A new frequency formula for closed circular cylindrical shells for a large variety of boundary conditions

doi:10.1016/0022-460X(80)90301-6

Journal of Sound and Vibration

Volume 70, Issue 3, 8 June 1980, Pages 309-317

https://doi.org/10.1016/0022-460X(80)90301-6 Get rights and content

Abstract

A new formula for the natural frequencies of circular cylindrical shells is presented for modes in which transverse deflections dominate. It is valid for all boundary conditions for which the roots of the analogous beam problem can be obtained. Good agreement with experimental data for a variety of boundary conditions is shown.

References (9)

A.W. Leissa
Vibration of shells
NASA SP-288
(1975)
W.C. Lyons et al.
Dynamic behavior of submerged reinforced cylindrical shells
Y.Y. Yu
Free vibrations of thin cylindrical shells having finite lengths with freely supported and clamped edges
Journal of Applied Mechanics
(1955)

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In recent years, acoustic black holes (ABHs) have revealed as a very effective method for the passive reduction of vibrations in flat plates and straight beams. Those essentially consist of circular indentations and/or side boundaries, whose thicknesses decrease to zero following a power-law profile. Nonetheless, many structures in the aeronautical and naval sectors, among others, involve cylindrical structures with periodic stiffeners or supports. Such periodic structures admit the propagation of Bloch-Floquet (BF) bending waves at some frequency passbands, which may result in strong vibration levels and noise radiation. In this work, we propose the design of annular ABHs to mitigate the problem. A wave finite element model is first used to determine the frequency passbands on an ideal, periodically simply supported cylindrical shell of infinite length, with embedded annular ABHs. Then, a long, but now finite simply supported cylindrical shell is presented to show how the transmissibility between two sections strongly diminishes with the inclusion of ABHs. That confirms annular ABHs as a very effective means of reducing BF wave propagation. Unfortunately, embedding ABHs on the shell weakens its structural rigidity so a thorough analysis is made on the effects of inserting stiffeners in the longitudinal direction to partially remedy that inconvenient. Quite surprisingly, it is shown that the inclusion of stiffeners can even enhance the performance of the ABHs for some configurations. To finally foresee the potential of annular ABHs for practical applications, a finite element simulation is made on a slightly more realistic geometry, resembling a very simplified version of the hull of a submarine vehicle or a gas tank.
A closed form solution for free vibration of orthotropic circular cylindrical shells with general boundary conditions
2019, Composites Part B: Engineering
Citation Excerpt :
Based on the formulation of an energy functional, Mustafa and Ali [19] presented a concise yet comprehensive method for the determination of natural frequencies of stiffened circular cylindrical shells. Other research papers on vibration analysis of isotropic circular cylindrical shells can be found in Refs. [20–36]. So far, the study has been confined to isotropic circular cylindrical shells.
In the past decades, the exact closed form solutions for the free vibration of thin orthotropic circular cylindrical shells have been merely restricted to some classical boundary conditions. Therefore, the target of the current paper is to present a new exact closed form solution for free vibration of orthotropic circular cylindrical shells with general boundary conditions by means of the method of reverberation-ray matrix (MRRM). Based on the Donnell–Mushtari shell theory, the wave solutions are constructed by the exact closed form solutions of the governing differential equations. The artificial spring technology is introduced to achieve the general boundary conditions of two end edges of shell. Hereby, the reverberation ray matrix can be easily obtained by using the MRRM together with the wave solutions, boundary conditions and dual coordinates of the orthotropic circular cylindrical shells. Then, the vibration results are obtained from the extrapolation method and golden section search (GSS) algorithm. By the comparison with other published methods and the finite element method, the accuracy of the present method is verified. On the basis of that, some new exact nature frequencies of the orthotropic circular cylindrical shells with general elastic restraints are shown which can serve as the benchmark data for the future computing method.
Free flexural vibration of a cylindrical shell horizontally immersed in shallow water using the wave propagation approach
2017, Ocean Engineering
Citation Excerpt :
Therefore, it could be concluded that the fundamental frequency for problem considered here is always with m = 1. This conclusion is also applicable for the free vibration of thin cylindrical shell in vacuum (Scedel, 1980), conveying fluid (Zhang, 2002a) or submerged in infinite fluid field (Zhang, 2002b). Besides, the location of valley for the U-shaped line will move towards high circumferential mode order, when the axial mode order increases.
