Nonlinear vibrations of rotating thin circular cylindrical shell

Nonlinear vibrations of thin circular cylindrical shells are investigated in this paper. Based on Love thin shell theory, the governing partial differential equations of motion for the rotating circular cylindrical shell are formulated using Hamilton principle. Taking into account the clamped-free boundary conditions, the partial differential system is truncated by using the Galerkin method. Sequentially, the effects of temperature, geometric parameters, circumferential wave number, axial half wave number and rotating speed on the nature frequency of the rotating circular cylindrical shell are studied. The dynamic responses of the rotating circular cylindrical shell are also investigated in time domain and frequency domain. Then, the effects of nonlinearity, excitation and damping on frequency responses of steady solution are investigated.     

The displacement solution of conical shell and its application

In this paper,the displacement solution method of the conical shell is presented.Fromthe differential equations in displacement form of conical shell and by introducing adisplacement function,U(s,θ),the differential equations are changed into an eight-ordersoluble partial differential equation about the displacement function U(s,θ)in which thecoefficients are variable.At the same time,the expressions of the displacement and internalforce components of the shell are also given by the displacement function.As special casesof this paper,the displacement function introduced by V.Z.Vlasov in circular cylindricalshell,the basic equation of the cylindrical shell and that of the circular plate are directlyderived.Under the arbitrary loads and boundary conditions,the general bending problem of theconical shell is reduced to finding the displacement function U(s,θ),and the generalsolution of the governing equation is obtained in generalized hypergeometric function,Forthe axisymmetric bending deformation of the     

带初始几何缺陷FGM固支圆柱壳的非线性动力学分析

本文主要研究了带初始几何缺陷的功能梯度固支圆柱壳在不同体积分数下的非线性动力学行为。假定该功能梯度圆柱壳材料的组分是沿厚度的方向呈梯度几何变化的。运用经典板壳理论、von-Karman几何非线性应变位移关系以及Hamilton原理,推导出两端固支FGM圆柱壳的偏微分非线性运动控制方程。本文考虑了圆柱壳的对称模态,利用Galerkin法对上述非线性动力学方程进行截断,得到常微分形式的非线性动力学方程。主要运用Runge-Kutta法进行数值仿真,并且画出了其最大lyapunov指数图,主要研究了面内载荷对振动响应的影响,并对比了不同体积分数对系统非线性动力学的影响。     

The exact solution for the general bending problems of conical shells on the elastic foundation

The general bending problem of conical shells on the elastic foundation (Winkler Medium) is not solved. In this paper, the displacement solution method for this problem is presented. From the governing differential equations in displacement form of conical shell and by introducing a displacement function U(s,θ), the differential equations are changed into an eight-order soluble partial differential equation about the displacement function U(s,θ) in which the coefficients are variable. At the same time, the expressions of the displacement and internal force components of the shell are also given by the displacement function U(s θ). As special cases of this paper, the displacement function introduced by V.S. Vlasov in circular cylindrical shell^[5], the basic equation of the cylindrical shell on the elastic foundation and that of the circular plates on the elastic foundation are directly derived.Under the arbitrary loads and boundary conditions, the general bending problem of the conical shell on the elastic foundation is reduced to find the displacement function U(s,θ).The general solution of the eight-order differential equation is obtained in series form. For the symmetric bending deformation of the conical shell on the elastic foundation, which has been widely usedinpractice,the detailed numerical results and boundary influence coefficients for edge loads have been obtained. These results have important meaning in analysis of conical shell combination construction on the elastic foundation,and provide a valuable judgement for the numerical solution accuracy of some of the same type of the existing problem.     

