D.V. Shick

On the normal acceleration sensitivity of contoured quartz resonators rigidly supported along rectangular edges

An analysis of the normal acceleration sensitivity of contoured AT‐ and SC‐cut quartz crystal resonators rigidly supported along rectangular edges is performed. The variational principle with all natural conditions for anisotropic static flexure is used in the calculation of the flexural biasing state. However, in this work the biasing shearing stresses with quadratic variation across the thickness of the plate are obtained a posteriori from the flexural solution in the conventional manner. The accompanying quadratically varying biasing shearing strains, which are very important in this work, are obtained from the constitutive equations along with all other quadratically varying strains resulting from the anisotropy. The calculated biasing deformation fields are employed in an existing perturbation equation along with the mode shapes of the contoured resonators to calculate the normal acceleration sensitivities. It is shown that the normal acceleration sensitivity vanishes for many cases and is less than ...

On the in‐plane acceleration sensitivity of ST‐cut quartz surface‐wave resonators with interior rectangular supports

An analysis of the in‐plane acceleration sensitivity of ST‐cut quartz surface‐wave resonators with the substrate extending beyond interior rectangular supports is performed. In the treatment the variational principle with all natural conditions is extended in such a way as to permit interior surfaces of discontinuity with spring supports at which the approximating solution functions need not satisfy the constraint‐type conditions. This extension is required for the approximation procedure used in the calculation of the biasing state. The resulting extensional biasing state is employed in an existing perturbation integral along with the proper continuous representation of the acoustic‐surface‐wave mode shape to calculate the in‐plane acceleration sensitivity. The calculated results indicate how to select the distances between the supports so as to minimize the in‐plane acceleration sensitivity for a given surface‐wave resonator.

An analysis of the in‐plane acceleration sensitivity of quartz surface‐wave resonators rigidly supported along the edges

An analysis of the in‐plane acceleration sensitivity of grooved ST‐ and SST‐cut‐quartz surface‐wave resonators rigidly supported along the edges is performed. The biasing deformation fields are obtained using a new variational approximation procedure which is very accurate, computationally efficient, and convenient to use. The calculated extensional biasing states are employed in an existing perturbation equation along with the proper continuous representation of the acoustic surface‐wave mode shape for arrays of reflecting grooves to calculate the in‐plane acceleration sensitivity for both cuts. The calculations reveal that the ST cut is slightly less sensitive than the SST cut and that both are a few parts in 1011 per g.

An analysis of the normal acceleration sensitivity of contoured quartz resonators rigidly supported along the edges

The authors perform an analysis of the normal acceleration sensitivity of contoured quartz resonators rigidly supported along rectangular edges. Calculated biasing deformation fields are used in an existing perturbation equation along with the equivalent trapped energy mode shapes for the contoured resonator to calculate the normal acceleration sensitivities. It is shown that the normal acceleration sensitivity vanishes for certain values of the planar aspect ratio for all modes considered in both the AT and SC cuts. In addition, the sensitivity has been calculated as a function of the orientation of the rectangle with respect to the nodal lines of the anharmonics of the contoured resonator for the case of square AT- and SC-cut plates. The calculations reveal that the SC cut has zero-crossings for this case while the AT cut does not and that the sensitivity is consistently lower for the square SC cut than the AT cut.<<ETX>>

An analysis of the normal acceleration sensitivity of contoured quartz resonators with simple rectangular supports

The normal acceleration sensitivity of contoured quartz resonators with the plate extending slightly beyond simple rectangular supports is analyzed. The biasing deformation fields are calculated by means of a novel variational approximation procedure that employs the variational principle with all natural conditions for anisotropic static flexure. The accompanying strains varying quadratically through the thickness are then determined recursively from the plate shearing stress resultants. The calculated biasing deformation fields are employed in an existing perturbation equation along with the equivalent trapped-energy mode shapes for the contoured resonator to calculate the normal acceleration sensitivities. Results are presented for a range of orientations and aspect ratios of the rectangular support configuration for both the AT and SC cuts.<<ETX>>

Forced thickness-extensional trapped energy vibrations of polarized ceramic plates

An analysis of PZT‐7A plates driven into thickness‐extensional trapped energy vibrations by the application of a voltage to strip electrodes is performed. The plate waves in the unelectroded region that carry energy away from the trapped mode are included in the treatment. The solution consists of a sum of the driven thickness solution and the pertinent plate eigensolutions for shorted electrodes in the electroded region and for zero surface charge in the unelectroded region. Since the sum of the solutions in each region satisfies the differential equations and boundary conditions on the major surfaces exactly, the substitution into the appropriate variational principle of linear piezoelectricity yields integral conditions across the thickness of the plate at the edge of the electrodes. The resulting system of inhomogeneous linear algebraic equations yields the response from very low frequency to slightly above the first thickness‐extensional frequency of the unelectroded plate.

An analysis of thickness‐extensional trapped energy mode polarized ceramic transducers

An analysis of PZT‐7A plates driven into thickness‐extensional trapped energy vibrations by the application of a voltage to strip electrodes and radiating into an adjacent fluid is performed. All previous treatments of thickness‐extensional trapped energy modes in piezoelectric plates have ignored the radiation into the surrounding medium and, hence, have been for resonators rather than transducers. The solution consists of a sum of a driven thickness solution and the pertinent plate eigensolutions for shorted electrodes in the electroded region and for zero surface charge in the unelectroded region, including in each instance the radiation into the adjacent fluid. Since the sum of the solutions in each region satisfies the differential equations and boundary conditions on the major surfaces including the fluid loading exactly, the substitution into the appropriate variational principle of linear piezoelectricity yields integral conditions across the thickness of the plate at the edge of the electrodes. T...