Dynamic Behavior of Cracked Timoshenko Beams Subjected to Sub-tangential Follower Force

In this paper, the purpose is to investigate the stability and variation of the natural frequency of a cracked Timoshenko cantilever beam subjected to sub-tangential follower force and a tip mass. In addition, an analysis of the flutter instability of a cantilever beam according to the change of slenderness ratio is studied. The governing differential equations of a Timoshenko beam, subjected to an end tangential follower force, are derived via Hamilton’s principle. The two coupled governing differential equations are reduced to one-fourth order ordinary differential equation in terms of the flexural displacement. Finally, the influence of the slenderness ratio, a tip mass on the critical follower force, and the natural frequency of a Timoshenko beam are investigated. It is found that a difference of only 29% between the first natural frequency of the Euler beam model and that of the Timoshenko beam exists in the special case. By using the results of this paper, a judgment base is obtained as regards the choice of cracked beam models for the effect of the slenderness ratio and sub-tangential follower force. Furthermore, the effect of crack on the dynamic behavior of beams with a tip mass is investigated.