A Finite Strip for the Vibration Analysis of Rotating Toroidal Shell Under Internal Pressure

In this paper, a finite strip for vibration analysis of rotating toroidal shells subjected to internal pressure is developed. The expressions for strain and kinetic energies are formulated in a previous paper in which vibrations of a toroidal shell with a closed cross section are analyzed using the Rayleigh–Ritz method (RRM) and Fourier series. In this paper, however, the variation of displacements u, v, and w with the meridional coordinate is modeled through a discretization with a number of finite strips. The variation of the displacements with the circumferential coordinate is taken into account exactly by using simple sine and cosine functions of the circumferential coordinate. A unique argument $nφ+ω t$ is used in order to be able to capture traveling modes due to the shell rotation. The finite strip properties, i.e., the stiffness matrix, the geometric stiffness matrix, and the mass matrices, are defined by employing bar and beam shape functions, and by minimizing the strain and kinetic energies. In order to improve the convergence of the results, also a strip of a higher-order is developed. The application of the finite strip method is illustrated in cases of toroidal shells with closed and open cross sections. The obtained results are compared with those determined by the RRM and the finite element method (FEM).