The free flexural vibration of a cylindrical shell horizontally immersed in shallow water is analyzed in low frequency range using the wave propagation approach. The effects of both the upper and lower fluid boundaries, i.e., free surface and seabed, are considered with the image method, and the scattering effect of cylindrical shell is neglected. Motions of cylindrical shell and fluid are modeled with the Flügge shell theory and wave equation, respectively. The accuracy of present method is verified through comparison with available experimental works. The modal added mass is introduced to show the influence of shallow water on coupled modal frequency. The influence of shallow water is significant in the case of small water depth and it's the contributions from fluid boundaries. The presence of free surface (seabed) will make negative (positive) contribution to the modal added mass and finally result in the increase (reduction) of coupled modal frequency. But the individual fluid boundary effect will be reduced gradually and finally negligible when the distance between cylindrical shell and fluid boundary increases. The influence of shallow water can not be negligible when the water depth is less than 8 times of shell radius. This work will help to select proper test environment for submerged cylindrical shell.
Free vibration analysis of a circular cylindrical shell using the Rayleigh-Ritz method and comparison of different shell theories
2015, Journal of Sound and Vibration
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However, neither presented explicit expressions for the mass and stiffness matrices. Soedel [24] used the Donnell–Mushtari theory and the beam functions for Galerkin׳s method. Mustafa and Ali [25] employed the Rayleigh–Ritz method and the Flügge theory for the free vibration analysis of stiffened circular cylindrical shells.
This study uses the Rayleigh–Ritz method to derive a dynamic model for the free vibration analysis of a circular cylindrical shell. In particular, explicit expressions for the mass and stiffness matrices are obtained to easily implement a computer simulation under different shell theories and boundary conditions. The dynamic model is constructed according to the Donnell–Mushtari theory, which is fully discussed herein, and then, dynamic models are constructed by using Sanders theory, Love–Timoshenko theory, Reissner theory, Flügge theory, and Vlasov theory. This paper also discusses the use of eigenfunctions of a uniform beam as admissible functions that produce compact expressions for the mass and stiffness matrices. The numerical results indicate that the Donnell–Mushtari theory is not sufficiently accurate to calculate the natural frequencies and that there is no discernible difference between the other shell theories considered in this study.
Fundamental frequency of a cantilever composite cylindrical shell
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An asymptotic analysis accounting for edge effect was undertaken by Louhghalam et al. to examine the dynamic characteristics of composite thin cylindrical shells [15]. Soedel [16] used Galerkin’s method with shape functions selected in the form of general beam mode shapes to derive a frequency formula for isotropic circular cylindrical shells with various boundary conditions. Effects of boundary conditions on the free vibration characteristics for a multi-layered cylindrical shell using the Ritz method where beam functions were used as the axial modal functions were studied by Lam and Loy [17].
Free vibrations of a cantilever composite circular cylindrical shell are considered in this paper. The edge of the shell is fully clamped at one end of the cylinder and is free at the open section of the other end. Variational equations of free vibrations are derived based on Hamilton’s principle and the problem is solved using the generalised Galerkin method. Analytical formulas enabling calculations of the fundamental frequency are obtained and verified by comparison with the results of a finite element modal analysis. The efficiency of the analytical solution is demonstrated using numerical examples including the design analysis of composite shells subject to constraints imposed on the fundamental frequency.
Dynamic characteristics of laminated thin cylindrical shells: Asymptotic analysis accounting for edge effect
2014, Composite Structures
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Yu’s equations are similar to the ones associated with lateral vibration of an Euler–Bernoulli beam. These equations are accurate for simply supported boundary conditions, but are less accurate for free edges where complex moment and shear conditions that characterize the edge effect need to be satisfied [35]. Later, Soedel [35] used Galerkin’s method with general beam mode shapes as the shape functions for isotropic cylinders with various boundary conditions.
The dynamic characteristics of composite thin cylindrical shells are examined through a systematic order-of-magnitude analysis. The analysis is used to eliminate terms of secondary importance, while retaining the dominant terms in the dispersion relation and boundary conditions. This results in analytical expressions that can describe the vibration of composite cylindrical shells with high accuracy for a wide range of frequencies. Furthermore, the asymptotic analysis is carried out in such a way that the dynamic edge effect is accounted for when determining the vibration mode shapes and the associated internal stresses. Numerical examples are also presented. It is shown that the proposed methodology gives closed-form and analytical results that are in close agreement with numerical solutions of the equations of motion.

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A new frequency formula for closed circular cylindrical shells for a large variety of boundary conditions

Abstract

Vibration of shells

NASA SP-288

Dynamic behavior of submerged reinforced cylindrical shells

Free vibrations of thin cylindrical shells having finite lengths with freely supported and clamped edges

Journal of Applied Mechanics