用Green函数求解扁球壳受环状线载和线偶作用的问题

本文从复变量形式的扁壳基本方程出发,通过建立复Green函数导出了在环状线载和线偶作用下扁球壳的位移和内力分布,通过积分可以求得轴对称的表面受变化分布载荷情况的解答,本文方法还可求得圆饭、圆柱壳等问题的解答,而且适用于各种轴对称边界条件。     

一类偏微分方程组的一般解及其在壳体理论中的应用

本文给出了构造一类偏微分方程组一般解的方法,用这种方法构造的一般解是完备的。最后,利用这个方法构造了壳体中的柱壳平衡方程和锥壳运动方程的一般解。     

旋转薄壁圆柱壳振型进动的非线性振动特性

选取在工程上常用的悬臂旋转薄壁圆柱壳为研究模型,首先推导出考虑阻尼的振型进动因子,然后根据Donnell's简化壳理论建立考虑科氏力,阻尼与几何大变形的非线性波动方程,采用Galerkin方法对波动方程进行离散化,得到模态坐标中相互耦合的三阶非线性微分方程组.应用Runge-Kutta法求解获得非线性幅频特性曲线,分析了不同模态组合下系统主模态(m=1,n=6,k=1)的共振响应.应用谐波平衡法对系统三阶非线性微分方程组解析分析,与数值解比较验证了解析解的正确性和有效性.最后分析了动力系统的运动稳定性.结果表明,节径数n和频率倍数k对于主模态共振响应的影响很小,而轴向半波数m对主共振的影响则相对较大,因此只需选取相邻的两个轴向模态(M=2)即可较为简洁,准确的描述主共振响应;谐波平衡法可以很好的解决三阶微分方程组的非线性问题,并且能够达到较为满意的精度.     

轴对称激励下力变厚度圆柱壳与液体的耦联振动

利用文献[1]的结果，研究了液体中的变厚度圆柱壳在轴对称激励力下的强迫振动问题，从基本微分方程出发，导出了问题的级数解，最后通过实例计算说明了该方法的应用。     

厚壁圆柱壳的弹性静力分析

本文研究了考虑横向剪切影响的弹性厚壁圆柱壳的静力问题。利用变分原理得到平衡微分方程组和相应的边界条件。将平衡方程组归并成一个高阶微分方程,用数值法求出它的特征根,得到问题的解。     

Exact piezothermoelastic solution of simply-supported orthotropic circular cylindrical panel in cylindrical bending

Summary This work presents an exact piezothermoelastic solution of infinitely long, simply supported, cylindrically orthotropic, piezoelectric, radially polarised, circular cylindrical shell panel in cylindrical bending under thermal and electrostatic excitation. The general solution of the governing differential equations is obtained by separation of variables. The displacements, electric potential and temperature are expanded in appropriate Fourier series in the circumferential coordinate to satisfy the boundary conditions at the simply-supported longitudinal edges. The governing equations reduce to Euler-Cauchy type of ordinary differential equations. Their general solution involves six constants for each Fourier component. These are solved from the algebraic equations obtained by satisfying the boundary conditions at the lateral surfaces. The solution of the inverse problem of inferring the applied temperature field from the given measured distribution of electrical potential difference between the lateral surfaces of the shell has also been presented. Numerical results are presented for typical thermal and electrostatic loadings for various values of radius to thickness ratio.     

An analytical study of the non-linear vibrations of cylindrical shells

The primary resonance response of simply supported circular cylindrical shells is investigated using the perturbation method. Donnell's non-linear shallow-shell theory is used to derive the governing partial differential equations of motion. The Galerkin technique is then employed to transform the equations of motion into a set of temporal ordinary differential equations. Considering only the primary resonance case, the method of multiple scales is used to study the periodic solutions and their stability. The necessary and sufficient conditions for appearance of the so-called companion mode are also discussed. To this end, a range of the possible multi-mode solution is obtained for response and excitation amplitudes and also excitation frequency as a function of damping, geometry and material properties of the shell. Parametric studies are performed to illustrate the effect of different values of thickness, length and material composition on the possibility of the companion mode participation in primary resonance response.     

