Analytical Solution for Vibration Analysis of Piezoelectric Shallow Spherical Sensors and Actuators

In this article, an exact analytical approach is presented to analyze free vibration of a thin piezoelectric spherical shell, using thin shallow shell theory. The piezoelectric spherical shell is modeled as a sensor or an actuator. The piezoelectric material is polarized through the thickness of the shell. Using the separation of variables method as well as some new potential functions, the equations of motion and Maxwell’s equation are exactly solved, simultaneously. First, the equation of the transverse displacement of the shell is separately obtained and after extracting the transverse displacement, other unknowns such as the in-plane displacements and electrical potential function are obtained. Then, applying mechanical and electrical boundary conditions, the natural frequencies of the shell are obtained for the sensor and actuator cases. In order to validate the accuracy of the present method, the obtained results are compared to those obtained by a finite element analysis in ABAQUS software. Also, the effects of various parameters such as inner radius to radius of curvature of the shell ratio, thickness to inner radius ratio, and different boundary conditions on the natural frequencies are considered. Results show that piezoelectricity effect causes an increase in strain energy of the structure leading to increasing the natural frequencies for both sensor and actuator shells. Also, by changing the conditions from actuator state to sensor one, the structure experiences an increase in the natural frequencies.