Izvorni znanstveni članak
https://doi.org/10.21857/y26kec30j9
Semi-analytical methods for vibration and stability analysis of pressurized and rotating toroidal shells based on the energy approach
Ivo Senjanović
; Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, Zagreb, Croatia
Neven Alujević
; Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, Zagreb, Croatia
Ivan Ćatipović
; Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, Zagreb, Croatia
Damjan Čakmak
; Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, Zagreb, Croatia
Nikola Vladimir
; Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, Zagreb, Croatia
Sažetak
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. The rigorous formulae for natural frequencies of in-plane and outof- plane free vibrations of a rotating ring are derived. An in-plane vibration mode of the ring is characterised by coupled flexural and extensional deformations, whereas an out-of-plane mode is distinguished by coupled flexural and torsional deformations. For the in-pane vibrations, the ring is considered to be a short top segment of a toroidal shell. The expressions for the ring strain and kinetic energies are deduced from the corresponding expressions for the torus. It is shown that the ring rotation causes the bifurcation of natural frequencies for the in-plane vibrations only. The bifurcation of natural frequencies of the out-of-plane vibrations does not occur. The derived analytical results are validated by a comparison with FEM and FSM (Finite Strip Method) results, as well as with experimental results available in the literature.
Ključne riječi
toroidal shell, ring; vibration; buckling; pressure; rotation; the Rayleigh-Ritz method; finite strip method
Hrčak ID:
240825
URI
Datum izdavanja:
31.1.2020.
Posjeta: 938 *