Geometrically nonlinear vibration of laminated composite cylindrical thin shells with non-continuous elastic boundary conditions

The geometrically nonlinear forced vibration response of non-continuous elastic-supported laminated composite thin cylindrical shells is investigated in this paper. Two kinds of non-continuous elastic supports are simulated by using artificial springs, which are point and arc constraints, respectively. By using a set of Chebyshev polynomials as the admissible displacement function, the nonlinear differential equation of motion of the shell subjected to periodic radial point loading is obtained through the Lagrange equations, in which the geometric nonlinearity is considered by using Donnell’s nonlinear shell theory. Then, these equations are solved by using the numerical method to obtain nonlinear amplitude–frequency response curves. The numerical results illustrate the effects of spring stiffness and constraint range on the nonlinear forced vibration of points-supported and arcs-supported laminated composite cylindrical shells. The results reveal that the geometric nonlinearity of the shell can be changed by adjusting the values of support stiffness and distribution areas of support, and the values of circumferential and radial stiffness have a more significant influence on amplitude–frequency response than the axial and torsional stiffness.

     

On the analyses of vibration and stability for orthotropic circular cylindrical shells conveying fluid

Based on flügge's thin orthotropic shell equations and a linear potential flow theory, buckling instability and flutter-type instability of orthotropic circular cylindrical shells conveying fluid are studied. By means of the expansion in beam-mode functions and Galerkin's method they can be reduced to solving a generalized complex eigenvalue problem. In calculating the generalized forces defined as Eq. (23), a direct numerical integration technique, which has proven to be more efficient than others, is used. For clamped-clamped orthotropic shells several numerical examples show that the effects of axial elastic modulus on the dynamic behaviors are quite different from those of circumferential elastic modulus.     

Physically Nonlinear Deformation of a Long Orthotropic Cylindrical Shell with Elliptic Cross-Section*

International Applied Mechanics - The stress–strain state of long cylindrical shells with elliptical cross-section is studied. The shells are made of a nonlinear elastic orthotropic organic...     

Torsional vibration and stability of functionally graded orthotropic cylindrical shells on elastic foundations

In this study, the torsional vibration and stability problems of functionally graded (FG) orthotropic cylindrical shells in the elastic medium, using the Galerkin method was investigated. Pasternak model is used to describe the reaction of the elastic medium on the cylindrical shell. Mixed boundary conditions are considered. The material properties and density of the orthotropic cylindrical shell are assumed to vary exponentially in the thickness direction. The basic equations of the FG orthotropic cylindrical shell under the torsional load resting on the Pasternak-type elastic foundation are derived. The expressions for the critical torsional load and dimensionless torsional frequency parameter of the FG orthotropic cylindrical shell resting on elastic foundations are obtained. The effects of variations of shell parameters, the exponential factor characterizing the degree of material gradient, orthotropy, foundation stiffness and shear subgrade modulus of the foundation on the critical torsional load and dimensionless torsional frequency parameter are examined.     

Natural vibrations of a cantilever thin elastic orthotropic cylindrical shell

The natural vibrations of a cantilever thin elastic orthotropic circular cylindrical shell are studied. Dispersion equations for the determination of possible natural frequencies of cantilever closed shells and open shells with Navier hinged boundary conditions at the longitudinal edges are derived from the classical dynamic theory of orthotropic cylindrical shells. It is proved that there are asymptotic relationships between these problems and the problems for a cantilever orthotropic strip plate and for a cantilever rectangular plate and the eigenvalue problem for a semi-infinite closed orthotropic cylindrical shell with free end and for the same but open shell with Navier hinged boundary conditions at the longitudinal edges. A procedure to identify types of vibrations is presented. Orthotropic cylindrical shells with different radii and lengths are used as an example to find approximate values of the dimensionless natural frequency and damping factor for vibration modes __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 5, pp. 68–91, May 2008.     

