Stochastic Analysis of Free Vibrations in Piezoelectric Bimorphs

Piezoelectric bimorph benders are a particular class of piezoelectric devices, which are characterized by the ability of producing flexural deformation greatly larger than the length or thickness deformation of a single piezoelectric layer. Piezoelectric bimorph benders were first developed by Sawyer in 1931 at the Brush Development Company. The performance of these actuators was rudimentary studied and improved much later, with the results of research on smart structures in 1980s. Piezoelectric benders have been used in different applications: in robotics, spoilers on missile fins, actuation for a quick-focusing lens, to control the vibration of a helicopter rotor blade and for many other purposes. Due to extensive dimensional reduction of devices and to high precision requested, the effect of erroneous parameter estimation and the fluctuation of parameters due to external reasons, sometimes, cannot be omitted. So, we consider mechanical, electrical and piezoelectric parameters as uniformly distributed around a nominal value and we calculate the distribution of natural frequencies of the device. We consider an efficient and accurate analytical model for piezoelectric bimorph. The model combines an equivalent single-layer theory for the mechanical displacements with layerwise-type approximation for the electric potential. First-order Timoshenko shear deformation theory kinematics and quadratic electric potentials are assumed in developing the analytical solution. Mechanical displacement and electric potential Fourier-series amplitudes are treated as fundamental variables, and full electromechanical coupling is maintained. Numerical analysis of simply supported bimorphs under free vibration conditions are presented for different length-to-thickness ratios (i.e., aspect ratio), and the results are verified by those obtained from the exact 2D solution. According to Timoshenko theory, a shear correction factor is introduced with a value proposed by Timoshenko (1922) and by Cowper (1966). Free vibration problem of simply supported piezoelectric bimorphs with series or parallel arrangement is investigated for the closed circuit condition, and the results for different length-to-thickness ratios are compared with those obtained from the exact 2D solution. Numerical examples are presented on bimorphs constituted by two orthotropic piezoceramic layers (PZT-5A material). The calculation of natural frequencies is based on a Weibull distribution, because it is capable to properly model a large class of stochastic behaviours. The effect of errors on the Weibull distribution of the natural frequencies is shown in terms of change of the Weibull parameters. The results show how the parameters errors are reflected on the natural frequencies and how an increment of the error is able to change the shape of the frequencies distribution.