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.     

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.     

Behavior of different ansätze in the generalized projection operator method

Using the recently reported generalized projection operator method for the nonlinear Schrödinger equation, we derive the generalized pulse parameters equations for ansätze like hyperbolic secant and raised cosine functions. In general, each choice of the phase factor θ in the projection operator gives a different set of ordinary differential equations. For θ = 0 or θ = π/2, the corresponding projection operator scheme is equivalent to the Lagrangian variation method or the bare approximation of the collective variable theory. We prove that because of the inherent symmetric property between the pulse parameters of a Gaussian ansätz results the same set of pulse parameters equations for any value of the generalized projection operator parameter θ. Finally we prove that after the substitution of the ansätze function, the Lagrange function simplifies to the same functional form irrespective of the ansätze used because of a special property shared by all the anätze chosen in this work.     

Method of solving nonlinear dynamic problems in viscoelasticity

The authors propose a method of solving Volterra's system of nonlinear integrodifferential equations. This method is based on the use of a power series. As an illustration, the authors consider the vibration of flexible viscoelastic cylindrical shells under impulsive and periodic loads.Institute of Cybernetics and Computer Center, Academy of Sciences of the Uzbek SSR, Tashkent. Translated from Mekhanika Polimerov, Vol. 9, No. 3, pp. 554–558, May–June, 1973.     

Formulation and evaluation of a finite element model for the linear analysis of stiffened composite cylindrical panels

A finite element model for linear static and free vibration analysis of composite cylindrical panels with composite stiffeners is presented. The proposed model is based on a cylindrical shell finite element, which uses a first-roder shear deformation theory. The stiffeners are curved beam elements based on Timoshenko and Saint-Venant assumptions for bending and torsion respectively. The two elements are developed in a cylindrical coordinate system and their stiffness matrices result from a hybrid-mixed formulation where the element assumed stress field is such that exact equilibrium equations are satisfied. The elements are free of membrane and shear locking with correct satisfaction of rigid body motions. Several examples dealing with stiffened isotropic and laminated plates and shells with eccentric as well as concentric stiffeners are analyzed showing the validity of the models.     

Analytical solution for cylindrical thin shells with normally intersecting nozzles due to external moments on the ends of shells

The stress analysis based on the theory of a thin shell is carried out for cylindrical shells with normally intersecting nozzles subjected to external moment loads on the ends of shells with a large diameter ratio(ρ ₀ «0. 8). Instead of the Donnell shallow shell equation, the modified Morley equation, which is applicable toρ ₀(R/T)^1/2 »1, is used for the analysis of the shell with cutout. The solution in terms of displacement function for the nozzle with a nonplanar end is based on the Goldenveizer equation. The boundary forces and displacements at the intersection are all transformed from Gaussian coordinates (α, β) on the shell, or Gaussian coordinates (ζ, θ) on the nozzle into three-di-mensional cylindrical coordinates(ρ,θ, z). Their expressions on the intersecting curve are periodic functions ofθ and expanded in Fourier series. Every harmonic of Fourier coefficients of boundary forces and displacements are obtained by numerical quadrature. The results obtained are in agreement with those from the three-dimensional finite element method and experiments.     

Simple regression in view of elliptical models

For the simple linear model Y = θ1 + βx + ? where the error vector follows the elliptically contoured distribution, we consider the unrestricted, restricted, preliminary test and shrinkage estimators for the intercept parameter, θ when it is suspected that the slope parameter β may be β_o. The exact bias and MSE expressions are derived and the mean-square relative efficiency is taken to determine the relative dominance properties of the proposed estimators in comparison. In the continuation, the optimal level of significance of the preliminary test estimator is tabulated and some graphical result are also displayed.     

