Deflection and critical velocity of rollers

A coordinate system with its origin at the center of the roller, and with the origin moving as the roller deflects, is used for solution of the fourth order differential equation of beam mechanics. The result is a continuous polynomial in terms of x, the distance from the center. The form of the equation is adaptable to further analyses such as finding the profile of a nip roller, predicting wrinkling of a web, or predicting the natural frequency of a live-shaft roller as demonstrated in this paper.The natural frequency of a live shaft roller is predicted by substituting values of deflection, caused by the weight of the roller determined either experimentally or analytically as in this paper, into a relationship derived by Rayleigh's method.Mechanical and Aerospace Engineerin