Nonlinear finite element analysis of composite shell under impact

Large deflection dynamic responses of laminated composite cylindrical shells under impact are analyzed by the geometrically nonlinear finite element method based on a generalized Sander’s shell theory with the first order transverse shear deformation and the von-Karman large deflection assumption. A modified indentation law with inelastic indentation is employed for the contact force. The nonlinear finite element equations of motion of shell and an impactor along with the contact laws are solved numerically using Newmark’s time marching integration scheme in conjunction with Akay type successive iteration in each step. The ply failure region of the laminated shell is estimated using the Tsai-Wu quadratic interaction criteria. Numerical results, including the contact force histories, deflections and strains are presented and compared with the ones by linear analysis. The effect of the radius of curvature on the composite shell behaviors is investigated and discussed.     

Dynamic response and damage of composite shell under impact

The dynamic response and damage of laminated composite cylindrical shell subjected to impact load is numerically investigated using the finite element method. A nine-node isoparametric quadrilateral element based on Sander's shell theory is developed, in which the transverse shear deformation is considered. A semi-empirical contact law that accounts for the permanent indentation is incorporated into the finite element program to evaluate the contact force. The Newmark time ingegration algorithm is used for solving the time dependent equations of the shell and the impactor. The Tsai-Wu failure criterion is used to estimate the failure of the laminated shell. Numerical results, including the contact force history, interlaminate damage zone, and failure indices in the shell are presented. Effects of curvature, impact velocity and mass of impactor on the composite shell behaviors are discussed.     

Layer-wise finite element analysis of functionally graded cylindrical shell under dynamic load

In this paper, a layer-wise finite element formulation is developed for the analysis of a functionally graded material (FGM) cylindrical shell with finite length under dynamic load. For this purpose, FGM cylinder is divided into many sub-layers and then the general layerwise laminate theory is formulated by introducing piecewise continuous approximations through the thickness for each state In this model the radial displacement is approximated linearly through each “mathematical” layer. The properties are controlled by volume fraction that is an exponential function of radius. The governing equations are derived from virtual work statement and solved by finite element method. The main contribution of the present study is to develop a discrete layerwise finite element for a 2-dimensional thick FGM cylindrical shell. Results are obtained for the time history of the displacement and stress components with different exponent “n” of functionally graded material. In addition, natural frequency and mean velocity of the radial wave propagation for different exponent “n” of functionally graded material (FGM) are studied and compared with similar ones currently obtained for FGM cylindrical shell of infinite length.     

Development of FAMD code to calculate the fluid added mass and damping of arbitrary structures submerged in confined

In this paper, the numerical finite element formulations were derived for the linearized Navier-Stokes’ equations with assumptions of two-dimensional incompressible, homogeneous viscous fluid field, and small oscillation and the FAMD (Fluid Added Mass and Damping) code was developed for practical applications calculating the fluid added mass and damping. In formulations, a fluid domain is discretized with C°-type quadratic quadrilateral elements containing eight nodes using a mixed interpolation method, i.e., the interpolation function for the velocity variable is approximated by a quadratic function based on all eight nodal points and the interpolation function for the pressure variable is approximated by a linear function based on the four nodal points at vertices. Using the developed code, the various characteristics of the fluid added mass and damping are investigated for the concentric cylindrical shell and the actual hexagon arrays of the liquid metal reactor cores.     

壳板组合结构功率传递分析

壳板耦合是圆柱壳组合结构中最常见的连接方式。为了研究壳板组合结构的功率传递规律,利用有效点导纳分析周向均布激励下圆柱壳表面振动响应的分布规律,得出圆柱壳振动响应理论计算公式;通过给出圆周线导纳的定义及其表达式,计算带端板圆柱壳的振动响应均方值之比。理论计算结果表明:壳板振动响应均方值之比随频率的增高逐渐增大,但其值始终小于1。实验测量结果与理论计算吻合,其值围绕理论值上下波动,表征了理论计算公式和分析结果的正确性和合理性。     

Frequency characteristics of a thin rotating cylindrical shell using the generalized differential quadrature method

In this paper, the generalized differential quadrature (GDQ) method is used for the first time to study the effects of boundary conditions on the frequency characteristics of a thin rotating cylindrical shell. The present analysis is based on Love-type shell theory and the governing equations of motion include the effects of initial hoop tension and the centrifugal and coriolis accelerations due to rotation. The displacement field is expressed as a product of unknown smooth continuous functions in the meridional direction and trigonometric functions along the circumferential direction so that the three-dimensional dynamic problem may be transformed mathematically into a one-dimensional problem. Based on this approach, the results are obtained for the effects of the boundary conditions on the frequency characteristics at different circumferential wave numbers and rotating speeds and various geometric properties; the effect of rotating speed on the relationship between frequency parameter and circumferential wave number is also discussed. To validate the accuracy and efficiency of the GDQ method, the results obtained are compared with those in the literature and very good agreement is achieved.     

