Resonance frequency analysis by galerkin's method for clamped‐free, stepped flexural mode quartz crystal resonators and its application to a quartz crystal tuning fork

Abstract

AbstractThis paper describes a stepped flexural mode quartz crystal resonator. The objective of this paper is to determine the resonant frequency, the motional series resistance R1, and the Q value of the stepped flexural mode tuning fork‐type quartz crystal resonator. The equation of motion is presented for a flexural mode resonator with nonuniform cross section with rotary inertia, and shear force is taken into consideration. Then, since the flexural mode quartz crystal resonator can be thought of as having two clamped‐free beams, the vibrational displacement computed from the boundary condition of clamped‐free ends and the condition of continuity of different cross sections are used to derive the frequency equation by Galerkin's method.In addition, the transcendental equation can be derived from the boundary conditions and the continuity condition. Further, from these relationships, the frequency constant (f·l1) and the resonant frequency are determined as functions of the width ratio W21 (width w2/‐width w1) and the length ratio L21 (length l2/length l1). As a result, it is shown that the frequency can be lowered without increasing the length of the resonator. These relationships are compared with the experimental results. It is shown that both agree very well. Hence, the present analysis results are effective for arbitrary frequency design. Finally, the typical values of the electrical equivalent circuit parameters are given. It is shown that, as L21 is increased, the motional series resistance R1 increases and the Q value decreases, although the characteristics are sufficient as a resonator. At the same time, the structure is resistant to vibration.