On the dynamics of piezoelectric cylindrical shells

Abstract

In the present paper new equations of motion are derived for the vibration of piezo-ceramic thin-walled cylindrical shells, generalizing Flügge's shell theory for this type of material. These new equations differ from the ones known from the literature in that here the electric field is not assumed as constant over the thickness but is obtained by solving an additional differential equation in the thickness direction. The shells are polarized in the radial direction and the electrodes are in the form of identical sectors. Such shells are used e.g. as stators in some piezoelectric ultrasonic travelling wave motors and it is therefore important to study their free and forced vibrations. The momentum and moment of momentum balance used in Flügge's shell theory are of course unchanged. The constitutive relations used in the theory of elastic shells are replaced by those of a linear piezoelectric material, so that additional field variables are introduced. These are subject to Maxwell's laws, which in particular have to be fulfilled by the electric field inside the shell. For a thin radially polarized shell, only dielectric displacements in the radial direction are taken into account. Due to the absence of free electric charges in dielectric media such as PZT, the divergence of the dielectric displacement vanishes. This condition leads to an ordinary differential equation of the Euler type in the radial electric field, which can be solved in closed form. Together with the thin shell assumptions and with the piezoceramic constitutive equations, this results in the equations of motion for thin piezoceramic shells with non-constant electric field over the thickness.