Timoshenko Theories in the Analysis of Cantilever Beams Subjected to End Mass and Dynamic End Moment

This paper investigates the effects of shear deformation on the flutter and divergence instabilities of a cantilever beam subjected to a concentrated mass and applied dynamic couple. The beam is modeled using classical and truncated Timoshenko beam theory, accounting for both shear deformation and rotary inertia. The inclusion of rotary inertia is shown to significantly influence the dynamic response, particularly for beams with greater thickness. According to Hamilton’s principle, the equations of motion for the cantilevered beam are derived, applying both classical and truncated Timoshenko beam theories. Auxiliary functions are utilized to solve the resulting system analytically. Various numerical examples are presented, illustrating typical results to demonstrate the effectiveness of the proposed approach. The numerical findings show significant convergence and computational effectiveness. The effect of the location of a concentrated mass and the dynamic couple applied at the free end is analyzed for various beam slenderness ratios and curvature positions, emphasizing their impact on modifying the critical instability limits. To highlight the significance of shear effects, a comparison is made between the outcomes of the Timoshenko model and those of the Euler-Bernoulli beam model, showing notable variations in the anticipated divergence and flutter stability characteristics. All the examples were executed using both classical theory and the truncated Timoshenko theory, and the findings indicated a remarkable level of convergence. Finally, a numerical comparisons with literature papers was performed. The results achieved showed strong alignment.