Control of bending and twisting vibration in the presence of thermal deformation

Vibration control of a beam which experiences a slowly changing through-thickness temperature variation is considered. This temperature variation produces thermal moments and curvatures that are assumed to change slowly with respect to the vibration frequencies. By proposing a Lyapunov controller which relies on a definition of strain energy without thermal expansion terms, stability of vibration control and removal of thermal curvatures are assured. By proposing a Lyapunov controller based on strain energy including thermal expansion terms, stability is assured but the thermal curvatures remain. The effect of temperature on the sensor output is included. A laminated beam with a sensor which fully covers its surface is used to illustrate the method. The continuous system is simulated using a discretized (finite element) approximation, and the successful operation of both controllers is demonstrated. The effects of the host beam stiffness and the ratio of actuator thickness to host beam thickness are determined.