engleski

Semi-analytical methods for vibration and stability analysis of pressurized and rotating toroidal shells based on the energy approach

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 nonlinear part. The non-linear part, which is subsequently linearized, is necessary in order to take into account the effects of centrifugal and pressure pre-tensions. The Green-Lagrange non-linear strains are considered. The kinetic energy is formulated taking into account the centrifugal and the Coriolis terms. The variation of displacements u, v and w in the circumferential direction is described exactly. The dependence of the displacements on the meridional coordinate is described through the Fourier series. The Rayleigh-Ritz method is applied to determine the Fourier coefficients. As a result thereof, 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 cases of a closed and open toroidal shell and a thin-walled toroidal ring. 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. Additionally, a finite strip for vibration analysis of rotating toroidal shells subjected to internal pressure is developed. The expressions for strain and kinetic energies are taken from the previous Rayleigh- Ritz method. The variation of displacements u, v and w with the meridional coordinate is modelled through a discretization with a number of finite strips. 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, the strip of a higher order is developed too. The application of the finite strip method is illustrated in case of closed toroidal shell. The obtained results are compared with those determined by the Rayleigh-Ritz method and the finite element method.

toroidal shell, ring ; vibration ; buckling ; pressure ; rotation ; the Rayleigh-Ritz method ; finite strip method

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