Optimal Positions for Multiple Flexoelectric Actuations on Beams

Inhomogeneous electric field induces internal stress in flexoelectric materials, which is known as the converse flexoelectric effect. The atomic force microscope (AFM) probe can be applied to actuate the flexoelectric patch laminated on flexible structures and, in turn, induce structural vibrations. In this study, multiple actuators (i.e., combination of multiple AFM probes and flexoelectric patches) are used to suppress the vibration of an elastic cantilever beam. The displacement of the beam under control is obtained by superposition of a mechanical force induced part and an actuator induced part. The optimal actuator positions are selected when the minimal tip displacement of the cantilever beam is achieved. In case studies, only three actuators are used and only first three beam modes are considered. Under resonance conditions where only one mode participates, the optimal positions are obtained and they only depend on the relationship between actuation factor and actuator’s position. The influence of the patch length is discussed, followed by the analysis of control voltages applied to actuators. When the control voltage is so small as to the induced displacement is far less than that induced by the mechanical force, the optimal actuation positions do not vary with the voltage magnitude; while when the voltage is relatively large the actuators keep adjusting their relative optimal positions with respect to the control voltage and the system becomes unstable. Furthermore, as long as the control voltage is relatively small below the stability threshold, the optimal positions do not change as the position where mechanical force placed along the beam. Under the small voltage condition, a smooth relationship between optimal positions and vibration frequency is obtained. There are four jump points of optimal positions and they are related to the tip displacement-actuator position curves under different frequencies. This study serves as the foundation of flexoelectric vibration control with multiple AFM/flexoelectric actuators on flexible structures.