Nonlinear modes of vibrations for simply supported cylindrical shell with geometrical nonlinearity

The system of three partial differential equations with respect to displacements (Donnell equations) is used to analyze nonlinear vibrations of a cylindrical shell. The Galerkin method is applied to every partial differential equation to obtain a finite-degree-of-freedom model of the shell. The system of ordinary differential equations with respect to the general coordinates of the radial shell displacements is derived. The nonlinear modes of free vibrations are calculated using the harmonic balance method. The stability analysis of periodic motions is performed.     

Elasticity Solution of a Laminated Clamped Cylindrical Shell with Piezoelectric Layer under Dynamic Load

Dynamic elasticity solution for a clamped, laminated cylindrical shell with two orthotropic layers bounded with a piezoelectric layer and subjected to impulse load distributed on inner surface is presented. The piezoelectric layer serves as sensor/actuator. The governing elasticity PDE equations are reduced to ordinary differential equations by means of Legendre polynomial expansion for displacement and electric potential in the axial direction. The resulting equations are transferred into state space form and reduced to an eigenvalue problem by using Galerkin's finite element in radial direction. The static and dynamic results are presented for [0/90/Piezo] lamination. The radius to thickness ratio effect on dynamic behavior is studied. The results are compared for different thickness ratios and applied electric loads with simply-supported shell results. Time responses for sensor and actuated shell are presented and natural frequencies are compared with simply-supported shell results.     

Levi functions for linear elliptic systems with variable coefficients including shell equations

This paper gives an algorithm to construct Levi functions of arbitrary degree for elliptic systems of linear partial differential equations with variable (real-analytic) coefficients. Further, an indirect method is described to transform elliptic boundary value problems into a system of integral equations. This method is applied to the shell equations in the non-shallow case. (In the shallow case the shell equations have constant coefficients.) Some questions of discretization are discussed and numerical results are presented.     

Enhanced mechanical properties of ZnO nanowire-reinforced nanocomposites: a size-scale effect

A multiscale approach is pursued to develop a shear-lag model in combination with core–surface and core–shell models for capturing size-scale effect on mechanical properties of ZnO nanowire (NW)-reinforced nanocomposites. Surface effects are represented by a zero-thickness (finite-thickness) surface with different elastic modulus from the central part of NW. The molecular dynamics technique is utilized for calculating thickness of the shell in the core–shell model. Linear elasticity for an axisymmetric problem and the cylindrical coordinate system is used to find the closed form of governing equations. The effect of different parameters, including diameter and aspect ratio of NWs, is studied to demonstrate the application of the developed model. Numerical results disclose that NWs with a larger aspect ratio and a smaller diameter can carry a larger portion of applied stress and are preferable in designing high-performance nanocomposites. This result is in agreement with the reported computational and experimental data.     

Supersonic flutter prediction of functionally graded cylindrical shells

The supersonic flutter analysis of simply supported FG cylindrical shell for different sets of in-plane boundary conditions is performed. The aeroelastic equations of motion are constructed using Love’s shell theory and von Karman–Donnell-type of kinematic nonlinearity coupled with linearized first-order potential (piston) theory. The material properties are assumed to be temperature-dependant and graded across the thickness of the shell according to a simple power law. The temperature distribution is assumed to vary in the thickness direction and is obtained by solving the steady-state heat conduction equation. The pre-stresses due to the thermal and mechanical loadings are obtained by exact solution of the equilibrium equations. The Galerkin method is used to solve the aeroelastic equations of motion employing appropriate displacement functions. The effects of internal pressure and temperature rise on the flutter boundaries of the simply supported FG cylinder with different values of power-law index are investigated.     

浸水旋转壳体振动特性的有限条—边界元分析

根据旋转壳体的结构特点,利用有限条法对其进行分析达到降维目的;对流场则提出了一种特殊的边界单元,这种边界元不仅能与结构的有限条单元协调,而且计算量小,精度高;在此基础上,通过凝聚降阶进一步减少流固耦合系统的控制方程。对典型浸水旋转壳的振动频率、模态以及响应作了全面分析,获得了理想的数值分析结果。     

大型复合材料夹芯筒屈曲分析中芯材剪切变形与壳体锥度的影响

针对某卫星结构中的大型复合材料夹芯承力筒进行屈曲分析。首先分别建立柱壳段、锥壳段和框段的几何方程、物理方程与应变能表达式,以及给定载荷条件下的外力势能表达式。然后根据位移边界条件采用正交完备的三角级数构造承力筒的位移模态,最后根据最小势能原理求解临界载荷。通过对比计及与不计及芯材剪切变形和壳体锥度时临界载荷的计算结果,发现对于该承力筒而言,芯材剪切变形与壳体锥度对临界载荷计算结果的影响均小于10%。该分析结果为承力筒在设计阶段的简化计算提供了理论依据。     

