Distributed Control of a Timoshenko Beam
Baras, John, S.
Date: November 09 - November 11, 1992
We consider a flexible beam clamped to a rigid base on one end with the other end free. In order to stabilize the beam vibration, we introduce active damping into the beam with feedback control using distributed actuators and sensors. We apply Tim Oshenko beam theory to model the substructure. Unlike the familiar Euler-Bernoulli beam model, the shear effect and the rotational inertia are included in the modeling of elastic deformation to obtain a more precise beam model. The piezoelectric theory is briefly reviewed. The piezoelectric ceramic material (PZT) is used to build the distributed actuator. The distributed sensor is made of piezoelectric polymer polyvinylidene fluoride (PVDF). The sensor and actuator are layers which are attached directly to both sides of the beam. Based on the constitutive properties and layer geometry, the models for sensor and actuator are developed. We then embed the static actuator and sensor models into the beam substructure to form the model of the composite beam. We design the feedback controller by Lyapunov direct methods based on the energy functional of the system. It is proved that the derived controller can extract energy from the system and increase system damping. The resulting closed loop system is asymptotically stable. Since this method does not depend on model truncation, we further analyze the combined effect of the distributed controller on the relevant vibration modes. It is possible to suppress the selected modes by choosing the appropriate actuator layout. It is also shown that by properly installing the sensor and determining the sensor shape function, we can further extract and manipulate the sensor signal for our control need.Download Full Paper