Thermal buckling analysis of functionally graded cylindrical shells

Thermal buckling behavior of cylindrical shell made of functionally graded material(FGM) is studied. The material constituents are composed of ceramic and metal.The material properties across the shell thickness are assumed to be graded according to a simple power law distribution in terms of the volume fraction rule of mixtures. Based on the Donnell shell theory, a system of dimensionless partial differential equations of buckling in terms of displacement components is derived. The method of separation of variables is used to transform the governing equations to ordinary differential equations(ODEs). A shooting method is used to search for the numerical solutions of the differential equations under two types of boundary conditions. Effects of the power law index, the dimensionless geometrical parameters, and the temperature ratio on the critical buckling temperature are discussed in detail.     

Axisymmetric Interaction Problem for a Sphere Pulsating Inside an Elastic Cylindrical Shell Filled with and Immersed Into a Liquid

An interaction problem is formulated for a spherical body oscillating in a prescribed manner inside a thin elastic cylindrical shell filled with a perfect compressible liquid and submerged in a dissimilar infinite perfect compressible liquid. The geometrical center of the sphere is on the cylinder axis. The solution is based on the possibility of representing the partial solutions of the Helmholtz equations written in cylindrical coordinates for both media in terms of the partial solutions written in spherical coordinates, and vice versa. Satisfying the boundary conditions on the sphere and shell surfaces results in an infinite system of linear algebraic equations. This system is used to determine the coefficients of the Fourier-series expansions of the velocity potentials in terms of Legendre polynomials. The hydrodynamic characteristics of both liquids and the shell deflections are determined. The results obtained are compared with those for a sphere oscillating on the axis of an elastic cylindrical shell filled with a compressible liquid (the ambient medium being neglected).     

An Exact Analysis for Free Vibration of a Composite Shell Structure-Hermetic Capsule

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The Subharmonic Bifurcation of a Viscoelastic Circular Cylindrical Shell

In this paper the nonlinear dynamic behavior of a viscoelastic circular cylindrical shell under a harmonic excitation applied at both ends is studied. The modified Flugge partial differential equations of motion are reduced to a system of finite degrees of freedom using the Galerkin method. The equations are solved by the Liapunov–Schmidt reduction procedure. In order to study 1/2 and 1/4 subharmonic parametric resonance of the shell, the transition sets in parameter plane and bifurcation diagrams are plotted for a number of situations. Results indicate that, for certain static loads, the shell may display jumps due to the presence of dynamic periodic load with small amplitude. Additionally, different physical situations are identified in which periodic oscillating phenomena can be observed, and where 1/4 subharmonic parametric resonance is simpler than the 1/2-one.     

含裂纹损伤充液圆柱壳的振动响应求解方法

薄壁圆柱壳流体冲击振动响应是一个复杂的流固耦合(FSI)动力学问题,对于薄壳状态监测与缺陷识别具有重要意义。基于Flügge壳体应力理论,得到壳体运动的高阶偏微分方程组(PDE),利用波传播方法获得圆柱壳系统振动响应。将壳体周围流体定义为理想声学介质,通过亥姆霍兹方程描述声压场,得到流固耦合条件下的薄壁圆柱壳受迫振动响应演变规律。针对薄壳裂纹损伤识别问题,基于断裂力学理论建立局部柔度矩阵,结合呼吸型线弹簧模型(LSM),构造裂纹附近应力及位移连续条件,获得含裂纹损伤充液圆柱壳的振动响应,进而给出一种基于振动能量流的裂纹损伤识别方法。研究结果表明：充液圆柱壳耦合系统在非线性激励下,位移响应在沿轴向、周向和径向的传播特性差异明显;裂纹的存在会导致结构局部柔度的降低和耦合系统固有频率下降;归一化输入功率流能够有效地对充液圆柱壳耦合系统进行结构裂纹损伤识别。研究结果可为充液薄壳振动响应方面的研究提供有益参考,也可为流固耦合条件下的结构裂纹损伤识别方面提供技术支持。     

