The effects of rotation, tip mass, and hub radius on the stability of beck's column

Abstract

The stability of a cantilever beam subjected to a follower force at its free end and rotating at a uniform angular velocity is investigated. The beam is assumed to be offset from the axis of rotation, carries a tip mass at its free end, and undergoes deflection in a direction perpendicular to the plane of rotation. The equations of motion are formulated within the Euler-Bernoulli and Timoshenko beam theories for the case of a Kelvin model viscoelastic beam. The associated adjoint boundary value problems are derived and appropriate adjoint variational principles are introduced. These variational principles are used for the purpose of determining approximately the values of the critical flutter load of the system as it depends upon its damping parameters, tip mass and its rotary inertia, hub radius, and speed of rotation. The variation of the critical flutter load with these parameters is revealed in a series of several graphs. The numerical results show that the critical load can be reduced significantly due to (a) the transverse and rotary inertia of the tip mass and (b) increasing values of the internal damping parameter associated with the transverse shear deformation of the rotating beam.