均布载荷作用下波纹扁壳的非线性振动

应用轴对称旋转扁壳的非线性大挠度动力学方程,研究了波纹扁壳在均布载荷作用下的非线性受迫振动问题．采用格林函数方法,将扁壳的非线性偏微分方程组化为非线性积分微分方程组．再使用展开法求出格林函数,即将格林函数展开为特征函数的级数形式,积分微分方程就成为具有退化核的形式,从而容易得到关于时间的非线性常微分方程组．针对单模态振形,得到了谐和激励作用下的幅频响应．作为算例,研究了正弦波纹扁球壳的非线性受迫振动现象．该文的解答可供波纹壳的设计参考．     

Nonlinear free dynamic response of laminated compressible cylindrical shell panels

The governing equations for a free dynamic response of a symmetrically laminated composite shell are used to analyze a nonlinear differential panel. The FEM and the Lindstedt–Poincare perturbation technique are invoked to construct a uniform asymptotic expansion of the solution to a nonlinear differential equation ofmotion. A comparison between numerical and finite-element methods for analyzing a symmetrically laminated graphite/epoxy shell panel is performed to show that the nonlinearities are of hardening type and are more repeated for smaller opening angles. It is also shown that large-amplitude motions are dominated by lower modes.     

Shear and Flexural Buckling Modes of a Spherical Sandwich Shell in a Centrosymmetric Temperature Field Inhomogeneous across the Thickness

Problems on buckling modes (BMs) are considered for a spherical sandwich shell with thin isotropic external layers and a transversely soft core of arbitrary thickness in a centrosymmetric temperature field inhomogeneous across the shell thickness. For their statement, the two-dimensional equations of the theory of moderate bending of thin Kirchhoff–Love shells are used for the external layers, with regard for their interaction with the core; for the core, maximum simplified geometrically nonlinear equations of thermoelasticity theory, in which a minimum number of nonlinear summands is retained to correctly describe its pure shear BM, are utilized. An exact analytical solution to the problem on initial centrosymmetric deformation of the shell is found, assuming that the temperature increments in the external layers are constant across their thickness. It is shown that the three-dimensional equations for the core, linearized in the neighborhood of the solution, can be integrated along the radial coordinate and reduced to two two-dimensional differential equations, which supplement the six equations that describe the neutral equilibrium of the external layers. It is established that the system of eight differential equations of stability, upon introduction of new unknowns in the form of scalar and vortical potentials, splits into two uncoupled sets of equations. The first of them has two kinds of solutions, by which the pure shear BM is described at an identical value of the parameter of critical temperature. The second system describes a mixed flexural BM, whose realization, at definite combinations of determining parameters of the shell and over wide ranges of their variation, is possible for critical parameters of temperature by orders of magnitude exceeding the similar parameter of shear BM.     

Creep stability of glass-reinforced plastic cylindrical shells under long-term external pressure

The behavior of a glass-reinforced plastic cylindrical shell under long-term hydrostatic pressure is investigated using the geometrically nonlinear equations of Timoshenko-type shell theory, which permit transverse shear strains to be taken into account. A system of nonlinear differential equations for describing the variation of the state of the shell with time under load is obtained and solved on a BÉSM-3M computer using a program written in Algol-60 and a "Signal" translator. Values of the critical time are obtained for various load levels.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 1, pp. 81–85, January–February, 1970.     

一类非线性发展方程组的定解问题

将不可压缩的广义neo-Hookean材料组成的超弹性圆柱壳径向对称运动的数学模型归结为一类非线性发展方程组的初边值问题.利用材料的不可压缩条件和边界条件求得了描述圆柱壳内表面径向运动的二阶非线性常微分方程.给出了微分方程的周期解(即圆柱壳的内表面产生非线性周期振动)的存在条件,讨论了材料参数和结构参数对方程的周期解的影响,并给出了相应的数值模拟.     

均布载荷作用下变厚度开顶扁球壳的非线性稳定问题

本文首先应用逐步加载法将具有硬中心的开顶扁球壳在均布载荷作用下的非线性微分方程组线性化,然后利用样条配点法解线性微分方程组,得到了临界载荷的数值．     

Static computation of a laminated nonuniform inclined shell in a temperature field

The finite difference method is used to obtain a solution of a nonlinear static problem for a laminated inclined rectangular shell in a plane acted on by a force load and a temperature field. The approximating system of nonlinear equations is obtained using an approximation of the equation of variations or systems of differential equations.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 27, 1988, pp. 86–89.     

