THEORETICAL AND EXPERIMENTAL STUDY ON FREE VIBRATION OF CANTILEVER TAPERED BEAM BASE ON NEW MODAL FUNCTION

A new type of modal function which is a linear combination of hypergeometric and Meijer-G functions is proposed for tapered cantilever beam vibration analysis. This modal function has advantages of real coefficients, accuracy and non-approximation. Two steps are carried out to verify the effectiveness and accuracy of this modal function. In the first step, the fundamental frequency as well as the shape of the modal function is validated. In the second step, the modal function is used to perform nonlinear vibration analysis, and its effectiveness is investigated. A comparison is made among the theoretical prediction, finite element method, finite element semi-analytical method and the experimental results, which demonstrates the accuracy of the calculated frequency and the modal shape, respectively. Subsequently, substituting the modal function into the governing equation of the vibrating tapered cantilever after the Galerkin procedure, the curvature nonlinear coefficient and the inertia nonlinear coefficient are obtained. The frequency-amplitude relationship is deduced by the energy balance method which has been examined by the experiment. The results show that the displacement-time relationship of a selected point on the beam which is detected by a laser sensor perfectively coincides with the theoretical prediction which is given by the modal function and the frequency-amplitude relationship. The results show the effectiveness of the presented method in predicting the deformation of the tapered cantilever during nonlinear vibration.