Effects of modal coupling on the dynamics of parametrically and directly excited cylindrical shells

Cylindrical shells exhibit a dense frequency spectrum, especially in the lowest frequency range. In addition, due to the circumferential symmetry, frequencies occur in pairs. So, in the vicinity of the lowest natural frequencies, several equal or nearly equal frequencies may occur, leading to multiple internal resonances. The aim of the present work is to investigate the dynamic behavior and stability of cylindrical shells under lateral and axial forcing with equal natural frequencies. The shell is modeled using the Donnell nonlinear shallow shell theory. A consistent modal solution for this problem is deduced and used to discretize the equations of motion by applying the Galerkin method. A parametric analysis is conducted to clarify the influence of the modal interaction among these nonlinear vibration modes on coexisting solutions, bifurcations, resonances curves and stability boundaries of the shell.     

Free vibration analysis of joined conical shells: Analytical and experimental study

Natural frequencies and mode shapes of two joined isotropic conical shells are presented in this study. The joined conical shells can be considered as the general case for joined cylindrical–conical shells, joined cylinder–plates or cone–plates, conical and cylindrical shells with stepped thicknesses and also annular plates. Governing equations are obtained using thin-walled shallow shell theory of Donnell and Hamilton׳s principle. The continuity conditions at the joining section of the cones are appropriate expressions among stress resultants and deformations. The equations are solved assuming trigonometric response in circumferential and series solution in meridional directions and all combinations of boundary conditions can be assumed in this method. The results are compared and validated with the available results in other investigations and also modal testing. The effects of semi-vertex angles and meridional lengths on the natural frequency and circumferential wave number of joined shells are investigated.     

Exact vibration frequencies of segmented axisymmetric shells

Axisymmetric shells are commonly used in all fields of engineering and their dynamic characteristics are of great importance in their overall performance. Segmented shells are shells that are built of several pieces that form together an axisymmetric shell with a joint axis. These can be cylindrical, conical or plate segments, which are connected to form a complete shell. In this research, the exact dynamic stiffness matrix for each segment is derived and used in the assembly of the complete structure dynamic stiffness matrix. Then the natural frequencies of vibrations are found as the frequencies that cause this matrix to become singular. Examples are given for the frequencies and modes of segmented shells made of conical, cylindrical, and plate segments.     

An improved treatment of mixed interpolation functions in eight-node assumed natural strain shell element for vibration analysis

In this article, we investigate the vibration analysis of plates and shells, using an eight-node shell element that allows for the effects of transverse shear deformation and rotary inertia. The natural frequencies of plates and shells are presented, and the forced vibration analysis of plates and shells subjected to arbitrary loading is carried out. In order to overcome membrane and shear locking phenomena, the assumed natural strain method is used. To improve the eight-node shell element for free and forced vibration analysis, a new combination of sampling points for assumed natural strain method was applied. The refined first-order shear deformation theory based on Reissner–Mindlin theory, which directly addresses the transverse shear deformation without a shear correction factor, is adopted for the development of a new eight-node assumed strain shell element with rotary inertia effect. In order to validate the finite element numerical solutions, the reference solutions of plates based on the first-order shear deformation theory are presented. Results of the present theory show good agreement with the reference solutions. In addition, the effect of damping is investigated on the forced vibration analysis of plates and shells.     

弹性地基上扁球壳的自由振动

本文研究双参数弹性地基上扁球壳的自由振动频率和振型，推导了计及地基惯性力影响的扁壳自由振动控制方程。方程的精确解以贝塞尔函数表达。同时，分别导出了壳的经向、环向和横向位移分量的表示式。根据所得的解，文中给出了固定边扁球壳的频率方程和数值结果。同时。还给出了对应于前三个频率的振型变化曲线。     

