Doubly rotated resonators for sensing the properties of liquids

When a doubly rotated resonator is operated in a liquid, the displacement of the surface is partly out of the plane of the plate of the resonator. The out-of-plane component of the displacement propagates a damped compressional wave into the liquid, and the in-plane component propagates a damped shear wave. In this paper, we report the measurements of the series resonant frequency and the motional arm resistance of doubly rotated quartz resonators (/spl theta//spl ap/35/spl deg/ and /spl phi/=7/spl deg/) in liquids to compare with singly rotated AT-cut resonators (/spl theta//spl ap/35/spl deg/ and /spl phi/=0/spl deg/). A modified Butterworth-Van Dyke (BVD) equivalent circuit model is suggested to analyze doubly rotated cut resonators under liquid loading.