Dynamic problems for and stress–strain state of inhomogeneous shell structures under stationary and nonstationary loads

This paper reviews studies and analyzes results on the effect of discrete ribs on the dynamic characteristics of rectangular plates and cylindrical shells. Use is made of the vibration equations derived from the classical theories of beams, plates, and shells. The effect of Pasternak’s elastic foundation on the critical velocities of a structurally orthotropic model of a ribbed cylindrical shell is determined. Nonstationary problems are solved for perforated and ribbed shells of revolution filled with a fluid or resting on an elastic foundation and subjected to moving or impulsive loads. Results from studies of the behavior of sandwich shell structures under impulsive loads of various types are presented     

Vibration analysis of nonhomogeneous orthotropic cylindrical shells including combined effect of shear deformation and rotary inertia

This paper is the result of an investigation on the vibration of non-homogeneous orthotropic cylindrical shells, based on the shear deformation theory. Assume that the Young’s moduli, shear moduli and density of the orthotropic material are continuous functions of the coordinate in the thickness direction. The basic equations of non-homogeneous orthotropic cylindrical shells with the shear deformation and rotary inertia are derived in the framework of Donnell-type shell theory. The ends of a non-homogeneous orthotropic cylindrical shell are considered as simply supported. The basic equations are reduced to the sixth-order algebraic equation for the frequency using the Galerkin method. Solving this algebraic equation, the lowest values of non-dimensional frequency parameters for non-homogeneous orthotropic cylindrical shells with and without shear deformation and rotary inertia are obtained. Calculations, effects of shear stresses and rotary inertia, orthotropy, non-homogeneity and shell geometry parameters on the lowest values of non-dimensional frequency parameter are described. The results are verified by comparing the obtained values with those in the existing literature.     

Nonlinear free vibration of spinning cylindrical shells with arbitrary boundary conditions

The aim of the present study is to investigate the nonlinear free vibration of spinning cylindrical shells under spinning and arbitrary boundary conditions. Artificial springs are used to simulate arbitrary boundary conditions. Sanders' shell theory is employed, and von Kármán nonlinear terms are considered in the theoretical modeling. By using Chebyshev polynomials as admissible functions, motion equations are derived with the Ritz method. Then, a direct iteration method is used to obtain the nonlinear vibration frequencies. The effects of the circumferential wave number, the boundary spring stiffness, and the spinning speed on the nonlinear vibration characteristics of the shells are highlighted. It is found that there exist sensitive intervals for the boundary spring stiffness, which makes the variation of the nonlinear frequency ratio more evident. The decline of the frequency ratio caused by the spinning speed is more significant for the higher vibration amplitude and the smaller boundary spring stiffness.     

Thermomechanical vibration analysis of a functionally graded shell with flowing fluid

This paper reports the results of an investigation into the vibration of functionally graded cylindrical shells with flowing fluid, embedded in an elastic medium, under mechanical and thermal loads. By considering rotary inertia, the first-order shear deformation theory (FSDT) and the fluid velocity potential, the dynamic equation of functionally graded cylindrical shells with flowing fluid is derived. Here, heat conduction equation along the thickness of the shell is applied to determine the temperature distribution and material properties are assumed to be graded distribution along the thickness direction according to a power-law in terms of the volume fractions of the constituents. The equations of eigenvalue problem are obtained by using a modal expansion method. In numerical examples, effects of material composition, thermal loading, static axial loading, flow velocity, medium stiffness and shell geometry parameters on the free vibration characteristics are described. The new features in this paper are helpful for the application and the design of functionally graded cylindrical shells containing fluid flow.     

Three-dimensional free vibration analysis of four-parameter continuous grading fiber reinforced cylindrical panels resting on Pasternak foundations

This research investigates three-dimensional free vibration analysis of four-parameter continuous grading fiber reinforced (CGFR) cylindrical panels resting on Pasternak foundations by using generalized power-law distribution. The functionally graded orthotropic panel is simply supported at the edges, and it is assumed to have an arbitrary variation of matrix volume fraction in the radial direction. A four-parameter power-law distribution presented in literature is proposed. Symmetric and asymmetric volume fraction profiles are presented. Suitable displacement functions that identically satisfy the boundary conditions at the simply supported edges are used to reduce the equilibrium equations to a set of coupled ordinary differential equations with variable coefficients, which are solved by generalized differential quadrature method, and natural frequency is obtained. The fast rate of convergence of the method is demonstrated, and to validate the results, comparisons are made with the available solutions for functionally graded isotropic shells with/without elastic foundations. The effect of the elastic foundation stiffness parameters and various geometrical parameters on the vibration behavior of the CGFR cylindrical panels is investigated. This work mainly contributes to illustrate the influence of the four parameters of power-law distributions on the vibration behavior of functionally graded orthotropic cylindrical panels resting on elastic foundation. This paper is also supposed to present useful results for continuous grading of matrix volume fraction in the thickness direction of a cylindrical panel on elastic foundation and comparison with similar discrete laminated composite cylindrical panel.     