Vibration of prestressed thin cylindrical shells conveying fluid

A general approach to modelling the vibration of prestressed thin cylindrical shells conveying fluid is presented. The steady flow of fluid is described by the classical potential flow theory, and the motion of the shell is represented by Sanders’ theory of thin shells. A strain–displacement relationship is deployed to derive the geometric stiffness matrix due to the initial stresses caused by hydrostatic pressure. Hydrodynamic pressure acting on the shell is developed through dynamic interfacial coupling conditions. The resulting equations governing the motion of the shell and fluid are solved by a finite element method. This model is subsequently used to investigate the small-vibration dynamic behaviour of prestressed thin cylindrical shells conveying fluid. It is validated by comparing the computed natural frequencies, within the linear region, with existing reported experimental results. The influence of initial tension, internal pressure, fluid flow velocity and the various geometric properties is also examined.     

一种新型后张拉整体成型网壳动力性能分析

提出一种新型后张拉整体成型局部双层球面网壳的单弦杆平板网架模型,并通过有限元软件ANSYS建立有限元模型,采用BlockLanczos法对3组缝隙值对应成型后局部双层球面网壳进行模态分析,得到网壳的自振频率和振型,并与相应成型后在单层部分环向补杆网壳进行对比分析。结果表明:成型后网壳的各阶自振频率有随成型前预留缝隙值的增大而增大的趋势;在网壳的单层部分进行环向补杆能有效提高结构刚度,使补杆网壳的各阶自振频率与非补杆网壳相比有显著提高。此外,补杆网壳与非补杆网壳都表现出复杂的动力性能,在进行网壳的动力响应分析时应特别注意。     

Associations between shell strength, shell morphology and heavy metals in the land snail Cepaea nemoralis (Gastropoda, Helicidae)

In snails there is an intimate relation between shell size, thickness, strength and calcium content that may be influenced by environmental factors such as predation and heavy metal pollution. The snail Cepaea nemoralis shows variability for shell colour and banding pattern, and frequencies of colour morphs are highly variable in natural populations. We used C. nemoralis to investigate (i) the relations between shell morphology, shell Ca and heavy metal content (Cd, Cr, Pb, Zn), and shell strength, (ii) differences in shell morphology and shell strength among localities and yellow and pink shells and (iii) whether snails from polluted sites show increased levels of heavy metals in their shell. Larger shells were heavier, thicker, needed a higher force to be crushed but did not have a higher Ca concentration. Cd and Zn concentrations were higher in shells from polluted plots compared to shells from unpolluted plots but Ca levels in the shell were comparable among plots. Zn concentration was negatively correlated with shell traits. Although there was substantial variation in shell strength, thickness and dry weight among localities, none of the shell traits differed between individuals from polluted and reference plots nor between colour morphs. Our results suggest that the effect of heavy metal pollution on shell strength and morphology is limited in the investigated populations.     

Bending and stretching actions in shallow domes. Part 1. Analytical derivations

In this two-part set of articles the response of shallow axisymmetric shells with circular plan is investigated with the aid of the two-surface shell theory. Such an approach has the unique advantage of elucidating, both qualitatively, and quantitatively, the complex—and key—interaction of bending and stretching actions in such shells in a simple and direct manner. Although discussion centres upon a shallow paraboloid of revolution, the general features emerging from the study are, obviously, relevant for the generic family of shallow domes (e.g. spherical) covering the various limiting cases of support around their circumference.This first paper begins with a two-surface exposition of the set of equations describing the behaviour of the shell. Next, analytical bending solutions to the specific practical problem wherein the shell is subjected to uniformly distributed vertical loading (corresponding, say, to its self-weight) are presented; and this is followed by the working out of closed-form expressions for the distributions of bending and stretching effects in theshell under consideration.     

