Free Vibrations Of Cylindrical Shells Research Articles

Free vibration of cylindrical shells and panels with cutout via the differential transform method

Semi-numerical methods hold significant value due to their efficiency in time-consuming calculations. Accordingly, this article proposes an effective strategy to derive the vibration parameters of cylindrical shells (CSs) with cutout utilizing a semi-analytical approach entitled differential transform method (DTM). To achieve this, the governing equations of motion are first introduced based on Sander’s theory of shells. These equations give rise to a coupled system of linear ordinary differential equations, which pose a challenge in solving the eigenvalue problem. It should be emphasized that the classical version of the differential transform method could not be directly applied to solve the eigenvalue problem related to the above-mentioned coupled differential equation. An innovative technique is employed and discussed in the paper to remedy these difficulties, which could be widely used in similar problems. As a result, the natural frequencies and mode shapes of some CSs with cutouts, various internal angles, and diverse material characteristics are attained. The outcomes undergo validation through the hybrid finite element method, and the proposed method’s accuracy is rigorously proved. In conclusion, the proposed DTM provides an efficient solution to the eigenvalue problem of coupled differential equations and offers a valuable tool for obtaining accurate vibration parameters of CSs.

Free Vibration of Composite Cylindrical Shells Based on Third-Order Shear Deformation Theory

The focus of this study is to analyse the free vibration of cylindrical shells under third-order shear deformation theory (TSDT). The constitutive equations of the cylindrical shells are obtained using third-order shear deformation theory (TSDT). The surface and traverse displacements are expected to have cubic and quadratic variation. Spline approximation is used to approximate the displacements and transverse rotations. The resulting generalized eigenvalue problem is solved for the frequency parameter to get as many eigenfrequencies as required starting from the least. From the eigenvectors, the spline coefficients are computed from which the mode shapes are constructed. The frequency of cylindrical shells is analysed by varying circumferential node number, length dimension, layer number, and different materials. The authenticity of the present formulation is established by comparing with the available FEM results.

Free Vibrations of Cylindrical Shells by The Nodal Line Finite Difference Method-Part-I Theory.(Dept.C)

In this paper, the nodal line finite difference method (NLFDM) is extended to the analysis of a thin circular cylindrical shell, as shown in Fig. 1, undergoing free vibrations [1]. The material of the circular cylindrical shell is assumed to be linear elastic isotropic.  The complexity of a theoretical analysis of a vibration problem depends largely on the number of degrees of freedom of the structural system in question. Since a thin circular cylindrical shell is a continuous system it has an infinite number of degrees of freedom. The NLFDM [10] transforms this continuous system into a system having a finite number of degrees of freedom. While carrying out the dynamic (or static) analysis by the NLFDM, the number of degrees of freedom depends on the number of the used nodal lines.  The natural frequencies and modes are treated in more detail because they are basic to understanding the dynamic response under any kind of excitation. It will be shown that the number of natural frequencies and that of normal modes, for axial wave number different from zero, are each equal to three times the number of nodal lines used.  The orthogonality relationships of the normal modes obtained by the nodal line finite difference method are also included. It should be remarked that these relationships are slightly different from those obtained by the finite element method; in fact the overall matrix of coefficients replaces the stiffness matrix, and any two different modes are orthogonal with resp ect to the unit matrix instead of the mass matrix.

Effects of local thinning defects and stepped thickness for free vibration of cylindrical shells using a symplectic exact solution approach

Based on the Reissner shell theory and a new symplectic method, an analytical exact solution approach is presented in this paper to deal with free vibration of stepped cylindrical shells with thickness variations in both the axial and circumferential directions. Such cylindrical shells are often applied for designing aeronautics and space structures and hence any local defects and local geometric variations will result in significant design consequences. For this reason, this paper intends to establish a cylindrical shell model with local wall defects including local wall thinning and wall thickness stepped variations. A new exact solution approach for free vibration of the model are obtained by superposing the eigensolutions of the perfect shell. To ensure validity of the present result and the universality of the method, the solutions for three forms of stepped shells are compared with published data and finite element analysis. The effects of geometric parameters of the local thinning defect on free vibration characteristics are also investigated and presented.

