Linear vibrations of a rotating elastic beam with an attached point mass

Abstract

The paper deals with linear vibrations of a thin elasticbeam with an attached point mass which is clamped radially to the inside of a rigid wheel rotating at a constant angular velocity. Material damping is also taken into consideration. A partial differential equation of motion with dynamical boundary conditions is derived. It is proved that in a sense the attached point mass plays a destabilizing role for any values of the problem parameters. The problem of the determination of critical parameter values is reduced to a second-order Sturm-Liouville problem with a singularity. Asymptotic representations of the stability threshold, as the ratio between the radius of the wheel and the length of the beam goes either to 0 or to +∞, are obtained. The dynamical stability of the rectilinear shape of the beam is studied by means of the direct Lyapunov method. A non-standard method of construction of Chetayev's function is proposed.