# Vibrations calculation of compound shells of revolution with circular frames by finite element method

### Mathematics. Physics. Mechanics

### Аuthors

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

e-mail: joyfulife@hanmail.net

### Abstract

Axisymmetric (*n*= 0) and non-axisymmetric (

*n*= 1,2,...) vibrations of compound elastic orthotropic shells of revolution with ring frames are considered (

*n*is the number of sine or cosine waves in the circumferential direction). The preliminary axisymmetric stress-strain state of the shell is taken into account. The deformation model of the shell is based on the Kirchhoff-Love hypothesis. The dimensions of the frame cross sections are considered like small in comparison with their radii. The eccentricities of the frame connections with the shells are taken into account. The ring frames with arbitrary solid cross sections which are assumed non-deformable are considered as curvilinear rods undergoing to tension-compression, bending in the two planes and torsion. The thin-walled frames with the arbitrary open or closed deformable cross section contours are considered as shells of revolution.

The finite element method (FEM) is used to calculate vibrations of a compound axisymmetric structure. The finite elements (FE) of the shells and thin-walled frames are considered as narrow ring conical belts. In the limits of the belt width the meridional and circumferential displacements are approximated by two-term linear functions and the normal displacement is approximated by four-term cubic polynomial. The shell of revolution poles are replaced by the circular plates or orifices with sufficiently small radius.

The amplitude values of the axial, radial and circumferential displacements as well as meridional angles of rotation for n-th harmonic on the FE junction circles and on the ring reference circles are considered as generalized coordinates. The equations of vibrations of the system in generalized coordinates are obtained by the Lagrange method taking into account the junction conditions between the rings and FE as well as the fixation conditions. Afterwards for practical calculations the obtained equations are transformed to uncoupled equations in the normal coordinates which represent to the motions for the normal modes of the system.

The developed by FEM algorithm is sufficiently general and can be used for calculation of vibrations of compound systems of arbitrary orthotropic shells of revolution with variable along the meridian thickness and with different ring frames. All coefficients of rigidity and inertia matrices of the FE and rings are written in formulas.

As an example free axisymmetric vibrations of a cylindrical shell fixed at the upper and connected at the lower via an elastic ring with a spherical shell with an attached massive load are considered. Calculations of the mode frequencies are fulfilled for two variants of the connecting ring with the same area but different shape of the cross section.

### Keywords:

shell of revolution, compound shells, circular frame, thin-walled frame, elastic vibrations, finite elements method### References

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