Nonlinear vibration of functionally graded circular cylindrical shells based on improved Donnell equations

In the present work, the study of the nonlinear vibration of a functionally graded cylindrical shell subjected to axial and transverse mechanical loads is presented. Material properties are graded in the thickness direction of the shell according to a simple power law distribution in terms of volume fractions of the material constituents. Governing equations are derived using improved Donnell shell theory ignoring the shallowness of cylindrical shells and kinematic nonlinearity is taken into consideration. One-term approximate solution is assumed to satisfy simply supported boundary conditions. The Galerkin method, the Volmir's assumption and fourth-order Runge–Kutta method are used for dynamical analysis of shells to give explicit expressions of natural frequencies, nonlinear frequency–amplitude relation and nonlinear dynamic responses. Numerical results show the effects of characteristics of functionally graded materials, pre-loaded axial compression and dimensional ratios on the dynamical behavior of shells. The proposed results are validated by comparing with those in the literature.     

Free vibration study of layered cylindrical shells by collocation with splines

The free vibration of circular cylindrical thin shells, made up of uniform layers of isotropic or specially orthotropic materials, is studied using point collocation method and employing spline function approximations. The equations of motion for the shell are derived by extending Love's first approximation theory. Assuming the solution in a separable form a system of coupled differential equations, in the longitudinal, circumferential and transverse displacement functions, is obtained. These functions are approximated by Bickley-type splines of suitable orders. The process of point collocation with suitable boundary conditions results in a generalized eigenvalue problem from which the values of a frequency parameter and the corresponding mode shapes of vibration, for specified values of the other parameters, are obtained. Two types of boundary conditions and four types of layers are considered. The effect of neglecting the coupling between the flexural and extensional displacements is analysed. The influences of the relative layer thickness, a length parameter and a total thickness parameter on the frequencies are studied. Both axisymmetric and asymmetric vibrations are investigated. The effect of the circumferential node number on the vibrational behaviour of the shell is also analysed.     

Free vibrational characteristics of isotropic coupled cylindrical-conical shells

This paper presents the free vibrational characteristics of isotropic coupled conical-cylindrical shells. The equations of motion for the cylindrical and conical shells are solved using two different methods. A wave solution is used to describe the displacements of the cylindrical shell, while the displacements of the conical sections are solved using a power series solution. Both Donnell-Mushtari and Flügge equations of motion are used and the limitations associated with each thin shell theory are discussed. Natural frequencies are presented for different boundary conditions. The effect of the boundary conditions and the influence of the semi-vertex cone angle are described. The results from the theoretical model presented here are compared with those obtained by previous researchers and from a finite element model.     

Free vibrations of elastic circular toroidal shells

In this paper, the free vibrations of elastic in vacuo circular toroidal shells under different boundary conditions are studied using the linear Sanders thin shell theory. Beam functions are used to describe the motion along the meridional direction whilst trigonometric functions are used to represent the deformation of the cross section. It is shown that both the natural frequencies and the mode shapes can be accurately predicted as long as the employed beam functions satisfy the boundary conditions at the ends of the shells. The dependence of the free vibration characteristics of an elastic toroidal shell upon boundary conditions and toroidal to cross-sectional radius ratio is also illustrated and explained in this paper.     

Nonlinear stability of cylindrical shells subjected to axial flow: Theory and experiments

This paper, is concerned with the nonlinear dynamics and stability of thin circular cylindrical shells clamped at both ends and subjected to axial fluid flow. In particular, it describes the development of a nonlinear theoretical model and presents theoretical results displaying the nonlinear behaviour of the clamped shell subjected to flowing fluid. The theoretical model employs the Donnell nonlinear shallow shell equations to describe the geometrically nonlinear structure. The clamped beam eigenfunctions are used to describe the axial variations of the shell deformation, automatically satisfying the boundary conditions and the circumferential continuity condition exactly. The fluid is assumed to be incompressible and inviscid, and the fluid–structure interaction is described by linear potential flow theory. The partial differential equation of motion is discretized using the Galerkin method and the final set of ordinary differential equations are integrated numerically using a pseudo-arclength continuation and collocation techniques and the Gear backward differentiation formula. A theoretical model for shells with simply supported ends is presented as well. Experiments are also described for (i) elastomer shells subjected to annular (external) air-flow and (ii) aluminium and plastic shells with internal water flow. The experimental results along with the theoretical ones indicate loss of stability by divergence with a subcritical nonlinear behaviour. Finally, theory and experiments are compared, showing good qualitative and reasonable quantitative agreement.     

