Vibrations Of Circular Cylindrical Shells Research Articles

Non-linear dynamics of cantilevered circular cylindrical shells with thickness stretch, containing quiescent fluid with small-amplitude sloshing

Studies on nonlinear vibrations of circular cylindrical shells containing fluid are mostly limited to thin simply supported shells, such that no results regarding the behavior of thick cantilevered shells with shear and thickness deformations can be found in the literature. In this article, for the first time, thin and thick circular cylindrical shells with clamped-free boundary conditions are modeled according to the third-order shear deformation theory with thickness stretch. The contained liquid satisfies the Laplace equation and linearized boundary conditions are imposed at the free surface and at the contact with the shell. Using Lagrange equations, the governing differential equations of the system in terms of shell-fluid interaction are derived. In linear free vibration analysis, it was found that increasing the fluid level inside the shell and decreasing the shell thickness, cause a significant rise in the fluid free-surface wave elevation. This increase is such that limits the application of linear sloshing theory. It was observed that the fluid presence can change the nonlinear behavior from softening to hardening type and intensifies the shell thickness deformation. In addition, the presence of contained liquid reduces the circumferential dynamic contraction of the shell caused by large amplitude vibrations.

Modal localization in vibrations of circular cylindrical shells with geometric imperfections

The present study aims to investigate the effect of geometric imperfections in circular cylindrical shells on the vibration characteristics. Perfect circular shells are characterized by the presence of double shell-like modes, i.e., modes having the same frequency with modal shape shifted of a quarter of wave-length in the circumferential direction. However, in the presence of geometric imperfections, the double natural frequencies split into a pair of distinct frequencies, the splitting is proportional to the level of imperfection. In some cases, the imperfections cause an interesting phenomenon on the modal shapes, which present a strong localization in the circumferential direction. The present study has been carried out by means of a semi-analytical approach. Theoretical formulation were derived based on Sanders–Koiter thin shell theory. The analytical results have been compared with those of standard finite element analyses. The results corresponding to the analysis of modal localization are novel and can be used as a benchmark for further studies.

Natural Vibrations of a Cylindrical Shell with Fluid Partly Resting on a Two-Parameter Elastic Foundation

This paper presents the results of studies on natural vibrations of circular cylindrical shells containing liquid and resting on an elastic foundation, which is described by the Pasternak two-parameter model. In the meridional direction, the elastic medium is nonuniform and represents an alternation of sections in which the foundation is present or absent. The behavior of the elastic structure and the compressible fluid is described in the framework of classical shell theory based on the Kirchhoff–Love hypothesis and the Euler equations. The equations of motion of the shell are reduced to a system of ordinary differential equations with respect to new unknowns. The wave equation written for pressure in the fluid also reduces to a system of ordinary differential equations using the straight line method. The solution of the formulated boundary value problem is found by the Godunov orthogonal sweep method. The validity of the results obtained is confirmed by comparison with the known numerical-analytical solutions. The dependences of the minimum vibration frequencies on the characteristics of elastic medium with variable nonuniformity along the length of the structure have been obtained for cylindrical shells with different boundary conditions. It has been found that the violation of smoothness of the derived dependences is caused by a change of the vibration mode with minimum frequency and is determined both by the ratio of the size of the elastic foundation to the entire length of the shell and its stiffness, and also by a combination of boundary conditions set at the edges of the thin-walled structure.

Natural frequencies analysis of functionally graded circular cylindrical shells

In the present work, a study on natural frequencies of functionally graded materials (FGM) circular cylindrical shells is presented. TheFGM is considered to be a mixture of two materials. The volumetric fractions are considered to vary in the radial direction (i.e., through the thickness) in compliance with a conventional power-law distribution. The equivalent material properties are estimated based on the Voigt model. The analysis of the FGM cylindrical shells is performed using the third-order shear deformation shell theory and the principle of virtual displacements. Moreover, the third-order shear deformation shell theory coupled with Carrera’s unified formulation is applied for the derivation of the governing equations associated with the free vibration of circular cylindrical shells. The accuracy of this method is examined by comparing the obtained numerical results with other previously published results. Additionally, parametric studies are performed for FGM cylindrical shells with several boundary conditions in order to show the effect of several design variables on the natural frequencies such as the power-law exponent, the circumferential wave number, the length to radius ratio and the thickness to radius ratio.