Vibrations of cylindrical shells with rectangular cross-section

The free vibrations of cylindrical shells with rectangular cross-section are studied in the work. The precision of Rayleigh method for various admissible functions and different boundary conditions is evaluated. Using finite element method, the natural frequencies in the case of deviation of the cross-section shape from square to rectangular are examined, and the vibration frequencies and modes for the shell joined with the plate are studied for various plate thicknesses. Additionally, fundamental frequency behavior is analyzed in the case of fixed support, as one of the cross-section sides approaches zero.

Effect of different viscoelastic models on free vibrations of thick cylindrical shells through FSDT under various boundary conditions

This paper investigates the free vibrations of cylindrical shells made of time-dependent materials for different viscoelastic models under various boundary conditions. During the extraction of equations, the displacement field is estimated through the first-order shear deformation theory taking into account the transverse normal strain effect. The constitutive equations follow Hooke\'s Law, and the kinematic relations are linear. The assumption of axisymmetric is included in the problem. The governing equations of thick viscoelastic cylindrical shell are determined for Maxwell, Kelvin-Voigt and the first and second types of Zener\'s models based on Hamilton\'s principle. The motion equations involve four coupled partial differential equations and an analytical method based on the elementary theory of differential equations is used for its solution. Relying on the results, the natural frequencies and mode shapes of viscoelastic shells are identified. Conducting a parametric study, we examine the effects of geometric and mechanical properties and boundary conditions, as well as the effect of transverse normal strain on natural frequencies. The results in this paper are compared against the results obtained from the finite elements analysis. The results suggest that solutions achieved from the two methods are ideally consistent in a special range.

A new simple method for shell vibration analysis with initial stress accounting

This paper presents a simple semi-analytical approach for free vibration of cylindrical shell with initial prestress based on equivalent load method and the Donell-Mushtari theory. In most practical applications, shells are subjected to static loadings causing internal stress field. The presence of such initial forces like internal pressure, axial force, centripetal force and torque moment significantly affects the natural frequency spectra. According to Calladin’s equivalent load method initial stress field create additional curvatures and can be added as additional terms to the basic equations. The results of presented method agree well with experimental data found in the literature. Effects of elastic support stiffness, the shell length and radius to thickness ration on natural frequencies are investigated.

Finite Element Vibration Analysis of Variable Thickness Thin Cylindrical FGM Shells under Various Boundary Conditions

Abstract This research aimed to study the analysis of free vibrations of cylindrical shells made of functionally graded materials (FGMs). The studied shells were thin and had been selected in a way that their thickness varies linearly along the length. A combination of different boundary conditions was applied, and vibrational analysis was performed through finite element method. Then, the effects of different parameters, including varying thickness along the length, shell length, and radius, on the frequency behavior of the shell were extracted. Afterward, the frequency behaviors of the shells made of different materials were compared. The numerical results obtained in this article have been verified by the results derived from the analytical relations of previous research, which indicated the high accuracy of the finite element method for the vibrational analysis of the shells with variable thickness. The analysis indicated significant effects of thickness changes and boundary conditions on the natural frequencies.

Free Vibrations of a Cylindrical Shell Partially Resting on Elastic Foundation

The free vibrations of a circular cylindrical shell resting on a two-parameter Pasternak elastic foundation are investigated. The elastic medium is inhomogeneous along the shell length, and the inhomogeneity represents an alternation of areas with and without the medium. The behavior of the shell is considered in the framework of the classical shell theory based on the Kirchhoff–Love hypotheses. The corresponding geometric and physical relations, together with the equations of motion, are reduced to a system of eight ordinary differential equations for new unknowns. The problem is solved by applying the Godunov orthogonal sweep method, and the differential equations are integrated using the fourth-order Runge–Kutta method. The natural frequencies are calculated by applying a stepwise iterative procedure, followed by a further refinement based on the bisection method. The results are validated by comparing them with available numerical-analytical solutions. For simply supported, clamped-clamped, and clamped-free cylindrical shells, the numerical results reveal that the lowest vibration frequencies depend on the elastic medium characteristics and the type of the inhomogeneity. It is shown that a violation in the smoothness of the curves is caused by variations in the lowest frequency mode, the ratio of the size of the elastic foundation to the total length of the shell, and its stiffness, as well as by a combination of the boundary conditions specified at the ends of the shell.

