Asymptotic analysis of radial vibration of thin piezoelectric disks

The one-dimensional radial vibration model of piezoelectric disks has been widely used to determine the relevant material coefficients from admittance measurements. However, the one-dimensional model assumes infinitely thin disks, and therefore cannot predict their axial displacements. We extend the one-dimensional model by performing an asymptotic analysis of the axisymmetric radial vibration of thin disks. The asymptotic expansions include the asymptotic axial displacement and the second-order corrections to the admittance and the radial displacement in the one-dimensional model. We verify the asymptotic expansions and the one-dimensional model with the Chebyshev tau method. In the one-dimensional model, the frequencies of the maximum admittance (Formula presented.) in the first and second radial modes are accurate to 1% for Pz27 disks with thickness-to-diameter ratios of 0.15 and 0.065, respectively. For a general piezoelectric disk in the forced vibration, the error of (Formula presented.) in the one-dimensional model can be estimated from the second-order correction of the asymptotic resonance frequency in the free vibration.