Weight vector study of velocity profile effects in straight-tube Coriolis flowmeters employing different circumferential modes

The weight vector theory is applied to evaluate the effects of velocity profile on the sensitivity of Coriolis mass flowmeters employing different circumferential modes of a straight measuring tube. The study is limited to fully developed, axisymmetric steady flow. The measuring tube is modelled as a thin, elastic shell with clamped ends, and the vibrational fluid motion inside the vibrating tube is determined using potential flow theory. By applying a power series expansion with respect to the aspect ratio of the tube, the general form of the weight function is simplified, allowing (approximate) analytical evaluations of profile effects. The results show that large velocity profile effects are expected for the higher circumferential (shell-type) modes and rather lower (but generally also significant) velocity profile effects for the first circumferential (beam-type) mode.     

Nonlinear dynamic response of rotating circular cylindrical shells with precession of vibrating shape—Part I: Numerical solution

The nonlinear dynamic response of a cantilever rotating circular cylindrical shell subjected to a harmonic excitation about one of the lowest natural frequency, corresponding to mode (m=1, n=6),where m indicates the number of axial half-waves and n indicates the number of circumferential waves, is investigated by using numerical method in this paper. The factor of precession of vibrating shape ? is obtained, with damping accounted for. The equation of motion is derived by using the Donnell’s nonlinear shallow-shell theory, and is general in the sense that it includes damping, Coriolis force and large-amplitude shell motion effects. The problem is reduced to a system of ordinary differential equations by means of the Galerkin method. Three different mode expansions are studied for finding the proper one which is more contracted and accurate to investigate the principal mode (i.e., m=1, n=6) response. From the present investigation, it can be found that for principal mode resonant response, there are two traveling waves with different linear frequencies due to the effect of precession of vibrating shape of rotating circular cylindrical shells; the effects of additional modes n and k (multiples of frequency) on the principal mode resonant response are insignificant compared with an additional mode m, showing that it is better to adopt two neighboring axial modes to study the principal resonant response of the system.     

Vibrations of cantilevered circular cylindrical shells: Shallow versus deep shell theory

Free vibrations of cantilevered circular cylindrical shells having rectangular plan-forms are studied in this paper by means of the Ritz method. The deep shell theory of Novozhilov and Goldenveizer is used and compared with the usual shallow shell theory for a wide range of shell parameters. A thorough convergence study is presented along with comparisons to previously published finite element solutions and experimental results. Accurately computed frequency parameters and mode shapes for various shell configurations are presented. The present paper appears to be the first comprehensive study presenting rigorous comparisons between the two shell theories in dealing with free vibrations of cantilevered cylindrical shells.     

基于ANSYS直齿圆柱齿轮有限元模态分析

研究了直齿圆柱齿轮的固有振动特性，采用有限元法建立了直齿圆柱齿轮的动力学模型，通过有限元分析软件ANSYS对齿轮进行模态分析，得到了齿轮的低阶固有振动频率和主振型，可以为齿轮系统的动态设计提供参考，同时也为齿轮系统的动态响应计算和分析奠定了基础。     

Transverse shear and rotary inertia effects on the stability analysis of functionally graded shells under combined static and periodic axial loadings

The effect of transverse shear and rotary inertias on the dynamic stability of functionally graded cylindrical shells subjected to combined static and periodic axial forces is investigated in this paper. Material properties of functionally graded cylindrical shells are considered temperature-dependent and are graded in the thickness direction according to a power-law distribution in terms of the volume fractions of the constituents. Numerical results for silicon nitride-nickel cylindrical shells are presented based on two different methods: the first-order shear deformation theory (FSDT) which considers the transverse shear strains and the rotary inertias, and the classical shell theory (CST). The results obtained show that the effect of transverse shear and rotary inertias on the dynamic stability of functionally graded cylindrical shells subjected to combined static and periodic axial forces is dependent on the shell’s material composition, environmental temperature, amplitude of static load, deformation mode, and the shell’s geometry parameters.     