Free vibration analysis of functionally graded cylindrical shells with different shell theories using semi-analytical method

In this study, the semi-analytical finite strip method is adopted to examine the free vibration of cylindrical shells made up of functionally graded material. The properties of functionally graded shells are assumed to be temperature-dependent and vary continuously in the thickness direction according to a simple power law distribution in terms of the volume fraction of ceramic and metal. The material properties of the shells and stiffeners are assumed to be continuously graded in the thickness direction. Theoretical formulations based on the smeared stiffeners technique and the classical shell theory with first-order shear deformation theory which accounts for through thickness shear flexibility are employed. The finite strip method is applied to five different shell theories, namely, Donnell, Reissner, Sanders, Novozhilov, and Teng. The approximate procedure is compared favorably with three-dimensional finite elements. Finally, a detailed numerical study is carried out to bring out the effects of power-law index of the functional graded material, stiffeners, and geometry of the shells on the difference between various shell theories. Finally, the importance of choosing the shell theory in simulating the functionally graded cylindrical shells is addressed.

Exact Solutions for Free Vibration of Cylindrical Shells by a Symplectic Approach

In engineering applications, the vibration analysis of cylindrical shells is highly significant for the design of engineering structures subjected to external dynamic excitations. In practice, the explicit expression would make a better and more quick design comparing to the numerical results. However, the studies on analytical method were concentrated on the semi-inverse method. To overcome the limitation, we introduce a new analytical treatment for the free vibration of cylindrical shells. Under the framework of the Hamiltonian system, an analytical symplectic approach is introduced into the free vibration of cylindrical shells. In the symplectic space, exact solutions are obtained in a rational and systematic way without any trial functions. Analytical frequency equation for arbitrary boundary conditions is derived. Highly accurate natural frequencies and analytical vibration mode functions are obtained simultaneously. In this paper, the governing equations for Donnell’s and Reissner’s shell theories can be successfully transformed into the Hamiltonian form by introducing a full state vector. The free vibration of cylindrical shells is reduced into a symplectic eigenproblem. Numerical results are compared with those previously published in literature and good agreement is observed. In addition, the proposed method can be extended to micro/nanoscale cylindrical shells.

An exact series solution for free vibration of cylindrical shell with arbitrary boundary conditions

У вигляді методу початкових параметрів записано точні формули для пошуку власних частот коливань циліндричної оболонки з довільними граничними умовами на основі теорії Донела-Муштарі. Також дано вирази з урахуванням деформаційної складової, показано що її вплив дуже незначний і ним можливо знехтувати. Наведено порівняння з літературними даними й показано, що отримані залежності добре описують експериментальні результати.

Free vibrations of cylindrical shells with arbitrary boundary conditions: A comparison study

In this paper, the free vibration characteristics of cylindrical shells with arbitrary boundary conditions are investigated. The Sanders shell theory is used to calculate the elastic strain energy. Artificial springs are implemented at the ends of the shells to represent the arbitrary boundary conditions. The shell displacements are expanded by three different sets of formulations, namely, the modified Fourier series, the Orthogonal polynomials, and the Chebyshev polynomials. A unified solution for the three different types of expansion functions is developed using the Rayleigh-Ritz method. The unified solution is validated by comparing with the available results in the literature. The accuracy, convergence rate, and computational efficiency of the three expansion functions are compared. Based on the comparison studies, the Chebyshev polynomials of high computational efficiency are selected to investigate the influence of boundary conditions on the free vibration characteristics of cylindrical shells.

Free vibration of carbon nanotubes reinforced (CNTR) and functionally graded shells and plates based on FSDT via discrete singular convolution method

Abstract In this study, free vibration analysis of conical and cylindrical shells and annular plates made of composite laminated and functionally graded materials (FGMs) is investigated. Carbon nanotubes reinforced (CNTR) composite case is also taken consideration for FGM. The equations of motion for conical shell are obtained via Hamilton's principle using the transverse shear deformation theory. To obtain the eigenvalue problem of the system, the method of discrete singular convolution is employed. Material properties are graded in the thickness direction according to a volume fraction power law and four-parameter power law indexes for FGM cases. Five types of distributions of CNTR material are also considered. To verify the accuracy of this method, comparisons of the present results are made with results available in the open literature. Free vibrations of cylindrical shells and annular plates with FGM are treated as special cases. Results are also presented for carbon nanotubes reinforced (CNTR) composite cylindrical shells and annular plates. It is found that the convergence and accuracy of the present DSC method is very good for vibration problem of shells with functionally graded materials (FMG) and CNTR functionally graded materials.

