Coupled Flexural–Torsional Forced Vibration Analysis of a Piezoelectrically Actuated Double-Cantilever Structure

In this paper, the forced coupled flexural–torsional vibration of a piezoelectrically actuated double-cantilever structure is investigated. The double-cantilever structure is composed of two uniform and identical Euler–Bernoulli cantilever beams connected by a rigid tip connection at their free ends. There is also a piezoelectric layer attached on the top surface of each cantilever beam. The characteristic equation for the coupled flexural–torsional vibrations of the structure is derived and solved to determine the natural frequencies. The time response to the forced vibrations of the structure is studied using the Galerkin approximation method. The effects of dimensional parameters, including the length of the cantilever beams and the length of the tip connection, and the piezoelectric input voltage on the coupled flexural–torsional natural frequencies and amplitude of vibrations of the structure are investigated analytically and experimentally. The results show that the coupled flexural–torsional fundamental frequency of the piezoelectrically actuated double-cantilever structure decreases as either the length of the cantilever beams or the tip connection is increased. Moreover, the amplitude of the coupled flexural–torsional vibrations of the structure is proportional to the piezoelectric input voltage; however, the slope of the curves depends on dimensional parameters. For a given voltage, the effect of either of the aforementioned dimensional parameters on the amplitude of vibrations depends on the other dimensional parameter such that there is a turning point in all the curves, whose location depends on the configuration of the structure.