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Research Papers

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

[+] Author and Article Information
Ivo Senjanović

Faculty of Mechanical Engineering and Naval
Architecture,
University of Zagreb,
Ivana Lučića 5,
Zagreb 10000, Croatia
e-mail: ivo.senjanovic@fsb.hr

Ivan Áatipović, Neven Alujević, Damjan Čakmak, Nikola Vladimir

Faculty of Mechanical Engineering and Naval
Architecture,
University of Zagreb,
Ivana Lučića 5,
Zagreb 10000, Croatia

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received July 20, 2018; final manuscript received October 7, 2018; published online November 19, 2018. Assoc. Editor: Stefano Lenci.

J. Vib. Acoust 141(2), 021013 (Nov 19, 2018) (17 pages) Paper No: VIB-18-1308; doi: 10.1115/1.4041734 History: Received July 20, 2018; Revised October 07, 2018

## Abstract

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).

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## References

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## Figures

Fig. 1

Rotating toroidal shell, main dimensions, and displacements

Fig. 2

Nodal displacements and forces of the finite strip

Fig. 3

Assembling of the finite strip stiffness matrices of a closed toroidal shell

Fig. 4

Modal displacements of closed toroidal shell cross sections, – · – U, - - - V, ––– W

Fig. 5

The first six natural modes of closed toroidal shell (ABAQUS)

Fig. 6

Natural modes of FEM model in the coordinate planes (ABAQUS)

Fig. 7

Convergence of natural frequencies, FS(3g, 3f)

Fig. 8

Convergence of natural frequencies, - - - ABAQUS, · · · NASTRAN

Fig. 9

Relative difference of FEM and RRM natural frequencies

Fig. 10

Natural frequencies of the rotating closed toroidal shell, asymmetric modes, ω0 = 80.73 Hz, ––– FSM, – – – FEM, – · – · – FSM Fcor. = 0

Fig. 11

Natural frequencies of the rotating closed toroidal shell, symmetric modes, ω0 = 80.73 Hz, ––– FSM, – – – FEM, – · – · – FSM Fcor. = 0

Fig. 12

Clamped toroidal shell

Fig. 13

The first six natural modes of clamped toroidal shell

Fig. 14

Modal displacements of clamped toroidal shell cross section, – · – U, - - - V, ––– W

Fig. 15

Tension forces of clamped toroidal shell, Ω = 48.61 Hz

Fig. 16

Natural frequencies of rotating clamped toroidal shell, ω0 = 48.61 Hz, n = 0

Fig. 17

Natural frequencies of rotating clamped toroidal shell, ω0 = 48.61 Hz, n = ±1

Fig. 18

Natural frequencies of rotating clamped toroidal shell, ω0 = 48.61 Hz, n = ±2

Fig. 19

Zoomed diagrams of natural frequencies of rotating clamped toroidal shell, ω0 = 48.61 Hz, n = ±2

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