Melnikov's method for chaos of a two-dimensional thin panel in subsonic flow with external excitation

In this brief communication, Melnikov's method is adopted to study the chaotic behaviors of a two-dimensional thin panel subjected to subsonic flow and external excitation. The nonlinear governing equations of the subsonic panel system are reduced to a series of ordinary differential equations by using Galerkin method. The critical parameters for chaos are obtained. It is found that the critical parameters obtained by the theoretical analysis are in agreement with the numerical simulations. The method suggested in this paper can also be extended for other fluid-structure dynamic systems, such as the fluid-conveying system.     

Postbuckling of pressure-loaded shear deformable laminated cylindrical panels

IntroductionInrecentyears,fiber_reinforcedcompositelaminatedpanelshavebeenwidelyusedintheaerospace,marine ,automobileandotherengineeringindustries .Theproblemofbucklingandpostbucklingofcylindricalpanelsunderaxialcompressionortorsionhasbeenextensivelystudied .Incontrast,theliteratureoncylindricalpanelsunderpressureloadingisrelativelyspares.Thesestudiesincludealinearbucklinganalysis (Singeretal.[1]) ,anonlinearbucklinganalysi(YamadaandCroll[2 ]) ,anelastoplasticbucklinganalysisusingreducedstif…     

Analytical solution for bending problem of moderately thick composite annular sector plates with general boundary conditions and loadings using multi-term extended Kantorovich method

An analytical solution for bending of composite sector plates is presented using multi-term extended Kantorovich method (MTEKM). The governing equations are derived using the displacement field of the first-order shear deformation theory and converted into two sets of coupled ordinary differential equations (ODEs) utilizing MTEKM. Next, an analytical iterative procedure is presented for solving the derived sets of ODEs based on state-space method. Numerous examples are studied by the present method, and as special cases, solid sector and rectangular plates are also investigated. Next, the results obtained by the present method are compared to those of finite element method and other results available in the literature. It is found that the present method has a high convergence rate as well as good accuracy in all cases.     

A variational analytical approach to time dependent flow of oldroyd B fluid in circular tube

In the present investigation the time dependent flow of an Oldroyd fluid B in a horizontal cylindrical pipe is stuided by the variational analytical approach developed by author. The time dependent problem is mathematically reduced to a partial differential equation of third order. Using the improved variational approach due to Kantorovich the partial differential equation can be reduced to a system of ordinary differential equations for different approximations. The ordinary differential equations are solved by the method of the Laplace transform which is led to an analytical form of the solutions. Project supported by TWAS and Chinese Academy of Sciences and the National Science Foundation of China     

具脱层轴对称层合圆柱壳的振动模态分析

对具环向贯穿脱层的轴对称层合圆柱壳进行振动模态分析.首先,采用Heaviside阶梯函数,构造了一种适合于脱层壳的位移模式.通过对脱层壳的能量分析,应用瑞利--里兹法后,得到用时间函数表示的系统振动控制方程,然后对其求解,得到脱层壳模态分析的特征方程式.算例中,讨论了不同的脱层位置、脱层大小和脱层深度对脱层壳振动模态的影响.     

Free vibration analysis of moderately thick functionally graded plates on elastic foundation using the extended Kantorovich method

Free vibration analysis of moderately thick rectangular FG plates on elastic foundation with various combinations of simply supported and clamped boundary conditions are studied. Winkler model is considered to describe the reaction of elastic foundation on the plate. Governing equations of motion are obtained based on the Mindlin plate theory. A semi-analytical solution is presented for the governing equations using the extended Kantorovich method together with infinite power series solution. Results are compared and validated with available results in the literature. Effects of elastic foundation, boundary conditions, material, and geometrical parameters on natural frequencies of the FG plates are investigated.     

Stability and vibration of shear deformable plates––first order and higher order analyses

This work presents the highly accurate numerical calculation of the natural frequencies and buckling loads for thick elastic rectangular plates with various combinations of boundary conditions. The Reissener–Mindlin first order shear deformation plate theory and the higher order shear deformation plate theory of Reddy have been applied to the plate’s analysis. The governing equations and the boundary conditions are derived using the dynamic version of the principle of minimum of the total energy. The solution is obtained by the extended Kantorovich method. This approach is combined with the exact element method for the vibration and stability analysis of compressed members, which provides for the derivation of the exact dynamic stiffness matrix including the effect of in-plane and inertia forces. The large number of numerical examples demonstrates the applicability and versatility of the present method. The results obtained by both shear deformation theories are compared with those obtained by the classical thin plate’s theory and with published results. Many new results are given too.     

