RAYLEIGH QUOTIENTS FOR COUPLED FREE-VIBRATIONS

Based upon the general thin shell theory and basic equations in fluid mechanics, three kinds of Rayleigh's quotients for coupled fluid-structure free vibrations are developed for arbitrary tanks fully or partially filled with (i) inviscid, irrotational and incompressible fluid, with the effects of free surface waves neglected, (ii) inviscid, irrotational and incompressible fluid with the effects of free surface waves considered, and (iii) inviscid, irrotational and compressible fluid, respectively. Theorems concerning stationary values of the respective Rayleigh quotients are proved by utilizing the generalized orthogonality relationships for the wet modes of these three kinds of liquid containing tank models. Thus the mathematical foundations for calculating approximately the eigenvalues and eigenvectors in fluid-structure interaction systems are presented. An example of applying the Rayleigh quotient to predicting a natural frequency of a cylindrical tank partially filled with fluid is provided and the result is compared with that available.