Numerical Free Vibration Analysis of Higher-Order Shear Deformable Beams

Free vibration analysis of Higher-Order shear deformation beam is studied using a numerical method, namely Differential Transform Method (DTM) which is capable of reducing the size of computational effort and can be applied to various types of differential equations. In this study, the applied DTM is different from those are in the textbook and is just considered as a part of calculation procedure. First, the governing differential quations of beam are derived in a general form considering the shear-free boundary conditions zero shear stress conditions at the top and bottom of a beam. Afterward, using DTM the derived equations governing beams, followed up Higher-Order shear deformation, Timoshenko and Bernoulli-Euler models are transformed to algebraic forms and a set of recurrence formulae is derived. Then, imposing the boundary conditions of the beam at hand, a set of algebraic equations are obtained and expressed in matrix form. Finally, the ransverse natural frequencies of the beam are calculated through an iterative procedure.Several numerical examples have been carried out to demonstrate the accuracy and competency of the present method and it is shown that the results obtained by the method are in good agreement with those in the literature. Afterward, thefree vibration of beams followed up different models (i.e. Bernoulli-Euler, Timoshenko and different Higher-Ordermodels) are taken into consideration.

Keywords:

Classical Beam Theory (CBT) , Timoshenko Beam Theory (TBT) , Higher-Order Beam Theory (HOBT) , Free Vibration , Differential Transform Method (DTM)