The Effect of Rotatory Inertia and of Shear Deformation on the Frequency and Normal Mode Equations of Uniform Beams With Simple End Conditions

A. KALNINS.3 Attention should be called to previous investigations of torsional vibrations of shells of revolution which are not cited in the paper. At least two other papers have dealt with the uncoupled purely torsional modes of arbitrary shells of revolution considered by the authors. Detailed studies of torsional modes of spherical shells are also available in the literature. A theory of vibration of shells of revolution was derived by Mathieu4 in 1882 who also discussed the torsional vibrations of shells of revolution but concluded that the purely torsional modes without axial symmetry are also possible. In the well-known paper of 1888 Love5 included a re-examination of some of M a t h ieu's results and derived a theorem which in effect states that for any surface of revolution there exists a system of uncoupled axially symmetric purely torsional modes of vibration and that no uonsymmetric purely torsional modes are possible. The first part of this theorem is evidently the same as the conclusion reached b y the authors that the torsional modes are uncoupled for shells of revolution. The governing equation of motion derived b y the authors for the axisymmetric circumferential displacement of shells of revolution also was previously given b y Love, 5 and his equation (40), except for a minus sign, is identical to equation (5a) of the paper. The discrepancy of the sign stems from a misprint in Love 's equation (40) as it can be easily seen b y comparing it to the second of his equations (36).