Thermoelastic vibration and buckling analysis of functionally graded piezoelectric cylindrical shells

This paper presents the report of an investigation into thermoelastic vibration and buckling characteristics of the functionally graded piezoelectric cylindrical, where the functionally graded piezoelectric cylindrical shell is made from a piezoelectric material having gradient change along the thickness, such as piezoelectricity and dielectric coefficient et al. Here, utilizing Hamilton’s principle and the Maxwell equation with a quadratic variation of the electric potential along the thickness direction of the cylindrical shells and the first-order shear deformation theory, and taking into account both the direct piezoelectric effect and the converse piezoelectric effect, the thermoelastic vibration and buckling characteristics of functionally graded piezoelectric cylindrical shells composed of BaTiO₃/PZT − 4, BaTiO₃/PZT − 5A and BaTiO₃/PVDF are, respectively, calculated. The effects of material composition (volume fraction exponent), thermal loading, external voltage applied and shell geometry parameters on the free vibration characteristics are described, and the axial critical load, critical temperature and critical control voltage are obtained.     

Global well‐posedness of 2D nonlinear Boussinesq equations with mixed partial viscosity and thermal diffusivity

In this paper, we discuss with the global well‐posedness of 2D anisotropic nonlinear Boussinesq equations with any two positive viscosities and one positive thermal diffusivity. More precisely, for three kinds of viscous combinations, we obtain the global well‐posedness without any assumption on the solution. For other three difficult cases, under the minimal regularity assumption, we also derive the unique global solution. To the authors' knowledge, our result is new even for the simplified model, that is, F(θ) = θe₂. Copyright © 2017 John Wiley & Sons, Ltd.     

On left (θ, ϕ)-derivations in BCI-algebras

The notion of (regular) left (θ, ϕ)-derivations of a BCI-algebra is introduced, some useful examples are discussed, and related properties are investigated. Conditions for a left (θ, ϕ)-derivation to be regular are provided. The concepts of a d_(θ,ϕ)-invariant left (θ, ϕ)-derivation and θ-ideal are introduced, and their relations are discussed. Furthermore, some more interesting results are established.     

Space–time symmetry violation,configuration mixing model and E-infinity theory

We have studied the time reversal symmetry violation on the bases of the configuration mixing model and E-infinity theory. With the use of the Cabibbo angle approximation, we have presented the transformation matrix in terms of the golden ratio (?), and shown that the time reversal symmetry violation is described by the configuration mixing of the unstable and stable manifolds (W^u, W^s). The magnitude of the mixing for the weak interaction field is given by the expression sin² θ_T(theor) ≈ sin⁴ θ_C(theor) ≈ (?)¹² = 3.105 × 10^?3, which is compared to the Kaon decay experiment ～2.3 × 10^?3. We have also discussed the space–time symmetry violation by using the CPT theorem.     

Vibration of a viscoelastic rectangular orthotropic plate on a viscoelastic foundation

The integrodifferential equation describing the vibration of a viscoelastic rectangular orthotropic plate resting on a viscoelastic foundation with two foundation moduli is investigated. The solution of the starting equation is obtained by the method of straight lines [1] in conjunction with the averaging method [2]. By setting c₁=0 or c₂=0 or their combinations in the solutions obtained we obtain various particular cases that are generalized in the present investigation.Institute of Cybernetics and Computer Center, Academy of Sciences of the Uzbek SSR, Tashkent. Institute of Mechanics and Earthquake Resistance of Structures, Academy of Sciences of the Uzbek SSR, Tashkent. Translated from Mekhanika Polimerov, No. 6, pp. 1087–1094, November–December, 1970.     

非完善加筋圆柱壳在外压和热荷载共同作用下的后屈曲

基于壳体屈曲的边界层理论,本文给出有限长加筋圆柱壳在侧向外压和均布热荷载共同作用下的后屈曲分析。分析中同时考虑壳体非线性前屈曲变形,大挠度和初始几何缺陷的影响。肋条的处理采用“平均刚度”法。采用奇异摄动方法导得壳体屈曲载荷关系曲线和后屈曲平衡路径,并给出完善和非完善,纵向加筋或环向加筋圆柱壳数值算例。     

Tangential limit values of a biharmonic poisson integral in a disk

Let C ₀ be a curve in a disk D={|z|<1} that is tangent to the circle at the point z=1, and let C _θ be the result of rotation of this curve about the origin z=0 by an angle θ. We construct a bounded function biharmonic in D that has a zero normal derivative on the boundary and for which the limit along C _θ does not exist for all θ, 0≤θ≤2π.     

