Abstract
Dynamic Finite Element (DFE) and conventional finite element formulations are developed to study the flexural - torsional vibration and stability of an isotropic, homogeneous and linearly elastic pre-loaded beam subjected to an axial load and end-moment. Various classical boundary conditions are considered. Elementary Euler - Bernoulli bending and St. Venant torsion beam theories were used as a starting point to develop the governing equations and the finite element solutions. The nonlinear Eigenvalue problem resulted from the DFE method was solved using a program code written in MATLAB and the natural frequencies and mode shapes of the system were determined form the Eigenvalues and Eigenvectors, respectively. Similarly, a linear Eigenvalue problem was formulated and solved using a MATLAB code for the conventional FEM method. The conventional FEM results were validated against those available in the literature and ANSYS simulations and the DFE results were compared with the FEM results. The results confirmed that tensile forces increased the natural frequencies, which indicates beam stiffening. On the contrary, compressive forces reduced the natural frequencies, suggesting a reduction in beam stiffness. Similarly, when an end-moment was applied the stiffness of the beam and the natural frequencies diminished. More importantly, when a force and end-moment were acting in combination, the results depended on the direction and magnitude of the axial force. Nevertheless, the stiffness of the beam is more sensitive to the changes in the magnitude and direction of the axial force compared to the moment. A buckling analysis of the beam was also carried out to determine the critical buckling end-moment and axial compressive force.
Highlights
To the best of the author’s knowledge, a conventional or dynamic finite element formulation has not yet been developed to model the geometrically coupled flexural – torsional free vibration of an Euler – Bernoulli beam subjected to an axial force and an end-moment simultaneously
Results from the Dynamic Finite Element (DFE) method confirm that unlike tensile force, application of compressive force causes a reduction in the stiffness of the beam which is accompanied by a simultaneous reduction in the natural frequencies of the system
The results determined using the DFE method are in agreement with the results found from the FE method
Summary
Beams are important and commonly used structures, since many components of airborne vehicles, such as wings and helicopter blades could be modeled as a simple beam or as a series of beams during the preliminary design stages. To the best of the author’s knowledge, a conventional or dynamic finite element formulation has not yet been developed to model the geometrically coupled flexural – torsional free vibration of an Euler – Bernoulli beam subjected to an axial force and an end-moment simultaneously. In what follows, a classical finite element solution and a DFE formulation are presented to investigate the stability and flexural – torsional vibration of a simple Euler – Bernoulli beam subjected to an axial load and an end-moment. Given the magnitude of aerospace components that are axially loaded and bending-torsion coupled that could be represented as beams to an acceptable degree of accuracy during the preliminary design stages, such as helicopter, propeller, compressor and turbine blades, the fact that engineers and designers could arrive at an acceptable ballpark for the vibrational characteristics within a fraction of the time, especially for higher modes, using an extremely coarser mesh in comparison to conventional FEM is a massive advantage as it avoids the difficulty of having to solve a very large Eigenvalue problem. The natural frequencies and mode shapes of the beam are evaluated
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