Optimizing vibration control in a cantilever beam with piezoelectric patches

In this work, the equation of motion of a cantilever beam with two piezoelectric patches, one acting as a sensor and the other as an actuator was first formulated, and the sensor induced voltage, representing the strain in the beam, was calculated. The beam governing equation was converted into a state space model and its response under active vibration control was studied through numerical simulations. Two types of control methods were used, velocity feedback control (VFC) and the Linear Quadratic Regulator (LQR). The effects of varying controller gains and weighting matrices on the beam vibration amplitude and settling time, as well as the induced voltage in the actuator were investigated. The LQR controller was found to be more effective than the VFC as the maximum induced actuator voltage was significantly lower. For the LQR controller weighting matrices Q and R, it was found that increasing Q reduces settling time and increases the actuator induced voltage, while increasing R, increases settling time. A calculation method for optimizing sensor placement and actuator length is also presented. The results indicate that the optimal actuator length is about 60% of the beam length.