Lateral Vibration Of A Centrifugally Tensioned Uniform Euler-Bernoulli Beam

Abstract

This paper describes the lateral vibration of a uniform Euler-Bernoulli beam that is doubly symmetric in cross-section and attached radially to the outside of a rotating hub. It is assumed that a principal axis of the beam is parallel to the axis of rotation and thus the out-of-plane and in-plane vibrations are uncoupled. The equation of motion is derived on the basis that the attachment at the hub is radially restrained. The general solution of the mode shape equation is expressed as the superposition of four linearly independent functions. Clamped, pinned and free boundary conditions are considered. It is shown that the natural frequencies depend not only on the natural and/or geometric boundary conditions but also on which of the two boundaries is radially free. The first three dimensionless natural frequencies are tabulated for out-of-plane vibration for six combinations of the simple boundary conditions and for a range of offset parameters and dimensionless rotational speed. From the tables it is possible to deduce the dimensionless natural frequencies for in-plane vibrations. For parameters not listed, interpolated results are accurate to within 0·3%. It is hoped that the tabulated results will serve as data for development of frequencies for problems with more complicated flexural rigidity and/or mass distribution.