The prediction of the elastic critical load of submerged elliptical cylindrical shell based on the vibro-acoustic model

Based on the vibro-acoustical model, an effective new approach to nondestructively predict the elastic critical hydrostatic pressure of a submerged elliptical cylindrical shell is presented in this paper. Based on the Goldenveizer–Novozhilov thin shell theory, the vibration equations considering hydrostatic pressures of outer fluid are written in the form of a matrix differential equation which is obtained by using the transfer matrix of the state vector of the shell. The fluid-loading term is represented as the form of Mathieu function. The data of the fundamental natural frequencies of the various elliptical cylindrical shells with different hydrostatic pressure and boundary conditions are obtained by solving the frequency equation using Lagrange interpolation method. The curve of the fundamental natural frequency squared versus hydrostatic pressure is drawn, which is approximately straight line. The elastic critical hydrostatic pressure is therefore obtained while the fundamental natural frequency is assumed to be zero according to the curve. The results obtained by the present approach show good agreement with published results.     

Vibration analysis of cylindrical shells in contact with an annular fluid region

An experimental and numerical investigation of the vibratory modal characteristics of a vertical thin-walled cylindrical shell containing water, or oil, in an adjacent coaxial region is presented. A system of two fluid-coupled cylindrical shells is studied. In addition to the effect of increasing liquid level in the shell, the effects of the thickness of the liquid layer in the annular region between the vibrating outer shell and another coaxial rigid cylinder, or between two thin coaxial cylindrical shells, are studied. Experimental results were obtained using the standard modal analysis technique and the time average holographic method. In parallel, computation of natural frequencies and mode shapes of the coupled fluid-structural vibrations were carried out by means of the finite element method operated within the framework of the ANSYS program package. The comparison between experimental results and finite element results was found to be reasonable in most of the cases studied. The fundamental influence of the viscosity of the fluid in the narrow annulus upon the dynamic behaviour of the shell is discussed.     

Nonlinear free vibration analysis of prestressed circular cylindrical shells on the Winkler/Pasternak foundation

Nonlinear free vibration analysis of prestressed circular cylindrical shells placed on Winkler/Pasternak foundation is investigated in the present paper. The nonlinearity is considered due to large deflections. Simultaneous effects of prestressed condition and elastic foundation on natural frequencies of the shells under various boundary conditions are examined extensively in this study. The nonlinear Sanders–Koiter shell theory is employed in order to derive strain–displacement relationships. The nonlinear classical Love's thin shell theory is also applied in some specific cases due to contrast the results. Beam modal functions are used to approximate spatial displacement field. The governing equations in linear state are solved by the Rayleigh–Ritz procedure. Perturbation methods are used to find the relationship between vibration amplitude and frequency in nonlinear state. Prestress state includes the effects of internal hydrostatic pressure and initial uniaxial tension. Results are compared with published theoretical and experimental data for some specific cases.     

均布荷载下多波连续球面扁壳的计算

本文根据扁壳的弯矩理论讨论了矩形底球面扁壳的解析解。给出的内力及位移分量的表达式可用来考虑连续壳体的边界影响。解析解采取单级数形式。通过数值计算的结果分析,证明解的收敛性较好,可满足工程设计的要求。已编制了输入数据简单且勿需较大内存的微机通用程序。     

The effect of geometric imperfections on the vibrations of anisotropic cylindrical shells

Analytical–numerical models to analyse the flexural vibration behaviour of anisotropic cylindrical shells are presented. The two models (denoted as Level-1 and Level-2 Analysis) have different levels of complexity and can be used to study the influence of important parameters, such as geometric imperfections, static loading, and boundary conditions. A specific anisotropic shell is used in the calculations in this paper. The influence of the imperfection shape and amplitude on the natural frequency is investigated for this shell via both the Level-1 and the Level-2 Analysis. Imperfections with the shape of the “lowest vibration mode” give a decrease of the natural frequency with increasing imperfection amplitude. The results of the Level-2 Analysis for the effect of imperfections on the natural frequency are in reasonable agreement with Finite Element calculations.     