Three-dimensional vibration analysis of a thermoelastic cylindrical panel with voids

In this paper, based on three-dimensional linear generalized thermoelasticity, an exact analysis of free vibration of a simply supported homogeneous isotropic, thermally conducting, cylindrical panel with voids initially at uniform temperature and undeformed state has been presented. Three displacement potential functions are introduced for solving the equations of motion, heat conduction and volume fraction field. The purely transverse wave gets decoupled from rest of motion and is not affected by thermal and volume fraction (voids) fields. After expanding the displacement potentials, volume fraction and temperature functions with orthogonal series, the equations of the considered vibration problem are reduced to five-second order coupled ordinary differential equations whose formal solution can be expressed by using Bessel functions with complex arguments. The corresponding results for thermoelastic panel without voids, elastic panel with and without voids have been deduced as special cases from the present analysis. In order to illustrate the analytical results, the numerical solutions of various relations and equations have been obtained to compute the lowest frequency as function of different cylindrical panel parameters. The computer simulated results have been presented graphically.     

圆柱扁壳的非线性动力响应的样条加权残值分析

本文对圆柱扁壳的非线性动力响应的问题,利用加相对加权残值法的子域方法将唐奈尔方程转化成时间因子作为示知函数的控制方程,运用样条配点法发成若干段,且保证在每时间段均连续将算例结果与有关文献比较的同时,还运用ＯＤＥ ＳＯＶＶＥＲ－ＣＯＬＳＹＳ程序作数值比较。     

Transient response analysis of one-dimensional distributed parameter systems

A semi-analytic method is presented for the analysis of transient response of one-dimensional distributed parameter systems. Replacing time differentials by finite difference, the governing partial differential equations are reduced to difference–differential equations. The solutions of derived ordinary differential equations are given in exact and closed form by distributed transfer function method. Complex systems that contain many one-dimensional sub-systems are also studied. Numerical results show that the efficiency and accuracy of the method are excellent.     

Postbuckling analysis of axially loaded functionally graded cylindrical panels in thermal environments

A postbuckling analysis is presented for a functionally graded cylindrical panel of finite length subjected to axial compression in thermal environments. Material properties are assumed to be temperature dependent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. The governing equations of a functionally graded cylindrical panel are based on Reddy’s higher order shear deformation shell theory with a von Kármán–Donnell-type of kinematic nonlinearity and including thermal effects. Two cases of the in-plane boundary conditions are considered. The nonlinear prebuckling deformations and initial geometric imperfections of the panel are both taken into account. A boundary layer theory of shell buckling, which includes the effects of nonlinear prebuckling deformations, large deflections in the postbuckling range, and initial geometric imperfections of the shell, is extended to the case of functionally graded cylindrical panels under axial compression. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling behavior of axially loaded, perfect and imperfect, functional graded cylindrical panels with two constituent materials and under different sets of thermal environments. The influences played by temperature rise, volume fraction distributions, the character of in-plane boundary conditions, transverse shear deformation, panel geometric parameters, as well as initial geometric imperfections are studied.     

Elastic nonlinear stability analysis of thin rectangular plates through a semi-analytical approach

A semi-analytical approach to the elastic nonlinear stability analysis of rectangular plates is developed. Arbitrary boundary conditions and general out-of-plane and in-plane loads are considered. The geometrically nonlinear formulation for the elastic rectangular plate is derived using the thin plate theory with the nonlinear von Kármán strains and the variational multi-term extended Kantorovich method. Emphasis is placed on the effect of destabilizing loads and on the derivation of the solution methodologies required for tracking a highly nonlinear equilibrium path, namely: parameter continuation and arc-length continuation procedures. These procedures, which are commonly used for the solution of discretized structural systems governed by nonlinear algebraic equations, are augmented and generalized for the direct application to the PDE. The boundary value problem that results from the arc-length continuation scheme and consists of coupled differential, integral, and algebraic equations is re-formulated in a form that allows the use of standard numerical BVP solvers. The performance of the continuation procedures and the convergence of the multi-term extended Kantorovich method are examined through the solution of the two-dimensional Bratu–Gelfand benchmark problem. The applicability of the proposed approach to the tracking of the nonlinear equilibrium path in the post-buckling range is demonstrated through numerical examples of rectangular plates with various boundary conditions.     

