Some Integrals Relating to the Vibration of a Cantilever Beam and Approximation for the Effect of Taper on Overtone Frequencies

Abstract

SummaryIn the investigation of approximate numerical values of overtone frequencies of a turbine blade it is desirable to know formulae for certain integrals of the modes of vibration of an ordinary cantilever beam. The first object of this paper is to obtain such formulae and to arrange them in tabulated form. By proceeding along the lines of the calculus of perturbations, these results may then be used to obtain new formulae which give second order approximations for the effects of uniform breadth and thickness tapers on the overtone frequencies. The theory gives good agreement with experiments for tapers which do not exceed about 0·5, which is present-day practice for turbine blading.