Akio Ishizaki

Three-Dimensional Analysis of Spurious Vibrations of Rectangular AT-cut Quartz Plates

We have developed a technique for the three-dimensional (3-D) analysis of all the spurious vibrations of rectangular AT-cut quartz plates. The 3-D displacements are described with a linear combination of modes guided along the X crystal axis of an AT-cut quartz strip. We accurately calculate each guided mode by applying the one-dimensional finite element method to the two-dimensional strip waveguide problem. To uniquely determine the amplitudes of guided modes in the 3-D solutions we employ the variational expression for the stress-free conditions on the X-length edges. The present technique makes it possible to perform 3-D analysis of spurious vibrations within a reasonable computational time. Frequency spectra for rectangular AT-cut quartz plates are computed as a numerical example and are compared with the numerical results obtained using an extended version of the previous method and with Kogas experimental data. Vibrational patterns of spurious modes are shown, and a convergence study is carried out.

Effect of Stepped Bi-Mesa Structures on Spurious Vibrations of AT-Cut Quartz Plates

In this paper, the effect of stepped bi-mesa structures on the coupling of vibrations in AT-cut quartz plates is analyzed theoretically. Since strong coupling of thickness-flexure modes are of interest, straight-crested waves propagating along the X axis are treated. Calculated results for the stepped bi-mesa structure show that the coupling strength as estimated from its mode chart decreases to about one-third that of a bi-mesa resonator. The characteristics of bi-mesa structures appear to be amenable to further improvement by the use of a stepped bi-mesa structure.

Analysis of spurious vibrations in mesa-shaped AT-cut quartz plates

The effect of mesa structures on the coupled vibrations of AT-cut quartz plates is analyzed based on two-dimensional equations of motion. The mesa structures are used to obtain effective energy trapping and to separate the thickness-shear (TS) and thickness-flexural (TF) modes spatially. First, the effect of the bi-mesa structure is analyzed. The calculated displacement field confirms that the TS mode is confined to the mesa portion and the TF mode is localized in the thinner portion. The coupling strength estimated from the mode chart decreases to about one-fifth that of conventional resonators. Although the coupling reduction between the TS and TF modes can be obtained when using the piano-mesa structure, the asymmetric geometry of the structure increases the coupling between the TS and extensional modes.

Three-dimensional analysis of spurious vibrations in rectangular AT-cut quartz plates

This paper presents the three-dimensional analysis of spurious vibrations in rectangular AT-cut quartz plates, based on exact equations of motion. According to the classical concept, three-dimensional displacements are described with linear combination of the eigenmodes guided along the X crystal axis. Guided eigenmodes are determined by applying the one-dimensional finite element method to the two-dimensional problem, and their amplitudes are determined using the variational technique for the three-dimensional problem. By means of the present method, we can perform the three-dimensional analysis of spurious vibrations with a high degree of accuracy within a reasonable computation time. As numerical examples, frequency spectra for rectangular AT-cut quartz plates are computed and compared with Kogas experimental data. A convergence study is carried out for the mode for which the frequency spectra are not satisfactory, and corresponding vibrational patterns are also shown.

Two-dimensional analysis using 1D-FEM for straight-crested waves in arbitrary anisotropic crystal plates and axisymmetric piezoelectric vibrations in ceramic disks

We have developed a two-dimensional analysis using the one-dimensional finite element method (1D-FEM) for straight-crested waves in arbitrary anisotropic crystal plates and axisymmetric piezoelectric vibrations in ceramic disks. The solution of the two-dimensional equations of motion is described with a linear combination of eigenmodes guided by a pair of parallel edges. 1D-FEM is effectively used to determine the guided eigenmodes and the their amplitudes. The method attains a high degree of accuracy in spite of a small matrix size as compared with two-dimensional (2D) FEM. As examples of the method, the frequency spectra of straight-crested waves in SC-cut quartz crystal plates and axisymmetric piezoelectric vibrations in barium titanate (Ba-TiO/sub 3/) disks are represented. A convergence study is presented for BaTiO/sub 3/ disks. The frequency spectra of straight-crested waves in AT-cut quartz plates are also calculated with the aim of examining the accuracy of Mindlins plate equations for those waves.

Three-dimensional analysis of forced vibrations of rectangular at-cut quartz plates

In the present paper, a previous technique using a hybrid method (analytical approach and numerical approach) is extended to deal with forced vibrations of rectangular AT-cut quartz plate resonators with partial electrodes. The mass loading of electrodes is considered with weak piezoelectric coupling. First rectangular acoustic waveguides with and without electrodes coating on the major faces are examined. The vibration modes of each waveguide are determined. The displacement of the resonator is defined with a linear combination of the modes. Second, the trapped-energy resonators containing a single-plated region and a double-unplated region are considered. A set of the amplitudes of the modes is derived from continuity conditions of displacement and stress at the interface between the regions, and stress-free boundary conditions at both ends of the resonator. Next, the input admittance characteristics of the resonators are calculated. Finally, the equivalent inductance of each resonance is determined from the obtained input admittance curve.