A comparison of the Timoshenko theory and the euler-bernoulli theory for control of laminated beams

In this paper, the governing equations and boundary conditions of laminated beam smart structures are presented. Sensor and actuator layers are included in the beam so as to facilitate vibration supression. Two mathematical models are presented: the shear-deformable (Timoshenko) model and the shear-indeformable (Euler-Bernoulli) model. The differential equations for a continuous beam 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 supression. 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 model to represent beam dynamics can lead to the design of an unstable controller.