Three-dimensional oscillations of a cantilever pipe conveying fluid

The aim of the study described in this paper is to investigate the two-dimensional (2-D) and three-dimensional (3-D) flutter of cantilevered pipes conveying fluid. Specifically, by means of a complete set of non-linear equations of motion, two questions are addressed: (i) whether for a system losing stability by either 2-D or 3-D flutter the motion remains of the same type as the flow velocity is increased substantially beyond the Hopf bifurcation precipitating the flutter; (ii) whether the bifurcational behaviour of a horizontal system and a vertical one (sufficiently long for gravity to have an important effect on the dynamics) are substantially similar. Stability maps and tables are used to delineate areas in a flow velocity versus mass parameter plane where 2-D or 3-D motions occur, and limit-cycle motions are illustrated by phase-plane plots, PSDs and cross-sectional diagrams showing whether the motion is circular (3-D) or planar (2-D).     

Dynamics of cantilevered pipes conveying fluid. Part 3: Three-dimensional dynamics in the presence of an end-mass

In this part, the last in a three-part study, the three-dimensional (3-D) dynamics of a cantilevered pipe conveying fluid is explored when an additional “point” mass is attached at the free end. For a typical case, the dynamical behaviour of this system is presented in the form of a bifurcation diagram, along with the time traces, phase-plane plots, PSD plots and Poincaré maps, showing planar periodic, quasiperiodic and chaotic oscillations, followed by 3-D quasiperiodic and chaotic motions. The effect on the results of the number of beam modes used in the Galerkin solution scheme is studied in some detail. The theoretical results are then compared with the results of a set of experiments done previously and good qualitative and quantitative agreement is observed.     

Detection and description of non-linear phenomena in experimental modal analysis via linearity plots

Ground vibration tests (GVTs) on aircraft prototypes are mainly performed to experimentally identify the structural dynamic behaviour in terms of a modal model. This assumes a linear dynamic behaviour of the structure. However, in the practice of ground vibration testing it is often observed that structures do not behave in a perfectly linear manner. Non-linearities can be determined, for example, by free play in junctions, hydraulic systems in control surfaces, or friction. This paper compiles measured, typical, non-linear phenomena from various GVTs on large aircraft. The standard procedure in GVTs nowadays is the application of the Harmonic Balance method which linearizes the dynamic behaviour on the level of excitation. The procedure requires a harmonic excitation of the structure which is usually performed during phase resonance testing. The non-linear behaviour is investigated in terms of linearity plots in which the resonance frequency of a mode is plotted as a function of the excitation level. The experimental data is then compatible with all post-processing procedures for the measured results, e.g. updating of the finite element model or flutter calculations. This paper shows measured linearity plots for some typical non-linear phenomena. In the second part of the paper analytical linearity plots for different non-linear stiffness and damping models are considered in order to investigate whether the type of non-linearity can be identified from measured linearity plots. The analytical linearity plots are discussed with respect to their application limits. The analytical linearity plots are used to interpret the experimental linearity plots stemming from various GVTs on different aircraft prototypes. Finally, the observability of non-linear stiffness and non-linear damping characteristics via linearity plots is assessed.     

求解变分型积分方程的一种新型数值方法——有限变分法

将作者提出的多虚拟裂纹扩展法(MVCE法)拓展为求解变分型积分方程问题的一种新型数值方法——有限变分法(FVM)。它的基本思想是,给定有限个(N个)局部变分模式,将所求解的未知量用适当的方法离散化,针对这N个局部变分模式列出N个方程,求解N个未知系数,从而求得未知量。单一未知变量FVM的最终方程组的系数矩阵通常是一个对称的窄带矩阵,对角元是大数,有很好的数值计算性能。用FVM求解了三维I型裂纹前缘的应力强度因子(SIF)分布。利用基于FVM的通用权函数法计算程序,可以高精度和高效率地求解表面力、体积力和温度载荷共同作用情况下三维裂纹前缘SIF的分布及其时间历程。FVM可以被推广到更广泛的领域,是一个求解变分型积分方程问题的普遍适用的新型数值方法。     