正交各向异性功能梯度材料开口圆柱壳的自由振动分析

区别于一般圆柱壳,开口圆柱壳沿周向是不封闭的,因此具有四个边界,本文根据轴向梁式振动和轴向曲拱振动特性对各种端部与侧边边界条件下的壳体提出统一的位移振型函数,并根据哈密顿原理建立了材料参数与空间坐标相关的正交各向异性开口圆柱壳的动力变分方程,求出了不同材料属性下开敞圆柱壳固有频率与振型解的一般解析表达式,适用于任意边界条件下不同材料的开敞圆柱壳自由振动分析．     

Study of vibration characteristics for orthotropic circular cylindrical shells using wave propagation approach and multivariate analysis

A study of free vibration of orthotropic circular cylindrical shells is presented. The vibration control equations of shells are based on Flügge classical thin shell theory. Wave approach is used in the analysis, in which the boundary conditions of shells can be simplified according to the associated beam. The free vibration frequencies of shells can be obtained from a frequency polynomial equation of order 6. The parametric analysis of the free vibration of orthotropic cylindrical shells is investigated using a statistical method. The effects of geometrical parameters and material characteristics upon frequencies are investigated here. Multivariate analysis (MVA) can be a useful tool for this parametric study. Some statistical characteristics, including correlation analysis and ANOVA are applied. ANOVA has been conducted to predict the statistical significance of the various factors. Calculations are performed in the Minitab statistical software. The results show that the L/R, h/R and m have larger effects on the lowest frequency. The importance of input parameters is ranked according to their contributions to the total variance. A knowledge and data visualization approach, Self-organizing mapping (SOM) is also adopted here for mining some intrinsic characteristics of shells.     

正弦波纹扁球壳的非线性强迫振动

波纹壳是传感器弹性元件的一类重要形式,也是精密仪器仪表弹性元件中的一类重要形式。由于波纹壳形状复杂、参数众多、厚度薄,对其进行非线性分析非常重要同时也是十分困难的。本文考虑一种在传感器弹性元件中有重要应用价值的正弦波纹浅球壳体,将这种壳体视为结构上的圆柱正交异性扁球壳,根据Andryewa的思想,分别得到了正弦波纹壳径向、环向在拉伸、弯曲下的等价的四个各向异性参数;建立了正弦波纹扁球壳的非线性强迫振动微分方程;得到了正弦波纹扁球壳非线性强迫振动的共振周期解及幅频特性曲线。     

The effect of material and geometry on the non-linear vibrations of orthotropic circular cylindrical shells

The extensive use of circular cylindrical shells in modern industrial applications has made their analysis an important research area in applied mechanics. In spite of a large number of papers on cylindrical shells, just a small number of these works is related to the analysis of orthotropic shells. However several modern and natural materials display orthotropic properties and also densely stiffened cylindrical shells can be treated as equivalent uniform orthotropic shells. In this work, the influence of both material properties and geometry on the non-linear vibrations and dynamic instability of an empty simply supported orthotropic circular cylindrical shell subjected to lateral time-dependent load is studied. Donnell׳s non-linear shallow shell theory is used to model the shell and a modal solution with six degrees of freedom is used to describe the lateral displacements of the shell. The Galerkin method is applied to derive the set of coupled non-linear ordinary differential equations of motion which are, in turn, solved by the Runge–Kutta method. The obtained results show that the material properties and geometric relations have a significant influence on the instability loads and resonance curves of the orthotropic shell.     