Free vibration of layered cylindrical shells filled with fluid

The vibration of the layered cylindrical shells filled with a quiescent, incompressible, and inviscid fluid is analyzed. The governing equations of the cylindrical shells are derived by Love’s approximation. The solutions of the displacement functions are assumed in a separable form to obtain a system of coupled differential equations in terms of the displacement functions. The displacement functions are approximated by Bickley-type splines. A generalized eigenvalue problem is obtained and solved numerically for the frequency parameter and an associated eigenvector of the spline coefficients. Two layered shells with three different types of materials under clamped-clamped (C-C) and simply supported (S-S) boundary conditions are considered. The variations of the frequency parameter with respect to the relative layer thickness, the length-to-radius ratio, the length-to-thickness ratio, and the circumferential node number are analyzed.     

范式理论与平均法的等价性分析

分析了范式理论与平均法的等价性。得到的结论是：对含有一对纯虚根的二维非线性系统，使用两种方法得到的结果是等价的，并提供了两个算例来证实其结论的正确性。虽然分析是针对一类二维非线性系统，但其结论同样适合于高维非线性系统。     

Analytical test functions for free vibration analysis of rotating non-homogeneous Timoshenko beams

In this paper, the governing equations for free vibration of a non-homogeneous rotating Timoshenko beam, having uniform cross-section, is studied using an inverse problem approach, for both cantilever and pinned-free boundary conditions. The bending displacement and the rotation due to bending are assumed to be simple polynomials which satisfy all four boundary conditions. It is found that for certain polynomial variations of the material mass density, elastic modulus and shear modulus, along the length of the beam, the assumed polynomials serve as simple closed form solutions to the coupled second order governing differential equations with variable coefficients. It is found that there are an infinite number of analytical polynomial functions possible for material mass density, shear modulus and elastic modulus distributions, which share the same frequency and mode shape for a particular mode. The derived results are intended to serve as benchmark solutions for testing approximate or numerical methods used for the vibration analysis of rotating non-homogeneous Timoshenko beams.     

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     

Nonlinear thermo-elastic buckling characteristics of cross-ply laminated joined conical–cylindrical shells

Here, the nonlinear thermo-elastic buckling/post-buckling characteristics of laminated circular conical–cylindrical/conical–cylindrical–conical joined shells subjected to uniform temperature rise are studied employing semi-analytical finite element approach. The nonlinear governing equations, considering geometric nonlinearity based on von Karman’s assumption for moderately large deformation, are solved using Newton–Raphson iteration procedure coupled with displacement control method to trace the pre-buckling/post-buckling equilibrium path. The presence of asymmetric perturbation in the form of small magnitude load spatially proportional to the linear buckling mode shape is assumed to initiate the bifurcation of the shell deformation. The study is carried out to highlight the influences of semi-cone angle, material properties and number of circumferential waves on the nonlinear thermo-elastic response of the different joined shell systems.     

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.     

多柔体系统数值分析的模型降噪方法

多柔体系统的动力学方程通常是一组刚性微分方程, 目前普遍采用的刚性微分方程数值解法主要通过数值阻尼滤除系统响应中的高频分量, 其求解效率难以令人满意. 为了降低多柔体系统动力学方程的刚性, 从而可采用ODE45等常规微分方程求解器进行求解, 研究了在建模过程中滤除高频振荡分量的方法. 在以当前时刻为起点的短时间内对柔性体的应力进行均匀化, 用均匀化后的应力计算柔性体的变形虚功率, 由此得到的系统动力学方程的解中不含过高频率的弹性振动, 并且可以通过调节均匀化时间区间的长度参数控制滤波的范围. 数值算例表明: 这种模型降噪方法的计算效率和精度均不低于刚性微分方程求解器, 并且在刚性微分方程求解器失效的情况下模型降噪方法仍有良好的精度和效率. 本文所提的模型降噪方法可成为求解多柔体系统动力学方程的新途径.