Displacements and rotations of practical vibrational modes of piezoelectric bimorph cantilever beams

Piezoelectric ceramic bimorph flexure beams are essential elements in micro- and nano-technology sensors, devices and systems, such as optical deflectors and modulators. Mode shape depends on drive frequency, so it is possible to adjust the shape to optimize a particular performance function. Two configurations are of particular practical importance for MEMS (micro-electro-mechanical structure/system) and MOMS (micro-opto-mechanical structure/system) sensor applications: the free end of a fixed-free cantilever resonator adjusted to translate without rotation, and to rotate without translation. Using simple Euler-Bernoulli beam theory applied to a piezoceramic bimorph configuration, expressions are found for beam displacement and slope as function of frequency. Relevant parameters are tabulated at frequencies of Rayleigh resonance and the piezo-antiresonances of zero displacement and slope, with numerical examples.