The R-function method used to solve nonlinear bending problems for orthotropic shallow shells on an elastic foundation

The paper proposes a method to solve geometrically nonlinear bending problems for thin orthotropic shallow shells and plates interacting with a Winkler–Pasternak foundation under transverse loading. This method is based on Ritz’s variational method and the R-function method. The developed algorithm and software are used to solve a number of test problems and to study complex-shaped shells. The effect of the shape of shells, the boundary conditions, the stiffness of the foundation, and the load distribution on the behavior of isotropic and orthotropic shells undergoing geometrically nonlinear bending is studied     

VIBRATION AND STABILITY OF RING-STIFFENED THIN-WALLED CYLINDRICAL SHELLS CONVEYING FLUID

Based on the Flügge shell theory,equations of motion of ring-stiffened thin-walled cylindrical shells conveying fluid are developed with the aid of the Hamilton's principle.Analysis is carried out on t...     

Non-linear dynamic analysis of symmetric and antisymmetric cross-ply laminated orthotropic thin shells

In this paper, the governing equations for non-linear free vibration of truncated, thin, laminated, orthotropic conical shells using the theory of large deformations with the Karman-Donnell-type of kinematic nonlinearity are derived. Applying superposition principle and Galerkin’s method, these equations are reduced to a time dependent non-linear differential equation. The frequency-amplitude relationship for the laminated orthotropic thin truncated conical shell is obtained using the method of weighted residuals. In the particular case, we can obtain the similar relationships for the single-layer and laminated orthotropic cylindrical shells, also. The influence played by geometrical parameters of the conical shell and physical parameters of the laminate (i.e. material properties, staking sequences and number of layers) on the non-linear vibration behavior of the conical shell is examined. It is noticed that the non-linear vibration of shells is highly dependent on laminate characteristics and, from these observations, it is concluded that specific configurations of laminates should be designed for each kind of application. Present results are compared with available data for special cases.     

接地惯容式减振器对悬臂输流管稳定性和动态响应的影响研究

输流管道广泛应用于机械、航空、核电和石油等重要工程领域.为防止管道结构因流致振动破坏造成的损失, 很有必要对其稳定性、动力学响应及其调控进行深入研究.本文提出一种由惯容器、弹簧和阻尼器并联组成的减振器模型, 研究了这种接地惯容减振器对悬臂输流管稳定性和非线性振动的影响. 首先, 基于哈密顿原理给出了带有接地惯容减振器非保守系统的非线性动力学模型; 然后, 利用高阶伽辽金方法对非线性方程进行离散化; 最后, 分别从线性和非线性角度分析了不同减振器参数下输流管道的被动控制效果, 着重讨论了惯容系数和减振器安装位置对悬臂管稳定性和动态响应的影响机制.线性理论模型的研究结果显示, 接地惯容减振器可显著影响悬臂管的失稳临界流速, 故通过调节减振器参数能有效提高输流管道的稳定性;惯容系数和弹簧刚度对系统稳定性的控制效果还与减振器的安装位置密切相关.非线性理论模型的分析结果显示, 惯容系数和减振器位置对输流管的非线性动态响应也有显著影响, 且这种影响还依赖于管道的流速取值; 在某些参数条件下, 减振器还可使输流管道由周期运动演化为复杂的混沌行为. 本文研究结果表明, 通过设计合理的惯容式减振器参数, 可提升悬臂输流管道的稳定性并有效抑制其颤振幅值.      

Spectral Problem for Shells with Fluid

Variational eigenvalue equations describing vibrations of orthotropic shells containing an ideal incompressible fluid are obtained. The vibration frequencies are assumed to be small, which makes it possible to use linear equations and to consider the boundary of the wet surface of the shell to be unchanged. The equations of anisotropic shells are based on the linear relations of multifield theory, which allows to obtain a more accurate model of anisotropic shells that satisfies the conditions of the finite-element method. The fluid flow is considered irrotational and is described using the Laplace equation. A finite-element algorithm is designed to determine the natural frequencies and modes of vibrations of an arbitrary multilayer orthotropic shell of revolution which is partially filled with an ideal incompressible fluid. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 6, pp. 128–135, November–December, 2005.     

Nonlinear vibrations of orthotropic shallow shells of revolution

A set of nonlinearly coupled algebraic and differential eigenvalue equations of nonlinear axisymmetric free vibration of orthotropic shallow thin spherical and conical shells are formulated.following an assumed time-mode approach suggested in this paper. Analytic solutions are presented and an asymptotic relation for the amplitude-frequency response of the shells is derived. The effects of geometrical and material parameters on vibrations of the shells are investigated.