Non-linear vibrations and instabilities of orthotropic cylindrical shells with internal flowing fluid

In this work, Donnell’s non-linear shallow shell equations are used to study the dynamic instability of perfect simply supported orthotropic cylindrical shells with internal flowing fluid and subjected to either a compressive axial static pre-load plus a harmonic axial load or a harmonic lateral pressure. The fluid is assumed to be non-viscous and incompressible and the flow, isentropic and irrotational. An expansion with eight degrees of freedom, containing the fundamental, companion, gyroscopic, and four axi-symmetric modes is used to describe the lateral displacement of the shell. The Galerkin method is used to obtain the non-linear equations of motion which are solved by the Runge-Kutta method. A detailed parametric analysis clarifies the influence of the orthotropic material properties on the non-linear buckling and vibration characteristics of the shell. Numerical methods are used to identify the effect of the fluid flow and applied loads control parameters on the bifurcations and stability of the shell motions.     

Research on acoustic-structure sensitivity using FEM and BEM

Acoustic-structure sensitivity is used to predict the change of acoustic pressure when a structure design variable is changed. The sensitivity is significant for reducing noise of structure. Using FEM (finite element method) and BEM (boundary element method) acoustic-structure sensitivity was formulated and presented. The dynamic response and response velocity sensitivity with respect to structure design variable were carried out by using structural FEM, the acoustic response and acoustic pressure sensitivity with respect to structure velocity were carried out by using acoustic BEM. Then, acoustic-structure sensitivity was computed by linking velocity sensitivity in FEM and acoustic sensitivity in BEM. This method was applied to an empty box as an example. Acoustic pressure sensitivity with respect to structure thickness achieved in frequency ranges 1–100 Hz, and its change rule along with stimulating frequency and design variable were analyzed. Results show that acoustic-structure sensitivity method linked with FEM and BEM is effective and correct. __________ Translated from Journal of Vibration Engineering, 2005, 18(3): 366–370 [译自: 振动工程学报]     

Vibration analysis of multi-supported curved panel using the periodic structure approach

This papers deals with the radial vibration of a row of cylindrical panels of finite length using the concept of wave propagation in periodic structures. For this study, the structure is considered as an assemblage of a number of identical cylindrically curved panels each of which will be referred to as a periodic element. For a given geometry dispersion curves of the propagation constant versus (non-dimensional) natural frequency have been drawn corresponding to the circumferential wave propagation. New conclusions that have emerged from this study are as follows. It is shown that by a proper choice of the periodic element the bounding frequencies and the corresponding modes in all the propagation bands can be determined. These have been shown to correspond to a single curved panel with all its edges simply supported. It is noted that there are no attenuation gaps in the entire frequency spectrum beyond the lowest bounding frequency. This is a unique feature of circumferential wave propagation around circular cylindrical shells and panels, as opposed to the wave propagation of periodically supported beams and rectangular panels without curvature. The natural frequency corresponding to every circumferential mode of the complete shell has been identified on the propagation constant curve. It has been observed that the natural frequencies of a cylindrically curved panel of a given curvature and length but of different circumferential arc length (corresponding to different angles subtended at the centre of any circular cross-section) may also be identified on the same propagation constant curve. Finally, it is shown that the same propagation constant curve may also be used to determine all the natural frequencies of a finite row of curved panels with the extreme edges simply supported. Wherever possible the numerical results have been compared with those obtained independently from finite element analysis and/or results available in the literature. Flutter analysis of multi-span curved panels using a wave approach is the ultimate objective of this work.     

Non-linear free vibration analysis of laminated orthotropic cylindrical shells

A method to predict the influence of geometric non-linearities on the natural frequencies of an empty laminated orthotropic cylindrical shell is presented in this paper. It is a hybrid of finite element and classical thin shell theories. Sanders—Koiter non-linear and strain-displacement relations are used. Displacement functions are evaluated using linearized equations of motion. Modal coefficients are then obtained for these displacement functions. Expressions for the mass, linear and non-linear stiffness matrices are derived through the finite element method (in terms of the elements of the elasticity matrix). The uncoupled equations are solved with the help of elliptic functions. The frequency variations are first determined as a function of shell amplitudes and then compared with the results in the literature.     