有限弹塑性变形的三维组集式本构模型

本文将文[1]中提出的三维组集式弹塑性本构模型推广应用于有限变形分析,导出了全量型和增量型本构关系在初始构形上的拉格朗日(Total Lagrange)形式和瞬时构形上的拉格朗日(Updated Lagrange)形式。文中对晶体单轴拉伸中的宏观剪切带进行了分析。预测结果与实验吻合。从而说明这种本构模型能够模拟有限变形中的几何非线性效应和晶体材料塑性变形中的宏观力学行为。     

求解三维裂纹前缘SIF分布时间历程的通用权函数法和有限变分法

将三维热权函数法扩展为适用于表面力、体积力和温度载荷的通用权函数法(UWF).推导出以变分型积分方程表达的UWF法基本方程,从变分的角度,将求解三维热权函数法基本方程的多虚拟裂纹扩展法(MVCE)改造为可以适用于一般的变分型积分方程的一类新型数值方法--有限变分法(FVM).在FVM中可以引入无穷多种线性无关的局部变分模式,可以根据计算要求在求解域中插入任意多个计算节点,单一型裂纹问题FVM所得到的最终方程组的系数矩阵总是一个对称的窄带矩阵,而且对角元总是大数,具有良好的数值计算性能.FVM对于SIF沿裂纹前缘急剧变化的复杂情况具有较好的数值模拟能力和较高的计算精度,利用自身一致性,可以求得三维裂纹前缘SIF的高精度解.     

On completeness for a 3-D linear theory of composite laminate

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3-D numerical modelling of greenwater loading on fixed ship-shaped FPSOs

This work is focused on the assessment of greenwater overtopping onto fixed ship-shaped FPSO models using three dimensional (3-D) Computational Fluid Dynamics (CFD) simulations. Good agreement between the numerical results and published experimental data from Barcellona et al. (2003) indicates that 3-D CFD is an effective tool which may be used to assess greenwater associated with an incident wave group. Different wall-sided bow shapes are investigated numerically, and the results provide insight into how the bow shape influences the evolution of on-deck flows, and the horizontal force on a vertical wall spanning the full deck width for incident waves approaching normal to the bow. It is found that the horizontal force, thus, the horizontal momentum of greenwater flows, is a result of the combined action of water-front velocities and the corresponding water volume impacting on the structures. For the bow shapes considered in this study, although clear differences in on-deck flow are observed for different bow shape, differences in force on the vertical wall are relatively small. Simulations for a 2-D vertical rectangular box that has the same longitudinal section as the 3-D model FPSOs give somewhat similar results to 3-D predictions along the centre-line. This similarity may be due partly to the fact that increases in on-deck flow velocity due to increased freeboard exceedance in the 2-D simulations compensates for a lack of focusing of the on-deck flow observed in the 3-D simulations. This finding has implications for understanding how computationally cheaper 2-D greenwater simulations relate to more realistic 3-D greenwater events.     

On the lower bounds for critical loads under large deformations in non-linear hyperelastic composites with imperfect interlaminar adhesion

The present paper investigates a mechanism of compressive fracture for heterogeneous incompressible non-linear materials with special kinds of defects of interfacial adhesion under large deformations. The analysis finds the lower bounds for the critical load. In order to calculate the bounds, the problem of the internal instability is considered within the scope of the exact statement based on the application of the model of a piecewise-homogeneous medium and the equations of the 3-D stability theory. The solution of the 3-D problem is found for the most general case accounting for large deformations and the biaxiality of compressive loads. The characteristic determinants are derived for the first four modes, which are more commonly observed. Special attention is given to the calculation of critical loads for hyperelastic layers described by a simplified version of Mooney's potential, namely the neo-Hookean potential.     