Nonlinear damped vibrations of three-phase CNT-FRC circular cylindrical shell

This study presents a semi-analytical solution of nonlinear vibrations of circular cylindrical shells made of carbon nanotube (CNT) fiber-reinforced composite (CNT-FRC). Vibrations are produced by a radial harmonic force and viscous structural damping is considered. The effective properties of a lamina of the CNT-FRC shell are evaluated in two steps. The elastic properties of randomly distributed CNTs in a polymeric matrix (i.e., hybrid matrix) are computed by the Eshelby-Mori-Tanaka/Voigt scheme to consider the CNTs agglomeration effect in the hybrid matrix. Then, the resulting hybrid matrix is reinforced with aligned fibers in order to prepare the lamina of the CNT-FRC shell; its effective properties are estimated by the Halpin Tsai homogenization approach. The CNT-FRC shell is modelled incorporating the von Kármán geometric nonlinearity and first-order shear deformation theory (FSDT). The nonlinear governing partial differential equations (PDEs) of the CNT-FRC shells are derived by the Hamilton’s principle. These PDEs are discretized into ordinary differential equations (ODEs) by using the Galerkin’s method. The ODEs are solved by incremental harmonic balance method (IHB) in conjunction with the arclength continuation method to obtain the frequency-amplitude response of the shell. The effect of different types of CNT agglomeration models, CNT mass fraction, agglomeration parameters and stacking sequence of laminates on the frequency-amplitude curves corresponding to forced and free nonlinear vibrations of the CNT-FRC shell are studied in detail.

Resonances and nonlinear vibrations of circular cylindrical shells, effects of thermal gradients

In this paper, the results of an experimental campaign focused on the vibrations of shells are presented. More specifically, the goal is to investigate the effect of thermal gradients across the shell thickness on the nonlinear dynamics. The shell is made of polymeric material and an aluminum mass is clamped on one end of the shell; the other shell end is clamped to an electrodynamic shaker, which provides a base harmonic excitation. Tests are performed in a controlled environment where a thermal gradient on the shell thickness is generated by means of a climatic chamber and an internal cartridge heater. Different temperature gradients and base excitation levels have been considered. The nonlinear dynamic scenario is analyzed through amplitude–frequency diagrams, bifurcation diagrams, waterfall diagrams, time histories, Fourier spectra, phase portraits, and Poincaré maps. Results show a strong effect of the temperature on the dynamic response of the shell: subharmonic, quasi-periodic, and chaotic vibrations take place as well as large amplitude vibrations, high sound levels are detected.

Free vibration of circular cylindrical shells with discrete circumferential stiffnesses

Gears with thin rims are broadly applied in aircraft and light-weighted applications. This study will derive the governing equations of cylindrical shells with discrete circumferential stiffnesses based on Hamilton's principle and Flügge shell theory. To derive the governing equations, the discrete circumferential stiffnesses which are displacement coupled are handled as external forces and expressed by Dirac delta function. The circumferential modal function contains multiple components of different circumferential wave number for the existence of discrete stiffnesses. The Galerkin method is adopted to discretize the governing equations. The natural frequencies and vibration modes are studied. The effects of different stiffness, the ratio of thickness and length to radius, number of discrete stiffnesses on natural frequencies and mode shapes are also investigated in this study.

Nonlinear vibrations of circular cylindrical shells with thermal effects: an experimental study

The nonlinear dynamics of a polymeric cylindrical shell carrying a top mass under axial harmonic excitation are experimentally investigated; the tests have been carried out in a controlled environment under several conditions of homogeneous temperature and excitation amplitude. The thermal effects on shells dynamics have been studied. The purpose of this paper is to fill an important gap in the literature regarding the effect of the temperature on the complex dynamics of shells. The cylindrical shell is excited in the axial direction by means of a seismic excitation provided by an electrodynamic shaker. The analysis is focused on the range of frequencies of excitation close to the first axisymmetric mode resonance; the base motion induces a parametric excitation. A saturation phenomenon of the top mass vibration is observed; the vibrating energy directly transferred from the shaker to the first axisymmetric mode is transferred to radial motion of the shell. The experimental data are examined and discussed in detail; a complete dynamic scenario is analyzed by means of: amplitude–frequency curves, bifurcation diagrams, spectrograms, Poincare maps, phase portraits, Fourier spectra and time histories. Results show that: (i) the temperature strongly affects the instability regions and the magnitude of the measured kinematic quantity, (ii) high environmental temperature leads to a more complex shell dynamics.

