Free Vibrations of Thin Cylindrical Shells Having Finite Lengths With Freely Supported and Clamped Edges

Abstract

Free vibrations of thin cylindrical shells having finite lengths are investigated on the basis of a set of three differential equations which are derived in a similar manner as Donnell obtained his equations for the bending and buckling problems. The equations can be solved readily after a simplifying assumption is introduced. In this manner the frequency equations are obtained for cylindrical shells with both edges freely supported, with both edges clamped, and with one edge freely supported and the other edge clamped. It is found that the lowest frequency given by the frequency equation is the smallest in the first case, larger in the third, and the largest in the second. The other two frequencies yielded by the frequency equation are approximately the same in all cases. As a result of the approximations, the characteristic equations for the three cases are found to be similar to the frequency equations for the lateral vibration of beams with similar end conditions. For the case of freely supported edges the normal functions obtained are identical in form with those assumed by Flügge and by Arnold and Warburton. For the same case, natural frequencies of one numerical example are computed by means of the present method, and the results are in good agreement with those obtained by these previous authors.