Nonlinear vibration and instability analysis of functionally graded CNT-reinforced cylindrical shells conveying viscous fluid resting on orthotropic Pasternak medium

In this study, nonlinear vibration and instability of embedded temperature-dependent cylindrical shell conveying viscous fluid resting on temperature-dependent orthotropic Pasternak medium are investigated. The equivalent material properties of nanocomposites are estimated using rule of mixture. Both cases of uniform distribution and functionally graded distribution patterns of reinforcements are considered. Based on orthotropic Mindlin shell theory, the governing equations are derived. Generalized differential quadrature method is applied for obtaining the frequency and critical fluid velocity of a system. The effects of different parameters, such as distribution type of single-walled carbon nanotubes (SWCNTs), volume fractions of SWCNTs, and Pasternak medium are discussed.     

载流悬臂柱壳的非线性数值分析

针对在电磁场和机械场耦合作用下的载流柱壳的非线性变形问题的数值解法进行了研究。给出了载流柱壳在耦合场作用下的几何方程、物理方程、二维电动力学方程、磁弹性非线性运动方程和洛仑兹力表达式,建立了差分格式和线性化迭代方程,给出了这些方程的数值解系统,并以悬臂柱壳为例,计算了该壳在电磁场和机械载荷耦合作用下的应力及变形;讨论了其应力及变形与外加电磁参量之间的关系;证明了变化电磁参量可以对壳的工作状态实施控制。     

Modeling arbitrary crack propagation in coupled shell/solid structures with X‐FEM

In this paper, the extended finite element method (X‐FEM) formulation for the modeling of arbitrary crack propagation in coupled shell/solid structures is developed based on the large deformation continuum‐based (CB) shell theory. The main features of the new method are as follows: (1) different kinematic equations are derived for different fibers in CB shell elements, including the fibers enriched by shifted jump function or crack tip functions and the fibers cut into two segments by the crack surface or connecting with solid elements. So the crack tip can locate inside the element, and the crack surface is not necessarily perpendicular to the middle surface. (2) The enhanced CB shell element is developed to realize the seamless transition of crack propagation between shell and solid structures. (3) A revised interaction integral is used to calculate the stress intensity factor (SIF) for shells, which avoids that the auxiliary fields for cracks in Mindlin–Reissner plates cannot satisfy exactly the equilibrium equations. Several numerical examples, including the calculation of SIF for the cracked plate under uniform bending and crack propagation between solid and shell structures are presented to demonstrate the performance of the developed method. Copyright © 2015 John Wiley & Sons, Ltd.     

Stress-deformable state of isotropic double curved shell with internal cracks and a circular hole

This work is devoted to the stress–strain state of isotropic double curved shell with defect system. The construction is weakened by two non-through thickness (internal) cracks of different length and by a circular hole located between cracks. In this study we use the line-spring model. Within the framework of this model cracks are modeled as mathematical cuts of shell’s middle surface. This leads to a two-dimensional problem. The problem is reduced to a system of eight boundary integral equations. To ensure the uniqueness of solution an additional equation is added. In the numerical solution of the problem special quadrature formulas for singular integrals of Cauchy type and the finite difference method are applied. The influence of defects on each other for double curved shell has been investigated. The given theoretical results can be used for the calculation of structural elements with holes, cracks on the strength and fracture toughness in various branches of engineering.     

Vibration control of laminated truncated conical shell via magnetostrictive layers

Abstract

In this study feedback control is applied to control the free vibration response of an isotropic truncated conical shell embedded with magnetostrictive layers. Classical shell theory is applied to derive the shell vibration equations. The results are derived based on the Galerkin method and the results are compared with published results and the results of finite element software in order to determine the accuracy of using method. The influence of several parameters such as the thickness of magnetostrictive layers, control gain, length and radius of the large edge of the shell on the vibration suppression of fundamental frequency is determined.     

The modified couple stress functionally graded cylindrical thin shell formulation

In this article, the functionally graded (FG) cylindrical thin shell formulation is developed by using modified couple stress theory. The equations of motion and classical and nonclassical boundary conditions are extracted based on Hamilton's principle. As a special case, the equations of motion in conjunction with the boundary conditions for simply supported FG cylindrical shell are obtained, and then Navier solution procedure is used for analysis free vibration of nano shell. Afterwards, the influences of different parameters like length scale parameter, distribution of FG properties, and length to radius ratio on dimensionless natural frequency are investigated and compared with classical theory.     

