Relationship between softening and stiffening effects in terms of Southwell coefficients

The Southwell coefficient (SC) represents an efficient way of presenting the relationship between rotating and non-rotating frequencies for rotating structures. For a given setting angle, a relationship between the in-plane frequencies and out-of-plane frequencies exists. In this paper, a proof of the relationship that exists between the in-plane and out-of-plane frequencies is given in terms of SC. With knowledge of the speed of rotation and the out-of-plane or in-plane frequencies along with this relationship in hand, the in-plane or out-of-plane frequency is easily investigated without recourse to further extensive calculations. Therefore, the Southwell relation provides a suitable tool for natural frequency estimates of rotating beams at a preliminary design stage.