Some dilemmas in energy approach to vibration and stability analysis of pressurized and rotating toroidal shells

In this paper some dilemmas related to the application of the Rayleigh – Ritz method (RRM) in conjunction with the Fourier series for vibration and stability analysis of pressurized and rotating toroidal shells are elucidated. The physical meaning of the strain and kinetic energy terms of different order of displacements in the energy balance equation are explained. Only the second order terms remain in the variation of equation of motion. In the RRM, a symmetric Coriolis mass matrix is obtained as a result of using the energy approach. In FEM, the matrix equation of motion is complex and the Coriolis mass matrix is antisymmetric. It is shown that by transferring the equation of motion from the complex into the real domain, its size is doubled and the total Coriolis matrix becomes symmetric. The influence of using the Green–Lagrange non–linear strains and the engineering strains on vibration and buckling of a toroidal shell is contrasted. It is observed that differences in the dynamic analysis, due to the two different non–linear strain formulations, is quite small. On the contrary, in buckling analysis the engineering strains give considerably higher value of the critical load