makes the design of anti-shock mountings easier. Also such vibrators as tuning forks and multiple mode resonators which have no definite nodes can be easily mounted. Characteristics of a typical mounting structure, in which a main bar vibrator is supported at one end by a collinear cantilever vibrator, are analyzed, based on the flexural vibration theory of a composite bar. Spectra of resonance frequency were calculated as functions of the length of the supporting vibrator and compared favorably with experimental values obtained for the following two vibrators. One vibrator was so designed that the main vibrator was able to vibrate in a free-free flexural mode and hence the length of the supporting vibrator was able to be a quarter wavelength. The calculation of displacement shows that a proper choice of the supporting length yields a displacement pattern closely matching the pattern of a perfect free-free bar. Another vibrator was designed so that the main vibrator was able to vibrate in a fixed-free flexural mode and hence the supporting length was able to be a half wavelength. Both theory and experiment show that only the second or higher fixed-free mode can be realized but not the first mode. Hence the quarter wavelength mounting is preferable to the half wavelength mounting from the view point of miniaturization. The determination of the supporting length based on the matching of displacement requires an interactive calculation. A straight forward determination, however, becomes possible if appropriate boundary conditions are assumed at the junction between the main vibrator and the supporting vibrator. Experiments using various vibrators show that the assumption of the sliding end (where both tangent and shearing stress are zero) gives better results than the assumption of the free end (where both moment and shearing stress are zero), which has been used in the mounting of high frequency quartz vibrators. The overall mounting characteristics of a complete vibrator were evaluated by the change of resonance frequency before and after the clamping of the base by a heavy vise. Quarter wavelength of a free-free bar, a tuning fork and a multiple mode vibrator with a rectangular cross section and half wavelength mounting of a cantilever vibrator were tested and yielded good results. A complete filter utilizing the multiple mode vibrator was also made, to show the compactness obtained by the present mounting technique.