Holographic Interferometry Study of the Free Vibrations of Cylindrical Shells of Constant and Variable Thickness

A holographic interferometry technique is used to determine the natural frequencies of isotropic circular cylindrical shells of constant and variable thickness and equivalent mass. It is established that the resonant frequencies of the cylindrical shell of variable thickness exceed those of the cylindrical shell of constant thickness of the same mass if circumferential halfwaves are few

Free vibration analysis of cylindrical shells using the Haar wavelet method

This paper presents a novel and efficient solution for free vibrations of thin cylindrical shells subjected to various boundary conditions by using the Haar wavelet discretization method. The Goldenveizer–Novozhilov shell theory is adopted to formulate the theoretical model. The displacements and their derivatives in the governing equations are represented by Haar wavelet series and their integrals in the axial direction and the Fourier series in the circumferential direction . The constants appearing from the integrating process are determined by boundary conditions and thus the partial differential equations are transformed into a set of algebraic equations . The frequency parameters of the cylindrical shells are obtained by solving the algebraic equations. The present solution is verified by comparing the numerical results with those previously published in literature. Very good agreement is observed. It is shown that accurate frequency parameters can be obtained by using a small number of collocation points and boundary conditions can be easily achieved. The advantages of this current solution method consist in its simplicity, fast convergence, low computational cost and high precision. • An efficient solution based on the Haar wavelet discretization method is developed for free vibration of cylindrical shells. • Each of displacements and their derivatives in the governing equations are expanded using Haar wavelet series and their integrals. • Accurate solution can be obtained using a small number of collocation points and boundary conditions can be easily achieved. • The advantage of this present method consists in its simplicity, fast convergence, low computational cost and high precision.

Free vibration analysis of circular cylindrical shell with non-uniform elastic boundary constraints

As far as the study of circular cylindrical shell is concerned, most of the existing works are limited to classical homogeneous boundary conditions. However, complicated boundary conditions, such as elastic and non-uniform boundary conditions are encountered in many engineering applications. Thus, it is of important practical significance to investigate the vibrations of cylindrical shell with this kind of boundary conditions. This paper is mainly concerned with the free vibrations of cylindrical shell with non-uniform elastic boundary constraints. The exact solution for the problem is obtained using improved Fourier series method, in which each of three displacements of the shell is represented by a Fourier series supplemented by several terms introduced to ensure and accelerate the convergence of the series expansions. The unknown expansions coefficients are treated as the generalized coordinates and determined using the Rayleigh–Ritz procedure. The change of the boundary conditions can be easily achieved by only varying the stiffness of the four sets of boundary springs at each end of the shell without the need of making any change to the solution procedure. The excellent accuracy of the current results is validated by comparison with the finite element method (FEM) results. Numerical results are presented to illustrate the effects of some interesting and practically important boundary restraints on free vibrations of circular cylindrical shell, including varying stiffness of boundary springs, point supported and partially supported boundary conditions.

Analysis of Fiberglass Winding Angle on Natural Frequency of Free Vibration of Cylindrical Shell with Asymmetric Boundary Conditions

In order to obtain approximate solution of natural frequencies for the free vibration of anisotropic circular cylindrical shells made of GFRP (glass fiber-reinforced plastic) with asymmetric boundary conditions, Love’s theory and energy method are used. Computation results show that the fundamental natural frequency comes from different vibration modes while the winding angle varies, the effect of number of axial half waves is stronger than number of circumferential waves on natural frequency of free vibration of anisotropic circular cylindrical shell. The effect of shell’s geometrical parameters is also investigated on natural frequencies.

Analysis of free vibration of cylindrical shells on the basis of three dimensional exact elasticity theory

In this paper we present a study of free vibration of cylindrical shells on the basis of three dimensional exact theory. The studied boundary condition is simple-simple supported boundary condition. Extensive frequency parameters have been obtained through resolving frequency equations. The results of frequency parameters have been obtained by computers programs and have been compared with the results of classical theories. Keywords: Free vibration, Cylindrical shells, Frequencies, Exact elasticity theory

Analysis of Free Vibration of Cylindrical Shells on the Basis of Three Dimensional Exact Elasticity Theory

In this paper we present a study of free vibration of cylindrical shells on the basis of three dimensional exact theory. The studied boundary condition is simple-simple supported boundary condition. Extensive frequency parameters have been obtained through resolving frequency equations. The results of frequency parameters have been obtained by computers programs and have been compared with the results of classical theories.