Vibration of bilayered cylindrical shells with layers of different materials

In this analysis, a comparative study for natural frequencies of two-layered cylindrical shells was presented with one layer composed of functionally graded material and the other layer of isotropic material. Love’s thin shell theory was exploited for the strain-displacement and curvature-displacement relationships. For governing frequency equations, the Rayleigh-Ritz method was utilized to minimize the Lagrangian functional in the form of an eigenvalue problem. Frequency spectra were computed for long, short, thick, and thin cylindrical shells by varying the nondimensional geometrical parameters, length-to-radius and thickness-to-radius ratios for a simply supported end condition. Influence of different configurations of cylindrical shells on the shell frequencies was studied. For validity, the results obtained were compared with some results of isotropic and single-layered functionally graded cylindrical shells from the literature.     

A frequency response function-based damage identification method for cylindrical shell structures

In this paper, a structural damage identification method (SDIM) is developed for cylindrical shells and the numerically simulated damage identification tests are conducted to study the feasibility of the proposed SDIM. The SDIM is derived from the frequency response function solved from the structural dynamic equations of damaged cylindrical shells. A damage distribution function is used to represent the distribution and magnitudes of the local damages within a cylindrical shell. In contrast with most existing modal parameters-based SDIMs which require the modal parameters measured in both intact and damaged states, the present SDIM requires only the FRF-data measured in the damaged state. By virtue of utilizing FRF-data, one is able to make the inverse problem of damage identification well-posed by choosing as many sets of excitation frequency and FRF measurement point as needed to obtain a sufficient number of equations.     

利用反演技术求解消声器内部结构参数

应用流体动力学理论研究消声器内部流场,建立消声器有限元模型,确定给定结构参数求解出口气流流速的正问题;在正问题基础上,将出口气流速度作为消声器结构参数变化时的响应,运用反演技术求解消声器内部结构参数,并将反演问题转化为最优问题,采用遗传算法和梯度法相结合的混合优化算法进行优化,得到了较理想的仿真实验结果,验证了反演技术应用于消声器设计的可行性.     

A finite element model for the analysis of flexible tube Coriolis mass flow meter using first order shear deformation shell theory

The numerical modelling of Coriolis Mass flow Meter (CFM) is essential for predicting its outcomes accurately in terms of sensitivity as well as exact mass flow rates. In the majority of mathematical and numerical modelling concerning the flexible structures, the authors neglect the dimensional and shape variation of the structure due to self-weight. The shell based on the First-order shear deformation shell theory (FSDST) is preferred in modelling shells compared to the beam model. The current work includes numerical modelling of CFM using eight noded isoparametric shell elements and twenty noded Acoustic fluid elements. The fluid energy describes as the potential, and the dynamic boundary condition is assumed utilising the displacement of structure and potential of the fluid. The fluid dynamic equation combining suitable numerical model, fluid-structure interaction module and cross-correlation technique helps to achieve the numerical modelling of CFM. The numerical model of CFM utilises the Newmark Beta method of numerical integration, and the response of two equidistant locations from the point of tube excitation is acquired. For the flexible tube conveying fluid, there exists sagging of tube due to the weight of tube and fluid. The Coriolis force and the external excitation force cause the fluid conveying tube to bend and twist, and as a result, the velocity responses picked from two equidistant points shows a difference in phase. The effect of sagging leads to a lower phase shift and time decay, and hence the sensitivity of the CFM is low for low pre-stretched flexible tubes. The pre-stretching of flexible tubes reduces the effect of sagging, facilitates to regain the cylindrical shape of the tube and increases the sensitivity of CFM. The result reveals that the shell element along with the three-dimensional acoustic fluid element provides the most accurate numerical model for the CFM and the change in sensitivity, as well as the change in mass flow measurements, can appropriately be analysed with the help of this numerical model. The amplitude of the velocity of the structure, measured from the two equidistant points, shows a difference. The severe variation in amplitude of velocity measured from two points is an implication of the out of plane deflection of the tube. For a CFM made up of metal tubes, the amplitude of velocity variation is minimal and ignored by the authors.     

The analysis of structural-acoustic coupling of an enclosure using Green’s function method

Currently researchers expect to control the internal noise of an enclosure by controlling the relative structural vibration. The method is called ANC (Active Noise Control). Sound pressure response and velocity response of the structure are two significant factors to evaluate the performance of the ANC. But previous researches have to depend on a three-dimensional model with regular shape. Based on the Green’s function theorem this paper proposed two formulas to describe the contributions of acoustic modes and structural modes. These two formulas are the essential basis for the analysis of structural-acoustic coupling of an enclosure. From them compact matrix formulas are deduced and a numerical simulation method is presented to calculate the responses of sound pressure and velocity. The numerical simulation method is verified by comparing the numerical result with the theoretical one based on a regular model. And the application of the method in an irregular model shows this method can be used to analyze any model with arbitrary shape.