Study of the bending modes in circular quartz resonators

An experimental and theoretical study of bending modes in a partially electroded circular piezoelectric quartz (AT-cut) with free edge is presented. The quartz is excited by a voltage pulse applied on the electrodes, and its surface is scanned by a laser vibrometer that measures the out-of-plane displacements. The classical theory of bending of thin disks is used to describe the flexural modes at frequencies lower than the first thickness shear resonance (6 MHz). A fairly good agreement is found between experimental and theoretical results for the forced mode shapes and for the resonance frequencies. However, it appears that the two springs used to maintain the disk in position introduce extra clamping conditions. Several source shapes were studied, among which a collection of an arbitrary number of forces is particularly useful. The two-dimensional wavenumber representation shows the presence of anisotropy related to the crystallographic axes at higher frequencies, which is not predicted by the model. The experimental phase velocities are compared to those given by the classical theory of disks and to those of Lamb A₀ mode. This study confirms the correspondence at low frequencies between the A₀ mode and the bending eigenmodes of a disk with finite size