Application of third order shear deformation theory in buckling analysis of 2D-functionally graded cylindrical shell reinforced by axial stiffeners

This paper presents buckling analysis of a two-dimensional functionally graded cylindrical shell reinforced by axial stiffeners (stringer) under combined compressive axial and transverse uniform distributive load. The shell material properties are graded in the direction of thickness and length according to a simple power law distribution in terms of the volume fractions of the constituents. Primarily, the third order shear deformation theory (TSDT) is used to derive the equilibrium and stability equations. Since there is no closed form solution, the numerical differential quadrature method, (DQM), is applied for solving the stability equations. Initially, the obtained results for an isotropic shell using DQM were verified against those given in the literature for simply supported boundary conditions. The effects of load, geometrical and stringer parameters along with FG power index in the various boundary conditions on the critical buckling load have been studied. The study of results confirms that, stringers have significant effects on critical buckling load.     

Nonlinear vibrations of laminated circular cylindrical shells: Comparison of different shell theories

The geometrically nonlinear forced vibrations of laminated circular cylindrical shells are studied by using the Amabili–Reddy higher-order shear deformation theory. An energy approach based on Lagrange equations, retaining modal damping, is used in order to obtain the equations of motion. An harmonic point excitation is applied in radial direction and simply supported boundary conditions are assumed. The equations of motion are studied by using the pseudo-arclength continuation method and bifurcation analysis. A one-to-one internal resonance is always present for a complete circular cylindrical shell, giving rise to pitchfork bifurcations of the nonlinear response with appearance of a second branch with travelling wave response and quasi-periodic vibrations. The numerical results obtained by using the Amabili–Reddy shell theory are compared to those obtained by using an higher-order shear deformation theory retaining only nonlinear term of von Kármán type and the Novozhilov classical shell theory.     

Supersonic flutter prediction of functionally graded conical shells

Aero-thermoelastic analysis of a simply supported functionally graded truncated conical shell subjected to supersonic air flow is performed to predict the flutter boundaries. The temperature-dependent properties of the FG shell are assumed to be graded through the thickness according to a simple rule of mixture and power-law function of volume fractions of material constituents. Through the thickness steady-state heat conduction is considered for thermal analysis. To perform the stability analysis, the general nonlinear equations of motion are first derived using the classical Love’s shell theory and the von Karman–Donnell-type of kinematic nonlinearity together with the linearized first-order piston theory for aerodynamic loading. Then the nonlinear equations of motion are linearized to obtain the linear equilibrium and aeroelastic equations. The equilibrium equations are solved using power series method to obtain the initial stresses induced by aerodynamic and thermal loadings. The results are then used as an input to the aeroelastic stability equations which are finally solved with the generalized Galerkin method. The flutter boundaries are obtained for the FG conical shells with different semi-vertex cone angles, different temperature distributions, and different volume fraction indices.     

Nonlinear vibrations of laminated circular cylindrical shells: Comparison of different shell theories

The geometrically nonlinear forced vibrations of laminated circular cylindrical shells are studied by using the Amabili–Reddy higher-order shear deformation theory. An energy approach based on Lagrange equations, retaining modal damping, is used in order to obtain the equations of motion. An harmonic point excitation is applied in radial direction and simply supported boundary conditions are assumed. The equations of motion are studied by using the pseudo-arclength continuation method and bifurcation analysis. A one-to-one internal resonance is always present for a complete circular cylindrical shell, giving rise to pitchfork bifurcations of the nonlinear response with appearance of a second branch with travelling wave response and quasi-periodic vibrations. The numerical results obtained by using the Amabili–Reddy shell theory are compared to those obtained by using an higher-order shear deformation theory retaining only nonlinear term of von Kármán type and the Novozhilov classical shell theory.     

螺杆冷水机组动态仿真

给出了螺杆冷水机组动态过程模型，其中，壳管式冷凝器和满液式蒸发器被分成制冷剂侧、管壁、壳体以及水侧等四个控制容积，针对每个控制容积建立了质量或能量平衡的微分方程，由于螺杆压缩机和电子膨胀阀具有较小的热惯性，因而采用稳态模型描述压缩过程和膨胀过程，通过“自适应步长”方法实现了系统仿真，仿真结果与测试数据吻合良好。     

粘弹性圆柱壳在轴向恒压下的动力稳定性

基于Timoshenko-Mindlin假设,得到考虑粘弹性的各向同性圆柱壳及纤维增强正交铺设层合圆柱壳在轴向恒压下的动力学方程。文中对两端简支的圆柱壳进行了分析,依Laplace变换,导出动力稳定的特征方程,由Routh-Hurwitz判据建立动力稳定性条件,对两类圆柱壳讨论了横向剪切变形的影响。