双参数弹性地基上圆底扁球壳的非线性弯曲

本文从Krmn型非线性基本微分方程出发,提出了将修正迭代法和伽辽金法联合应用,分析了Pasternak弹性地基上周边固定凹圆底扁球壳在均匀压力作用下的非线性弯曲问题,给出了荷载与挠度间的数学表达式,其所得结果与已有文献结果吻合较好,且简明、计算量小.     

扁锥面网壳非线性动力分岔与混沌运动

对曲面为正三角形网格的3向扁锥面单层网壳,用拟壳法建立了轴对称非线性动力学方程．在几何非线性范围内给出了协调方程．网壳在周边固定条件下,通过Galerkin作用得到一个含2次、3次的非线性微分方程,通过求Floquet指数讨论了分岔问题．为了研究混沌运动,对一类非线性动力系统的自由振动方程进行了求解,继之给出了单层扁锥面网壳非线性自由振动微分方程的准确解,通过求Melnikov函数,给出了发生混沌的临界条件,通过数值仿真也证实了混沌运动的存在．     

Existence of solutions for a system of equations of nonlinear elastic materials

The question of the existence of solutions for the system of nonlinear partial differential equations governing the arbitrarily heated and prestressed nonhomogeneous shell is considered. We give the sufficient conditions for the existence of at least one solution by way of a fixed point theorem in an appropriate function space.     

The dynamic stability and nonlinear vibration analysis of stiffened functionally graded cylindrical shells

A theoretical model is developed to study the dynamic stability and nonlinear vibrations of the stiffened functionally graded (FG) cylindrical shell in thermal environment. Von Kármán nonlinear theory, first-order shear deformation theory, smearing stiffener approach and Bolotin method are used to model stiffened FG cylindrical shells. Galerkin method and modal analysis technique is utilized to obtain the discrete nonlinear ordinary differential equations. Based on the static condensation method, a reduction model is presented. The effects of thermal environment, stiffeners number, material characteristics on the dynamic stability, transient responses and primary resonance responses are examined.     

Dynamic stability of glass-reinforced plastic shells

The problem of the dynamic stability of circular-cylindrical glass-reinforced plastic shells subjected to external transverse pressure is examined in the nonlinear formulation. After the Lagrange equations have been constructed, the problem reduces to the integration of a system of ordinary differential equations with aperiodic coefficients. The integration has been carried out numerically on a computer for various loading rates and shell parameters. Analogous problems for isotropic metal shells were examined in [1–4]. A review of the subject may be found in [5].Mekhanika Polimerov, Vol. 4, No. 1, pp. 109–115, 1968     

带有集中激励的一类非线性边值问题的奇摄动

本文综合利用近代分析和奇异摄动方法，讨论了一类具有集中激励的非线性向量微分方程组初值总是 的渐近性质，构造了这类问题的一致有效渐近解。并利用这个方法求解几何参数的扁球壳在集中力作用下的非线性稳定性问题，得到了较好的结果。     

Efficient hybrid method for solving special type of nonlinear partial differential equations

In this article, an efficient hybrid method has been developed for solving some special type of nonlinear partial differential equations. Hybrid method is based on tanh–coth method, quasilinearization technique and Haar wavelet method. Nonlinear partial differential equations have been converted into a nonlinear ordinary differential equation by choosing some suitable variable transformations. Quasilinearization technique is used to linearize the nonlinear ordinary differential equation and then the Haar wavelet method is applied to linearized ordinary differential equation. A tanh–coth method has been used to obtain the exact solutions of nonlinear ordinary differential equations. It is easier to handle nonlinear ordinary differential equations in comparison to nonlinear partial differential equations. A distinct feature of the proposed method is their simple applicability in a variety of two‐ and three‐dimensional nonlinear partial differential equations. Numerical examples show better accuracy of the proposed method as compared with the methods described in past. Error analysis and stability of the proposed method have been discussed.     

