Analysis of a monolithic, two-dimensional array of quartz crystal microbalances loaded by mass layers with nonuniform thickness

We study free thickness-shear vibrations of a monolithic, two-dimensional, and periodic array of quartz crystal microbalances loaded by mass layers with gradually varying thickness. A theoretical analysis is performed using Mindlin´s two-dimensional plate equation. It is shown that the problem is mathematically governed by Mathieu´s equation with a spatially varying coefficient. A periodic solution for resonant frequencies and modes is obtained and used to examine the effects of the mass layers. Results show that the vibration may be trapped or untrapped under the mass layers. The trapped modes decay differently in the two in-plane directions of the plate. The mode shapes and the decay rate of the trapped modes are sensitive to the mass layer thickness.