Vibration Studies of Circular Cylindrical Shells Using Self-Organizing Maps (SOM) Approach and Multivariate Analysis

Network and multivariate approach are employed to research the free vibration of circular cylindrical shells in present work. An analytical procedure is developed for finding the frequencies and vibration modes of circular cylindrical shells with shear diaphragm boundary conditions, based on Flugge’s classic thin shell theory. Many design samples could be obtained with the help of design of experiment technique. The information behind these design samples can be investigated by the self-organizing map (SOM) approach. SOM is a tool for data mining. It can map high-dimensional data to two-dimensional plots. When the dataset is large, the clustering of samples is needed then. The clustering of samples can be obtained by combining SOM and traditional multivariate approach, such as hierarchical and partitive techniques. The samples in different clusters would have their own characteristics. The dimension-reduction approaches are used for finding the structure of data set. The variable significance of each cluster is examined based on standard deviation. A classification method, LVQ, is also used for finding discrimination rules of vibration modes. The rules from LVQ are compared with traditional decision-tree approach.

Influence an Additional Inclusion on Dynamic Performance of Machine

Research was carried out into the role of the small concentrated mass of frequencies and forms regarding free bending vibrations of circular cylindrical shell. The quantitative result, which was obtained by the method of final element, are compared with the well-known theoretical and experimental data. It has been established, that the effect of declining the least of split own frequencies depends heavily on the geometric and wave parameters of the shell. The limit of applicability of a well-known formula to assess the lowest frequencies of spectrum has been examined. It has been shown that the interaction of bending and radial vibrations must be taken into account already in linear functions.

Free vibrations of circular cylindrical shells with a small added concentrated mass

The effect of a small added mass on the frequency and shape of free vibrations of a thin shell is studied using shallow shell theory. The proposed mathematical model assumes that mass asymmetry even in a linear formulation leads to coupled radial flexural vibrations. The interaction of shape-generating waves is studied using modal equations obtained by the Bubnov–Galerkin method. Splitting of the flexural frequency spectrum is found, which is caused not only by the added mass but also by the wave-formation parameters of the shell. The ranges of the relative lengths and shell thicknesses are determined in which the interaction of flexural and radial vibrations can be neglected.

Free vibration of bi-material cylindrical shells

Based on the Sanders thin shell theory, this paper presents an exact solution for the vibration of circular cylindrical shell made of two different materials. The shell is sub-divided into two segments and the state-space technique is employed to derive the homogenous differential equations. Then continuity conditions are applied where the material of the cylindrical shell changes. Shells with various combinations of end boundary conditions are analyzed by the proposed method. Finally, solving different examples, the effect of geometric parameters as well as BCs on the vibration of the bi-material shell is studied.

Fluid–structure interaction for nonlinear response of shells conveying pulsatile flow

Circular cylindrical shells with flexible boundary conditions conveying pulsatile flow and subjected to pulsatile pressure are investigated. The equations of motion are obtained based on the nonlinear Novozhilov shell theory via Lagrangian approach. The flow is set in motion by a pulsatile pressure gradient. The fluid is modeled as a Newtonian pulsatile flow and it is formulated using a hybrid model that contains the unsteady effects obtained from the linear potential flow theory and the pulsatile viscous effects obtained from the unsteady time-averaged Navier–Stokes equations. A numerical bifurcation analysis employs a refined reduced order model to investigate the dynamic behavior. The case of shells containing quiescent fluid subjected to the action of a pulsatile transmural pressure is also addressed. Geometrically nonlinear vibration response to pulsatile flow and transmural pressure are here presented via frequency-response curves and time histories. The vibrations involving both a driven mode and a companion mode, which appear due to the axial symmetry, are also investigated. This theoretical framework represents a pioneering study that could be of great interest for biomedical applications. In particular, in the future, a more refined model of the one here presented will possibly be applied to reproduce the dynamic behavior of vascular prostheses used for repairing and replacing damaged and diseased thoracic aorta in cases of aneurysm, dissection or coarctation. For this purpose, a pulsatile time-dependent blood flow model is here considered by applying physiological waveforms of velocity and pressure during the heart beating period. This study provides, for the first time in literature, a fully coupled fluid–structure interaction model with deep insights in the nonlinear vibrations of circular cylindrical shells subjected to pulsatile pressure and pulsatile flow.

