Distributed control of laminated beams: Timoshenko theory vs. Euler-Bernoulli theory

In this paper, the governing equations and boundary conditions of laminated beam-like components of smart structures are reviewed. Sensor and actuator layers are included in the beam so as to facilitate vibration suppression. Two mathematical models, namely the shear-deformable (Timoshenko) model and the shear-indeformable (Euler-Bernoulli) model, are presented. The differential equations of the continuous system are approximated by utilizing finite element techniques for both models. A cantilever laminated beam with and without a tip mass is investigated to assess the validity and the accuracy of the two models when used for vibration suppression. Comparison between the two models is presented to show the advantages and the limitations of each of the models. Since the Timoshenko beam theory is higher order than the Euler- Bernoulli theory, it is known to be superior in predicting the transient response of the beam. The superiority of the Timoshenko model is more pronounced for beams with a low aspect ratio. It is shown that use of an Euler-Bernoulli based controller to suppress beam vibration can lead to instability caused by the inadvertent excitation of unmodelled modes.