State-Space Modeling and Active Vibration Control of Smart Flexible Cantilever Beam with the Use of Finite Element Method

Abstract

The aim of this study is to design a Linear Quadratic Regulator (LQR) controller for the active vibration control of a smart flexible cantilever beam. The mathematical model of the smart beam was created on the basis of the Euler-Bernoulli beam theory and the piezoelectric theory. State-space and finite element models used in the LQR controller design were developed. In the finite element model of the smart beam containing piezoelectric sensors and actuators, the beam was divided into ten finite elements. Each element had two nodes and two degrees of freedom were defined for each node, transverse displacement, and rotation. Two Piezoelectric ceramic lead Zirconate Titanate (PZT) patches were affixed to the upper and lower surfaces of the beam element as pairs of sensors and actuators. The location of the piezoelectric sensor and actuator pair changed and they were consecutively placed on the fixed part, the middle part, and the free end of the beam. In each case, the design of the LQR controller was made considering the first three dominant vibratory modes of the beam. The effect of the position of the sensor-actuator pair on the beam on the vibration damping capability of the controller was investigated. The best damping performance was found when the sensor-actuator pair was placed at the fixed end.