On decay and blow-up of the solution for a viscoelastic wave equation with boundary damping and source terms

In this paper we consider the decay and blow-up properties of a viscoelastic wave equation with boundary damping and source terms. We first extend the decay result (for the case of linear damping) obtained by Lu et al. (On a viscoelastic equation with nonlinear boundary damping and source terms: Global existence and decay of the solution, Nonlinear Analysis: Real World Applications 12 (1) (2011), 295-303) to the nonlinear damping case under weaker assumption on the relaxation function g(t). Then, we give an exponential decay result without the relation between g^′(t) and g(t) for the linear damping case, provided that ‖g‖_L¹(0,∞) is small enough. Finally, we establish two blow-up results: one is for certain solutions with nonpositive initial energy as well as positive initial energy for both the linear and nonlinear damping cases, the other is for certain solutions with arbitrarily positive initial energy for the linear damping case.     

被动约束层阻尼圆柱壳振动和阻尼分析的一种新矩阵方法

基于线弹性薄壳理论和线粘弹性理论,考虑粘弹性层的剪切耗能作用和各层间的相互作用力,导出了被动约束层阻尼层合圆柱壳在谐激励作用下的一阶常微分矩阵控制方程．然后,借助作者提出的齐次扩容精细积分技术建立了一种新的矩阵方法,并利用该方法研究了层合圆柱壳的振动特性和阻尼特性．该方法与已提出的以位移及其导数作为状态向量的传统传递矩阵法的根本区别在于,控制方程中的状态向量中包含了层合壳的全部位移和整合内力变量,因此,可以方便地适用于各种位移和内力边界条件以及部分环状覆盖约束层阻尼圆柱壳的动态分析．数值算例与解析解和有限元解的结果比较有力说明了该方法的正确性和有效性．     

双层柱壳在流场中辐射声场压力的解析解

应用Donnell壳体理论,对加强内外壳体的横向构件,利用交界面的变形协调条件,等价为作用在壳体上的反力和反力矩,把双层柱壳振动辐射声场压力的求解,归结为求解结构动力方程、流场Helmholtz方程、流体和结构交界面上连续性条件组成的声-流体-结构的耦合振动方程.通过复杂的求解方法,可直接求得双层柱壳近场声压.     

加肋圆柱壳在轴压作用下的屈曲和后屈曲

本文讨论完善和非完善的,纵向加肋和正交加肋圆柱壳在轴压作用下的屈曲和后屈曲性态.依据文[1]提供的圆柱薄壳屈曲的边界层理论及其分析方法,给出了加肋圆柱壳在轴压作用下的屈曲和后屈曲理论分析.本文同时讨论肋骨与壳板材料不同时对加肋圆柱壳屈曲和后屈曲性态的影响.     

Approximate solutions of a nonlinear oscillator typified as a mass attached to a stretched elastic wire by the homotopy perturbation method

The homotopy perturbation method is used to solve the nonlinear differential equation that governs the nonlinear oscillations of a system typified as a mass attached to a stretched elastic wire. The restoring force for this oscillator has an irrational term with a parameter λ that characterizes the system (0 ? λ ? 1). For λ = 1 and small values of x, the restoring force does not have a dominant term proportional to x. We find this perturbation method works very well for the whole range of parameters involved, and excellent agreement of the approximate frequencies and periodic solutions with the exact ones has been demonstrated and discussed. Only one iteration leads to high accuracy of the solutions and the maximal relative error for the approximate frequency is less than 2.2% for small and large values of oscillation amplitude. This error corresponds to λ = 1, while for λ < 1 the relative error is much lower. For example, its value is as low as 0.062% for λ = 0.5.