MirTahmaseb Towliat Kashani - Aerospace Eng., Ryerson University
Heenkenda Jayasinghe - Aerospace Eng., Ryerson University
Seyed M. Hashemi - Aerospace Eng., Ryerson University

A frequency-dependent, Dynamic Finite Element (DFE) formulation is developed to study thecoupled flexural–torsional vibration of a preloaded, linearly elastic, slender beam subjected toan axial load and end moment. Euler-Bernoulli bending and St. Venant torsion beam theoriesare used to develop the governing equations and the DFE solution. A nonlinear Eigenvalueproblem is formulated and the Eigensolutions are determined using a dedicated Matlab® code.The DFE frequency results, for an illustrative example, are validated against conventional FiniteElement Method (FEM). Various classical boundary conditions and loadings are considered. Asexpected, tensile forces increase natural frequencies indicating beam stiffening, compared tocompressive forces that reduce the beams stiffness, seen by a reduction in natural frequency.When a force and an end moment are acting in combination, the moment reduces the stiffness ofthe beam. Nevertheless, the stiffness of the beam is found to be more sensitive to the changes inthe magnitude of the axial force compared to the moment. The presented DFE procedure resultsin much higher convergence rates compared to the conventional FEM. Based on the presenteddynamic shape functions, the weighted residual method, and the principle of virtual work, it isalso possible to further extend the DFE method to more complex structural elements.

coupled flexural-torsional vibration; FEM; DSM; DFE