Vibration of doubly curved shallow shells with arbitrary boundaries

The first comprehensive study of shallow shell vibrations subjected to as many as 21 possible boundary conditions is presented. Thin shallow shell theory is used. Relatively accurate results for natural frequencies of doubly-curved shallow shells have been obtained. These can be used for benchmarking by researchers as well as reference data for practicing engineers. The Ritz method is used to solve for natural vibrations of these shells with arbitrary boundary conditions. Natural frequencies are presented for various shell curvatures including spherical, cylindrical and hyperbolic paraboloidal shells.     

双层周期加肋有限长圆柱壳声散射精细特征研究

研究了双层周期性加肋有限长圆柱壳在水中的声散射特性. 壳体振动用薄壳理论的Donnell 方程描述,环肋振动用相互独立的薄板纯弯曲振动和平面应力状态下的振动方程描述,忽略弦间流体对环肋轴向力的作用. 数值计算给出远场收发合置情况下的周向目标强度和角度-频率谱图,并据此进行机理分析. 计算结果表明远场散射声场中除壳体弹性贡献外,弦间流体以及环肋与内外壳的相互作用对散射声场的贡献也是很重要的,并且在角度-频率谱中出现了舷间流体引起的流体附加波以及周期环肋引起的Bragg散射等回波精细特征,其中流体附加波是双层加肋圆柱壳声散射最重要的散射精细特征,是以往单层圆柱壳声散射所不具有的现象. 最后通过实验对理论推导进行了验证,实验与理论基本符合. 关键词： 声散射 圆柱壳 环肋 流体附加波     

Effect of elastic end rings on the eigenfrequencies of thin cylinders

The effect of elastic end rings on the eigenfrequencies of thin cylindrical shells is studied by using an exact solution of the linear eigenvalue problem. The out-of-plane and torsional rigidities of the rings are responsible for the overall shell stiffness. Considerable mode interaction exists for modes with low circumferential wave numbers when the mass of the ring is comparable with that of the shell. The hypothetical simply supported and clamped boundary conditions are practically impossible to realize with a finite-mass ring for relatively short and thin shells.     

MICRO-SENSING CHARACTERISTICS AND MODAL VOLTAGES OF LINEAR/NON-LINEAR TOROIDAL SHELLS

Toroidal shells belong to the shells of revolution family. Dynamic sensing signals and their distributed characteristics of spatially distributed sensors or neurons laminated on thin toroidal shell structures are investigated in this study. Spatially distributed modal voltages and signal patterns are related to the meridional and circumferential membrane/bending strains, based on the direct piezoelectricity, the Gauss theorem, the Maxwell principle and the open-circuit assumption; linear and non-linear toroidal shells are defined based on the thin shell theory and the von Karman geometric non-linearity. With the simplified mode shape functions defined by the Donnell-Mushtari-Vlasov theory, modal-dependent distributed signals and detailed signal components of spatially distributed sensors or neurons are defined and these signals are quantitatively illustrated. Signal distributions basically reveal distinct modal characteristics of toroidal shells. Parametric studies suggest that the dominating signal component results from the meridional membrane strains. Shell dimensions, materials, boundary conditions, natural modes, sensor locations/distributions/sizes, modal strain components, etc., all influence the spatially distributed modal voltages and signal generations.     

Free vibration of a conical shell with variable thickness

An analysis is presented for the free vibration of a truncated conical shell with variable thickness by use of the transfer matrix approach. The applicability of the classical thin shell theory is assumed and the governing equations of vibration of a conical shell are written as a coupled set of first order differential equations by using the transfer matrix of the shell. Once the matrix has been determined by quadrature of the equations, the natural frequencies and the mode shapes of vibration are calculated numerically in terms of the elements of the matrix under any combination of boundary conditions at the edges. The method is applied to truncated conical shells with linearly, parabolically or exponentially varying thickness, and the effects of the semi-vertex angle, truncated length and varying thickness on the vibration are studied.     

