OUT-OF-PLANE VIBRATION OF A UNIFORM EULER-BERNOULLI BEAM ATTACHED TO THE INSIDE OF A ROTATING RIM

Abstract

This paper describes the out-of-plane vibration of a uniform Euler-Bernoulli beam one end of which is radially restrained (clamped or pinned) on the inside of a rotating rigid rim and the other end is radially unrestrained (clamped, pinned or free). Depending on the root offset parameter, the centrifugal axial force distribution may be wholly tensile or partly compressive and partly tensile or wholly compressive. The general solution of the mode shape differential equation is expressed as the superposition of four converging polynomial functions. Six combinations of clamped, pinned and free boundary conditions are considered, and the corresponding frequency equation in expressed in closed form, the roots of which give the natural frequencies. The first three out-of-plane dimensionless natural frequencies for typical combinations of the root offset parameter and rotational speed are presented in tabular form. Beyond a value of the root offset parameter, the frequencies increase and then decrease with increase in rotational speed. This aspect is discussed for the six combinations of the boundary conditions. It is possible for the rotational speed and a natural frequency to be equal (a “tuned” state) and for the beam to buckle at a critical rotational speed. These aspects are addressed and some representative results tabulated.