Free Vibrations of Circular Cylindrical Shells with a Small Added Concentrated Mass

Free Vibrations of Circular Cylindrical Shells with a Small Added Concentrated Mass

Comparison Study on the Exact Dynamic Stiffness Method for Free Vibration of Thin and Moderately Thick Circular Cylindrical Shells

Comparison study on free vibration of circular cylindrical shells between thin and moderately thick shell theories when using the exact dynamic stiffness method (DSM) formulation is presented. Firstly, both the thin and moderately thick dynamic stiffness formulations are examined. Based on the strain and kinetic energy, the vibration governing equations are expressed in the Hamilton form for both thin and moderately thick circular cylindrical shells. The dynamic stiffness is assembled in a similar way as that in classic skeletal theory. With the employment of the Wittrick-Williams algorithm, natural frequencies of circular cylindrical shells can be obtained. A FORTRAN code is written and used to compute the modal characteristics. Numerical examples are presented, verifying the proposed computational framework. Since the DSM is an exact approach, the advantages of high accuracy, no-missing frequencies, and good adaptability to various geometries and boundary conditions are demonstrated. Comprehensive parametric studies on the thickness to radius ratio (h/r) and the length to radius ratio (L/r) are performed. Applicable ranges of h/r are found for both thin and moderately thick DSM formulations, and influences of L/r on frequencies are also investigated. The following conclusions are reached: frequencies of moderately thick shells can be considered as alternatives to those of thin shells with high accuracy where h/r is small and L/r is large, without any observation of shear locking.

Free vibration analysis of cylindrical shells partially resting on an elastic foundation

The present study aims at the investigation of free vibration analysis of thin cylindrical shells surrounded by an elastic medium, which is distributed over a particular length of the shell. The Love’s shell theory is utilized along with the Winkler foundation to obtain the governing equations of motion. An exact series expansion method of solution is employed for any arbitrary boundary conditions. The excellent accuracy of the obtained results is validated by comparing to the available literature. Moreover, several case studies are performed to provide an insight into the influence of some practically important parameters on free vibrations of circular cylindrical shells, including stiffness and length of the partial foundation. Finally, it is concluded that the proposed method can accurately determine the exact solution to the free vibration of shells partially surrounded by elastic foundations with any boundary conditions.

Simple models for linear and nonlinear modal vibration of circular cylindrical shells

Beginning shortly after the Second World War, ONR sponsored projects at Columbia University, New York, that analyzed vibration and shock response of shell structures. This work necessitated ingenuity because it preceded the advent of numerical modeling tools. A central theme was energy methods. This paper will review two works of this type. Baron and Bleich [J. Appl. Mech. 21, 178–184 (1954)] used the mechanical energies of the Flugge-Byrne-Lur’ye shell theory and a fundamental property of the modal energies to obtain accurate formulas for the three vibration branches and associated modal displacements of a simply supported cylindrical shell. This author [Ginsberg, J. Appl. Mech. 40, 471–477 (1973)] used these modes to analyze finite amplitude effects at resonances. Examination of the order of magnitude of terms in the mechanical energies led to identification of nonlinear mode coupling. The resulting differential equations for the modal coordinates were solved by a singular perturbation technique. The ou...

Numerical and analytical investigation of free vibrations of circular cylindrical shells with added mass linearly distributed along generatrix

Numerical and analytical investigation of free vibrations of circular cylindrical shells with added mass linearly distributed along generatrix

Rapid heating induced vibration of circular cylindrical shells with magnetostrictive functionally graded material

The vibration and transient response of rapid heating on inner surface of the functionally graded material (FGM) circular cylindrical shells with outer magnetostrictive layer is investigated and computed by using the generalized differential quadrature (GDQ) method. The effects of heat flux value, power law index value, environmental temperature value and control gain value on Terfenol-D FGM circular cylindrical shell subjected to two edges clamped condition due to the not very high temperature fluid rapidly flow into the circular cylindrical shells from one side to the end of axial length direction are analyzed. The higher amplitudes of displacement and thermal stress can be obtained under the higher rapid heat flux value. With suitable product of coil constant and control gain value can reduce the amplitudes of displacement and thermal stress into a smaller value. The displacement of Terfenol-D FGM circular cylindrical shell versus the Terfenol-D thickness is stable for all power law index values. The Terfenol-D FGM circular cylindrical shell can stand against the higher temperature of environment with some values of power law index under rapid heating.

Exact solutions for free vibration of circular cylindrical shells with classical boundary conditions

The obtaining of exact solutions for free vibration of circular cylindrical shells with classical boundary conditions, except for all edges being shear diaphragm, was known to be difficult or impossible. This work presents those exact solutions based on the Donnell–Mushtari shell theory . They are shown to be in simple and compact form, and not as complex as they have generally been regarded. The computed natural frequencies are in excellent agreement with those in literature and those of the highly accurate semi-analytical differential quadrature finite element method developed in this study. The exact solutions can be used as benchmarks for researchers to check their numerical methods and for engineers to design such shell structures. Two of the four eigenvalues are found to be much larger than the other two for some boundary conditions. High accurate membrane approximate solutions are also presented for the circular cylindrical shells.