Vibration characteristics of thin rotating cylindrical shells with various boundary conditions

An analysis is presented for the vibration characteristics of thin rotating cylindrical shells with various boundary conditions by use of Fourier series expansion method. Based on Sanders’ shell equations, the governing equations of motion which take into account the effects of centrifugal and Coriolis forces as well as the initial hoop tension due to rotating are derived. The displacement field is expressed as a product of Fourier series expressions which represents the axial modal displacements and trigonometric functions which represents the circumferential modal displacements. Stokes’ transformation is employed to derive the derivatives of the Fourier series expressions. Then, through the process of formula derivation, an explicit expression of the exact frequency equation can be obtained for a thin rotating cylinder with classical boundary conditions of any type. Once the frequency equation has been determined, the frequencies are calculated numerically. To validate the present analysis, comparisons between the results of the present method and previous studies are performed and very good agreement is achieved. Finally, the method is applied to investigate the vibration characteristics of thin rotating cylindrical shells under various boundaries, and the results are presented.     

Vibration and stability of a circular cylindrical shell subjected to a torque

An analysis is presented for the vibration and stability of a circular cylindrical shell subjected to a torque. The displacements of a circular shell are written in a series of beam eigenfunctions satisfying the boundary conditions. The kinetic and strain energies of the shell are evaluated analytically, and the frequency eauation of the shell is derived by the Ritz method. The method is applied to circular cylindrical shells under two types of boundary conditions at the edges; the natural frequencies and the divergence torques are calculated numerically, and the effects of the thickness ratio, length ratio and edge conditions on the vibration and stability are studied.     

Large deformation of liquid capsules enclosed by thin shells immersed in the fluid

The deformation of a liquid capsule enclosed by a thin shell in a simple shear flow is studied numerically using an implicit immersed boundary method. We present a thin-shell model for computing the forces acting on the shell middle surface during the deformation within the framework of the Kirchhoff–Love theory of thin shells. This thin-shell model takes full account of finite-deformation kinematics which allows thickness stretching as well as large deflections and bending strains. For hyperelastic materials, the plane-stress assumption is used to compute the hydrostatic pressure and the incompressibility condition yields the thickness strain component and the corresponding change in the thickness. The stresses developing over the cross-section of the shell are integrated over the thickness to yield the stress and moment resultants which are then used to compute the forces acting on the shell middle surface. The immersed boundary method is employed for calculating the hydrodynamics and fluid–structure interaction effects. The location of the thin shell is updated implicitly using the Newton–Krylov method. The present numerical technique has been validated by several examples including an inflation of a spherical shell and deformations of spherical and oblate spheroidal capsules in the shear flow.     

Vibrations of segmented cylindrical shells by a fourier series component mode method

The vibrations of a multi-segment cylindrical shell with a common mean radius are studied. The shell is uniform for any segment but the material and geometric properties may vary from segment to segment. The solution is based on the component mode method coupled with Fourier series and Lagrange multipliers. It is shown that a single segment shell with boundary conditions of free support without tangential constraint is sufficient for an arbitrary shell with arbitrary boundary conditions. Results are presented for simply supported shells and clamped-free shells for two segments with different length and thickness.     

Free vibrations of cylindrical shells with elastic-support boundary conditions

This paper concerns the free vibrations of cylindrical shells with elastic boundary conditions. Based on the Flügge classical thin shell theory, the equations of motion for the cylindrical shells are solved by the method of wave propagations. The wave numbers are obtained by directly solving an eighth order equation. The elastic-support boundary conditions can be arbitrarily specified in terms of 8 independent sets of distributed springs. All the classical homogeneous boundary conditions can be considered as the special cases when the stiffness for each set of springs is equal to either infinity or zero. The present solutions are validated by the results previously given by other researchers and/or obtained using finite element models. The effects on the frequency parameters of elastic restraints are investigated for shells of different geometrical characteristics.     

Free vibration analysis of functionally graded conical shells and annular plates using the Haar wavelet method

The main aim of this paper is to provide a simple yet efficient solution for the free vibration analysis of functionally graded (FG) conical shells and annular plates. A solution approach based on Haar wavelet is introduced and the first-order shear deformation shell theory is adopted to formulate the theoretical model. The material properties of the shells are assumed to vary continuously in the thickness direction according to general four-parameter power-law distributions in terms of volume fractions of the constituents. The separation of variables is first performed; then Haar wavelet discretization is applied with respect to the axial direction and Fourier series is assumed with respect to the circumferential direction. The constants appearing from the integrating process are determined by boundary conditions, and thus the partial differential equations are transformed into algebraic equations. Then natural frequencies of the FG shells are obtained by solving algebraic equations. Accuracy and reliability of the current method are validated by comparing the present results with the existing solutions. Effects of some geometrical and material parameters on the natural frequencies of shells are discussed and some selected mode shapes are given for illustrative purposes. It’s found that accurate frequencies 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 and excellent accuracy.     