Vibrations of doubly cantilevered laminated-composite thin shallow shells

Vibrations of laminated-composite thin shallow shells that are doubly cantilevered (i.e. with two adjacent edges clamped and the remaining free) are analyzed. The Ritz method, with algebraic polynomial displacement functions, is used. Convergence studies are made which demonstrate that reasonably accurate results are obtained by using 147 displacement terms for spherical, circular cylindrical, and hyperbolic paraboloidal shallow shells and 49 terms for plates. The effect of the material, fiber orientation, and curvature upon the frequencies is studied. The difference in behavior between these shells and those studied earlier (completely free, cantilevered, and simply supported) is outlined.     

Vibration, buckling and collapse of ovaloid toroidal tanks

A shell-theory finite element analysis is carried out for toroidal tanks with ovaloid cross-section. The analysis serves to determine the natural frequencies, and the buckling and collapse pressures. A variety of support conditions are considered, including lines of support at the inner and outer equator of the tank. For validation, comparison is made with previously published results for stress, vibration, and buckling of elliptical toroidal shells. Finally, a parametric study is carried out to determine the influence on the natural frequency, and buckling and collapse pressures, of shell size, shell thickness, material properties, and support conditions.     

预应力局部单双层扁网壳的参数分析与近似优化

预应力局部单、双层扁网壳具有较好的室内视觉效果和良好的经济性。其结构参数的确定属多级别优化问题,实施困难。为满足初步设计的需要,提出采用均匀设计法进行结构的参数分析与近似优化。它可利用现有软件来实现。作为算例,对一个柱支承预应力扁网壳进行了参数分析,得到了网壳用钢量、变形和极限荷载系数与三个主要设计参数(单层部分材料设计应力折减系数、单层部分范围和预应力值)之间的近似函数关系,并找出了满足规范要求的各参数之间的较理想的最优水平组合。设计检验说明了分析结果的可靠性。该方法可为解决类似优化问题提供参考。     

Effects of edge constraints upon shallow shell frequencies

The effects of changing edge constraints upon the frequencies of shallow shells with rectangular planforms are studied. Attention is focused upon a single edge, with the other three edges remaining completely free. For that edge clamped, simply supported, and free edge conditions are imposed; however, each of these has four possibilities depending upon the existence of either or both types of membrane constraint along the single edge. The Ritz method, assuming algebraic polynomials as displacement functions, is used to obtain accurate results. Frequencies for three types of shallow shells (circular cylindrical, spherical, hyperbolic paraboloidal) are obtained, for two shallowness ratios and two thickness ratios each. Careful attention is paid to the number of rigid body modes (zero frequencies) present, for these enter quite strongly into considerations of the effects of changing edge constraints.     

Strain-based finite element for the natural frequencies of cylindrical shells

The free-vibration natural frequencies of complete cylindrical shells, having different end conditions, and cylindrical panels have been investigated. The finite element method is used in which a strain-based cylindrical shell element is employed. This element satisfies the exact representation of rigid-body modes of displacements and all components of strains are assumed to be independent insofar as it is allowed by the compatibility equations. The element has only external degrees of freedom, five at each corner node, to avoid difficulties associated with higher-order elements. The stiffness and mass matrices are obtained using numerical integrations by the deployment of Gauss-Legendre quadrature points.     

Shallow shell finite element for the large deflection geometrically nonlinear analysis of shells and plates

An investigation into the large deflection, geometrically nonlinear behaviour of shells is carried out in the present paper. The finite element method is used in conjunction with linearised incrementation and the Newton-Raphson iterative technique.The finite element used is based on independent strain assumptions insofar as it is allowed by the compatibility equations. Strain-displacement relationships based on shallow shell formulation are used and applied to an element having three principal curvatures. The resulting element has the only essential external degrees of freedom, satisfies the exact requirement of strain free rigid body modes of displacements and can be used for the representation of cylindrical, spherical and hyperbolic paraboloid shells.Complex load-deflection curves are obtained for cylindrical and spherical shells by incrementing loads as well as deflections. The relative behaviour of cylindrical and spherical panels having the same overall dimensions are also discussed in terms of stiffness, instability and snap-through phenomena.