Dynamic deflection and capacitance of micromachined piezoelectric benders

Micromachined piezoelectric heterogeneous bimorphs have been fabricated using the techniques of IC fabrication. The voltage across the piezoelectric element of a heterogeneous bender causes it to contract or expand while the passive element is compressed or elongated and responds with a counterforce, thus building up a moment, which bends the structure. This dynamic moment has been calculated, and the corresponding Euler-Bernoulli equation has been written with a time-dependent voltage as a source term in the inhomogeneous part of the equation thus becoming a Rayleigh equation. The bending resistance has been determined using the transformation of area method. Once the solution of this equation is found the deformation can be calculated and the resulting dielectric displacement can be determined. From the latter we can determine the charge that flows into the electrodes, by integrating over the surface area. The relation between the charge on the electrodes and the driving voltage is the capacitance C. The capacitance has the expected resonance behavior: the bimorph goes through resonance at frequencies which are the solutions of the implicit relation cosΩL coshΩL=-1. The deflection of the bimorph has also been calculated and follows a similar behavior