A comparison of shell theories for large-amplitude vibrations of circular cylindrical shells: Lagrangian approach

Large-amplitude (geometrically non-linear) vibrations of circular cylindrical shells subjected to radial harmonic excitation in the spectral neighbourhood of the lowest resonances are investigated. The Lagrange equations of motion are obtained by an energy approach, retaining damping through Rayleigh's dissipation function. Four different non-linear thin shell theories, namely Donnell's, Sanders-Koiter, Flügge-Lur’e-Byrne and Novozhilov's theories, which neglect rotary inertia and shear deformation, are used to calculate the elastic strain energy. The formulation is also valid for orthotropic and symmetric cross-ply laminated composite shells. The large-amplitude response of perfect and imperfect, simply supported circular cylindrical shells to harmonic excitation in the spectral neighbourhood of the lowest natural frequency is computed for all these shell theories. Numerical responses obtained by using these four non-linear shell theories are also compared to results obtained by using the Donnell's non-linear shallow-shell equation of motion. A validation of calculations by comparison with experimental results is also performed. Both empty and fluid-filled shells are investigated by using a potential fluid model. The effects of radial pressure and axial load are also studied. Boundary conditions for simply supported shells are exactly satisfied. Different expansions involving from 14 to 48 generalized co-ordinates, associated with natural modes of simply supported shells, are used. The non-linear equations of motion are studied by using a code based on an arclength continuation method allowing bifurcation analysis.     

Polynomial versus trigonometric expansions for nonlinear vibrations of circular cylindrical shells with different boundary conditions

Large-amplitude (geometrically nonlinear) forced vibrations of circular cylindrical shells with different boundary conditions are investigated. The Sanders-Koiter nonlinear shell theory, which includes in-plane inertia, is used to calculate the elastic strain energy. The shell displacements (longitudinal, circumferential and radial) are expanded by means of a double mixed series: harmonic functions for the circumferential variable and three different formulations for the longitudinal variable; these three different formulations are: (a) Chebyshev orthogonal polynomials, (b) power polynomials, and (c) trigonometric functions. The same formulation is applied to study different boundary conditions; results are presented for simply supported and clamped shells. The analysis is performed in two steps: first a liner analysis is performed to identify natural modes, which are then used in the nonlinear analysis as generalized coordinates. The Lagrangian approach is applied to obtain a system of nonlinear ordinary differential equations. Different expansions involving from 14 to 34 generalized coordinates, associated with natural modes of both simply supported and clamped-clamped shells, are used to study the convergence of the solution. The nonlinear equations of motion are studied by using arclength continuation method and bifurcation analysis. Numerical responses obtained in the spectral neighborhood of the lowest natural frequency are compared with results available in literature.     

考虑铺层角的复合材料圆柱壳自由振动三维弹性准确解

为了获得任意角度铺层的多层复合材料圆柱壳的自由振动准确解,在三维弹性理论的基础上,结合分层理论和状态空间法,建立横向位移和应力的传递矩阵,轴向和环向位移采用双螺旋模式的位移函数,对任意角度铺层复合材料圆柱壳简支边界条件下的自由振动进行了理论推导,得到了自由振动方程的精确形式。与文献理论解和有限元计算结果对比,结果表明,关注频率在2倍的环频率以下时,薄壳的固有频率计算精度能控制在1%以内,厚壳的固有频率计算精度能控制在2%以内。对于厚壳的计算可将壳体沿厚度方向划分为多层来处理,这样能有效提高计算精度。计算分析了铺层角对壳体固有频率的影响,环向模态数较低时,固有频率随着铺层角的增加呈抛物线变化趋势;环向模态数较高时,固有频率随着铺层角的增大单调递增。该理论方法同样适用于均质各向同性壳和正交各向异性圆柱壳。      

Free vibration of joined conical-cylindrical shells

An analysis is presented for the free vibration of joined conical-cylindrical shells. The governing equations of vibration of a conical shell, including a cylindrical shell as a special case, are written as a coupled set of first order differential equations by using the transfer matrix of the shell. Once the matrix has been determined, the entire structure matrix is obtained by the product of the transfer matrices of the shells and the point matrix at the joint, and the frequency equation is derived with terms of the elements of the structure matrix under the boundary conditions. The method has been applied to a joined truncated conical-cylindrical shell and an annular plate-cylindrical shell system, and the natural frequencies and the mode shapes of vibration calculated numerically. The results are presented.