Analytic and finite element solutions for active vibration control of a cantilever beam using piezoceramic sensor and actuator patches

An analytical solution to the equation of motion for a controlled beam with piezoceramic sensor and actuator patches is proposed. Active vibration control is implemented using closed-loop feedback where the sensor detects a strain when the beam is vibrating and the output voltage signal of the sensor is amplified and sent to the actuator. The signal sent to the actuator attempts to counteract the vibration of the beam and control its motion.;Since the mass and stiffness of the piezoceramic patches are significant, they cannot be neglected. The equation of motion for the controlled structure includes Heaviside functions and derivatives of the Heaviside function due to finite patch lengths. This makes the equation of motion difficult to solve using conventional methods. An integral equation is introduced, where the eigensolutions of the integral equation are eigensolutions of the differential equation of motion for the controlled beam. The solution of the integral equation is transformed to the solution of a set of linear equations, which can be solved numerically after being reduced to a finite set of linear algebraic equations.;A finite element model of a controlled beam is also formulated. The model contains modified beam element mass and stiffness matrices to account for the piezo patches and control effect.;Several case studies are presented which examine the use of velocity and displacement feedback control. The first three natural frequencies and mode shapes are found using the finite element and integral equation solutions. The results from the integral equation solution match very closely the results from the finite element solution. Lastly, the tip displacement of the controlled beam is compared to the tip displacement of the uncontrolled beam for some of the case studies.