Interaction Between an Oscillating Sphere and a Thin Elastic Cylindrical Shell Filled with a Compressible Liquid. Internal Axisymmetric Problem

The problem on the interaction between a spherical body that oscillates in a prescribed manner and a thin elastic cylindrical shell filled with an ideal compressible liquid is formulated. It is assumed that the geometrical center of the sphere is located on the cylinder axis. The problem is solved based on the possibility of representing a partial solution of the Helmholtz equation written in cylindrical coordinates in terms of partial solutions in spherical coordinates, and vice versa. By satisfying the boundary conditions on the surfaces of the sphere and the shell, we obtain an infinite system of linear algebraic equations to determine the coefficients of expansion of the liquid-velocity potential into a Fourier series in terms of Legendre polynomials. The hydrodynamic characteristics of the liquid filling the cylindrical shell are determined and compared with the cases where a sphere oscillates in an infinite liquid and in a rigid cylindrical vessel     

Substructure computational algorithm for exact analytic method

In[1],the exact analytic method for the solution of differential equation with variablecoefficients was suggested and an analytic expression of solution was given by initialparameter algorithm.But to some problems such as the bending,free vibration andbuckling of nonhomogeneous long cylinders,it is difficult to obtain their solutions by theinitial parameter algorithm on computer.In this paper,the substructure computationalalgorithm for the exact analytic method is presented through the bending of non-homogeneous long cylindrical shell.This substructure algorithm can be applied to solve theproblems which can not be calculated by the initial parameter algorithm on computer.Finally,the problems can be reduced to solving a low order system of algebraic equationslike the initial parameter algorithm.Numerical examples are given and compared with theinitial para-algorithm at the end of the paper,which confirms the correcthess of thesubstructure computational algorithm.     

An exact solution for the steady-state thermoelastic response of functionally graded orthotropic cylindrical shells

We analyze the steady-state response of a functionally graded thick cylindrical shell subjected to thermal and mechanical loads. The functionally graded shell is simply supported at the edges and it is assumed to have an arbitrary variation of material properties in the radial direction. The three-dimensional steady-state heat conduction and thermoelasticity equations, simplified to the case of generalized plane strain deformations in the axial direction, are solved analytically. Suitable temperature and displacement functions that identically satisfy the boundary conditions at the simply supported edges are used to reduce the thermoelastic equilibrium equations to a set of coupled ordinary differential equations with variable coefficients, which are then solved by the power series method. In the present formulation, the cylindrical shell is assumed to be made of an orthotropic material, although the analytical solution is also valid for isotropic materials. Results are presented for two-constituent isotropic and fiber-reinforced functionally graded shells that have a smooth variation of material volume fractions, and/or in-plane fiber orientations, through the radial direction. The cylindrical shells are also analyzed using the Flügge and the Donnell shell theories. Displacements and stresses from the shell theories are compared with the three-dimensional exact solution to delineate the effects of transverse shear deformation, shell thickness and angular span.     

Nonlinear dynamic responses of sandwich functionally graded porous cylindrical shells embedded in elastic media under 1:1 internal resonance

In this article, the nonlinear dynamic responses of sandwich functionally graded(FG) porous cylindrical shell embedded in elastic media are investigated. The shell studied here consists of three layers, of which the outer and inner skins are made of solid metal, while the core is FG porous metal foam. Partial differential equations are derived by utilizing the improved Donnell's nonlinear shell theory and Hamilton's principle. Afterwards, the Galerkin method is used to transform the governing equations into nonlinear ordinary differential equations, and an approximate analytical solution is obtained by using the multiple scales method. The effects of various system parameters,specifically, the radial load, core thickness, foam type, foam coefficient, structure damping,and Winkler-Pasternak foundation parameters on nonlinear internal resonance of the sandwich FG porous thin shells are evaluated.