Geometrically non-linear dynamics of composite four-bar mechanisms

This work intends to demonstrate the importance of a geometrically nonlinear cross-sectional analysis of certain composite beam-based four-bar mechanisms in predicting system dynamic characteristics. All component bars of the mechanism are made of fiber reinforced laminates and have thin rectangular cross-sections. They could, in general, be pre-twisted and/or possess initial curvature, either by design or by defect. They are linked to each other by means of revolute joints. We restrict ourselves to linear materials with small strains within each elastic body (beam). Each component of the mechanism is modeled as a beam based on geometrically non-linear 3-D elasticity theory. The component problems are thus split into 2-D analyses of reference beam cross-sections and non-linear 1-D analyses along the three beam reference curves. For the thin rectangular cross-sections considered here, the 2-D cross-sectional non-linearity is also overwhelming. This can be perceived from the fact that such sections constitute a limiting case between thin-walled open and closed sections, thus inviting the non-linear phenomena observed in both. The strong elastic couplings of anisotropic composite laminates complicate the model further. However, a powerful mathematical tool called the Variational Asymptotic Method (VAM) not only enables such a dimensional reduction, but also provides asymptotically correct analytical solutions to the non-linear cross-sectional analysis. Such closed-form solutions are used here in conjunction with numerical techniques for the rest of the problem to predict multi-body dynamic responses more quickly and accurately than would otherwise be possible. The analysis methodology can be viewed as a three-step procedure: First, the cross-sectional properties of each bar of the mechanism is determined analytically based on an asymptotic procedure, starting from Classical Laminated Shell Theory (CLST) and taking advantage of its thin strip geometry. Second, the dynamic response of the non-linear, flexible four-bar mechanism is simulated by treating each bar as a 1-D beam, discretized using finite elements, and employing energy-preserving and -decaying time integration schemes for unconditional stability. Finally, local 3-D deformations and stresses in the entire system are recovered, based on the 1-D responses predicted in the previous step. With the model, tools and procedure in place, we identify and investigate a few four-bar mechanism problems where the cross-sectional non-linearities are significant in predicting better and critical system dynamic characteristics. This is carried out by varying stacking sequences (i.e. the arrangement of ply orientations within a laminate) and material properties, and speculating on the dominating diagonal and coupling terms in the closed-form non-linear beam stiffness matrix. A numerical example is presented which illustrates the importance of 2-D cross-sectional non-linearities and the behavior of the system is also observed by using commercial software (I-DEAS + NASTRAN + ADAMS).     

3-D inviscid self-excited vibrations of a blade row in the last stage turbine

Presented here is a three-dimensional (3-D) nonlinear time-marching method for the aeroelastic behaviour of an oscillating turbine blade row. The approach has been based in the solution of a coupled fluid–structure problem where the aerodynamic and structural dynamic equations are integrated simultaneously in time. This provides the correct formulation of a coupled problem, as the interblade phase angle (IBPA) at which stability (instability) would occur is also a part of solution. The ideal gas flow around multiple interblade passages (with periodicity in the entire annulus) is described by the unsteady Euler equations in conservative form, which are integrated by using the explicit monotonic second-order accurate Godunov–Kolgan finite-volume scheme and a moving hybrid H–O (or H–H) grid. The fluid and the structural equations are solved using the modal superposition method. An aeroelasticity prediction of a turbine blade of 0.765 m is presented. The natural frequencies and modal shapes of the blade were calculated by using 3-D finite element models. The instability regions for five mode shapes and the distribution of the aerodamping coefficient along the blade length were shown for harmonic oscillations with an assumed IBPA. The coupled fluid–structure oscillations in which the IBPA is part of the solution are shown.     

风力机叶片的三维非定常气动特性估算

结合动量-叶素理论、非定常空气动力和动态失速模型来计算风力机叶片的二维非定常气动特性,并在此基础上经过适当修正后考虑三维旋转效应的非定常气动特性。分析比较二维和三维两种计算结果,给出更为合理的计算叶片非定常气动特性的方法。计算结果表明,风力机叶片的三维非定常气动特性计算结果与二维时的计算结果相比有较大改善。     

Non-planar responses of cantilevered pipes conveying fluid with intermediate motion constraints

In this paper, the nonlinear responses of a loosely constrained cantilevered pipe conveying fluid in the context of three-dimensional (3-D) dynamics are investigated. The pipe is allowed to oscillate in two perpendicular principal planes, and hence its 3-D motions are possible. Two types of motion constraints are considered. One type of constraints is the tube support plate (TSP) which comprises a plate with drilled holes for the pipe to pass through. A second type of constraints consists of two parallel bars (TPBs). The restraining force between the pipe and motion constraints is modeled by a smoothened-trilinear spring. In the theoretical analysis, the 3-D version of nonlinear equations is discretized via Galerkin’s method, and the resulting set of equations is solved using a fourth-order Runge–Kutta integration algorithm. The dynamical behaviors of the pipe system for varying flow velocities are presented in the form of bifurcation diagrams, time traces, power spectra diagrams and phase plots. Results show that both types of motion constraints have a significant influence on the dynamic responses of the cantilevered pipe. Compared to previous work dealing with the loosely constrained pipe with motions restricted to a plane, both planar and non-planar oscillations are explored in this 3-D version of pipe system. With increasing flow velocity, it is shown that both periodic and quasi-periodic motions can occur in the system of a cantilever with TPBs constraints. For a cantilevered pipe with TSP constraints, periodic, chaotic, quasi-periodic and sticking behaviors are detected. Of particular interest of this work is that quasi-periodic motions may be induced in the pipe system with either TPBs or TSP constraints, which have not been observed in the 2-D version of the same system. The results obtained in this work highlight the importance of consideration of the non-planar oscillations in cantilevered pipes subjected to loose constraints.     

