Vibration analysis of rotating toroidal shell by the Rayleigh-Ritz method and Fourier series

In this self-contained paper free vibrations of a pressurised toroidal shell, rotating around its axis of symmetry, are considered. Extensional and bending strain-displacement relationships are derived from general expressions for a thin shell of revolution. The strain and kinetic energies are determined in the co-rotating reference frame. The strain energy is first specified for large deformations and then split into a linear and a non-linear part. The nonlinear part, which is afterwards linearized, is necessary in order to take into account the effects of centrifugal and pressure pre-tensions. Both the Green-Lagrange nonlinear strains and the engineering strains are considered. The kinetic energy is formulated taking into account centrifugal and Coriolis terms. The variation of displacements u, v and w in the circumferential direction is described exactly. This is done by assuming appropriate trigonometric functions with a unique argument nφ+ωt in order to allow for rotating mode shapes. The dependence of the displacements on the meridional coordinate is described through Fourier series. The Rayleigh-Ritz method is applied to determine the Fourier coefficients. As a result, an ordinary stiffness matrix, a geometric stiffness matrix due to pressurisation and centrifugal forces, and three inertia matrices incorporating squares of natural frequencies, products of rotational speed and natural frequencies and squares of the rotational speed are derived. The application of the developed procedure is illustrated in the cases of a closed toroidal shell and a thin-walled toroidal ring. With the increase of the rotation speed the natural frequencies of most natural modes are split into two (bifurcate). The corresponding stationary modes are split into two modes rotating forwards and backwards around the circumference with different speeds. The obtained results are compared with FEM results and a very good agreement is observed. The advantage of the proposed semi-analytical method is high accuracy and low CPU time-consumption in case of small pre-stress deformation for realistic structures. The illustrated numerical examples can be used as benchmark for validation of numerical methods.