Universal theory of dynamical chaos in nonlinear dissipative systems of differential equations

The article presents a new universal theory of dynamical chaos in nonlinear dissipative systems of differential equations, including autonomous and nonautonomous ordinary differential equations (ODE), partial differential equations, and delay differential equations. The theory relies on four remarkable results: Feigenbaum’s period doubling theory for cycles of one-dimensional unimodal maps, Sharkovskii’s theory of birth of cycles of arbitrary period up to cycle of period three in one-dimensional unimodal maps, Magnitskii’s theory of rotor singular point in two-dimensional nonautonomous ODE systems, acting as a bridge between one-dimensional maps and differential equations, and Magnitskii’s theory of homoclinic bifurcation cascade that follows the Sharkovskii cascade. All the theoretical propositions are rigorously proved and illustrated with numerous analytical examples and numerical computations, which are presented for all classical chaotic nonlinear dissipative systems of differential equations.     

Symmetry analysis and some exact solutions of cylindrically symmetric null fields in general relativity

The symmetry reduction method based on the Fréchet derivative of the differential operators is applied to investigate symmetries of the Field equations in general relativity corresponding to cylindrically symmetric space–time, that is a coupled system of nonlinear partial differential equations of second order. More specifically, this technique yields invariant transformation that reduce the given system of partial differential equations to a system of nonlinear ordinary differential equations. Some of the reduced systems are further studied for exact solutions.     

Simplified equations and solutions for the free vibration of an orthotropic oval cylindrical shell with variable thickness

Based on the framework of the Flügge's shell theory, the transfer matrix approach and the Romberg integration method, this paper presents the vibration behavior of an isotropic and orthotropic oval cylindrical shell with parabolically varying thickness along its circumference. The governing equations of motion of the shell, which have variable coefficients are formulated and solved. The analysis is formulated to overcome the mathematical difficulties related to mode coupling, which comes from variable curvature and thickness of shell. The vibration equations of the shell are reduced to eight first‐order differential equations in the circumferential coordinate and by using the transfer matrix of the shell, these equations can be written in a matrix differential equation. The proposed model is adopted to get the vibration frequencies and the corresponding mode shapes for the symmetrical and antisymmetrical modes of vibration. The sensitivity of the frequency parameters and the bending deformations to the shell geometry, ovality parameter, thickness ratio, and orthotropic parameters corresponding to different type modes of vibration is investigated. Copyright © 2011 John Wiley & Sons, Ltd.     

Ermakov–Pinney and Emden–Fowler Equations: New Solutions from Novel Bäcklund Transformations

We study the class of nonlinear ordinary differential equations y″ y = F(z, y²), where F is a smooth function. Various ordinary differential equations with a well-known importance for applications belong to this class of nonlinear ordinary differential equations. Indeed, the Emden–Fowler equation, the Ermakov–Pinney equation, and the generalized Ermakov equations are among them. We construct Bäcklund transformations and auto-Bäcklund transformations: starting from a trivial solution, these last transformations induce the construction of a ladder of new solutions admitted by the given differential equations. Notably, the highly nonlinear structure of this class of nonlinear ordinary differential equations implies that numerical methods are very difficult to apply.     

A novel numerical solution strategy for solving nonlinear free and forced vibration problems of cylindrical shells

Nonlinear vibration analysis of circular cylindrical shells has received considerable attention from researchers for many decades. Analytical approaches developed to solve such problem, even not involved simplifying assumptions, are still far from sufficiency, and an efficient numerical scheme capable of solving the problem is worthy of development. The present article aims at devising a novel numerical solution strategy to describe the nonlinear free and forced vibrations of cylindrical shells. For this purpose, the energy functional of the structure is derived based on the first-order shear deformation theory and the von–Kármán geometric nonlinearity. The governing equations are discretized employing the generalized differential quadrature (GDQ) method and periodic differential operators along axial and circumferential directions, respectively. Then, based on Hamilton's principle and by the use of variational differential quadrature (VDQ) method, the discretized nonlinear governing equations are obtained. Finally, a time periodic discretization is performed and the frequency response of the cylindrical shell with different boundary conditions is determined by applying the pseudo-arc length continuation method. After revealing the efficiency and accuracy of the proposed numerical approach, comprehensive results are presented to study the influences of the model parameters such as thickness-to-radius, length-to-radius ratios and boundary conditions on the nonlinear vibration behavior of the cylindrical shells. The results indicate that variation of fundamental vibrational mode shape significantly affects frequency response curves of cylindrical shells.