Non-linear dynamic phenomena in the behaviour of a railway wheelset model

A dicone moving on a pair of cylindrical rails can be considered as a simplified model of a railway wheelset. Taking into account the non-linear friction laws of rolling contact, the equations of motion for this non-linear mechanical system result in a set of differential-algebraic equations. Previous simulations performed with the differential-algebraic solver DASSL, [2], and experiments, [7], indicated non-linear phenomena such as limit-cycles, bifurcations as well as chaotic behaviour. In this paper the non-linear phenomena are investigated in more detail with the aid of special in-house software and the path-following algorithm PATH [10]. We apply Poincaré sections and Poincaré maps to describe the structure of periodic, quasiperiodic and chaotic motions. The analyses show that part of the chaotic behaviour of the non-linear system can be fully understood as a non-linear iterative process. The resulting stretching and folding processes are illustrated by series of Poincaré sections.     

Electromagnetically conducting elastic plates in a magnetic field: modeling and dynamic implications

The basic field equations and boundary conditions necessary for the dynamic approach of electromagnetically conducting flat plates subjected to an external magnetic field are derived.Whereas the structural equations include the geometrical non-linearities of elastic isotropic plates, the electromagnetic equations are used in a linearized form that is obtained from their non-linear counterpart by applying the small disturbance concept. In this context, it was shown that the governing equations involve the bending-stretching coupling arising from both the geometrical non-linear approach of the problem and the inclusion of the Lorentz ponderomotive forces that are reduced to a 2-D plate counterpart.A number of special cases are considered, implications of the external magnetic field on non-linear/linear eigenfrequencies are highlighted, and pertinent conclusions are outlined.     

贴片云纹干涉法

将全息胶片粘贴到测试表面,利用全息干涉两次曝光法,获得高灵敏度二维云纹;曲于实现了制栅和测试过程的合一,胶片与试件分离后处理,本方法具有简易、省时、灵活,试验费用低,适用面广,可用于现场在位测试等优点。     

Prediction of the creep behaviour of polyethylene and molybdenum from stress relaxation experiments

In many applications it is useful to be able to convert observed creep data of a material to corresponding stress relaxation data or vice versa. If the material exhibits non-linear viscoelasticity such a conversion can be rather difficult. In this paper two semi-empirical flow equations, the power law and the exponential law, are used to convert stress relaxation data into corresponding creep behaviour data. These two flow equations are often used to describe non-linear viscoelastic behaviour. The procedure adopted here is based on the assumption that the creep data during the experiment decrease due to an increase in the internal stress level, thus decreasing the effective stress for flow. The conversion method is applied to high density polyethylene and polycrystalline molybdenum at room temperature. In general predictions using the power law are in better agreement with the experimental results than predictions using the exponential formula. The concepts of secondary and ceasing creep are discussed in terms of build-up of internal stress during the creep process.     

Solitary wavepackets from oscillatory non-linear equations with damping

In this paper, an infinite family of solutions describing solitary wave packets with a finite number of nodes is presented. These structures arise from the study of damping in the framework of non-linear ordinary differential equations with oscillatory behaviour. Usually one expects to find effects of this kind in physical systems described by a set of partial differential equations. The standard argument is that the non-linear term acts against the dispersive flux and this balance explains the appearance of solitary waves. Here we show that the non-linear oscillatory behaviour can also balance the effect of damping in special cases. The theory used to discriminate among the various possibilities is plain Painlevé analysis. Several physical applications are briefly discussed.