Analytical solutions to flexural vibration of slender piezoelectric multilayer cantilevers

Abstract

The modeling of vibration of piezoelectric cantilevers has often been based on passive cantilevers of a homogeneous material. Although piezoelectric cantilevers and passive cantilevers share certain characteristics, this method has caused confusion in incorporating the piezoelectric moment into the differential equation of motion. The extended Hamilton's principle is a fundamental approach to modeling flexural vibration of multilayer piezoelectric cantilevers. Previous works demonstrated derivation of the differential equation of motion using this approach; however, proper analytical solutions were not reported. This was partly due to the fact that the differential equation derived by the extended Hamilton's principle is a boundary-value problem with nonhomogeneous boundary conditions which cannot be solved by modal analysis. In the present study, an analytical solution to the boundary-value problem was obtained by transforming it into a new problem with homogeneous boundary conditions. After the transformation, modal analysis was used to solve the new boundary-value problem. The analytical solutions for unimorphs and bimorphs were verified with three-dimensional finite element analysis (FEA). Deflection profiles and frequency response functions under voltage, uniform pressure and tip force were compared. Discrepancies between the analytical results and FEA results were within 3.5%. Following model validation, parametric studies were conducted to investigate the effects of thickness of electrodes and piezoelectric layers, and the piezoelectric coupling coefficient d 31 on the performance of piezoelectric cantilever actuators.