A linear stability approach to vortex-induced vibrations and waves

The motion induced by vortex shedding on slender flexible structures subjected to cross-flow is considered here. This phenomenon of vortex-induced vibration (VIV) is analysed by considering the linear stability of a coupled system that includes the structure dynamics and the wake dynamics. The latter is modelled by a continuum of wake oscillators, distributed along the span of the structure. In the case of uniform flows over a straight tensioned cable, VIV are found to arise as an instability related to the merging of two waves. In the case of a cable of finite length, the selection of modes that experience lock-in with the wake is found using the same stability argument. In non-uniform flows, several unstable wave systems are identified, and competition between them is discussed. Comparison is then made with existing experimental and computational data of VIV of slender structures under uniform and non-uniform flows. Phenomena previously identified in these systems, such as mode switching when the flow velocity is varied, time sharing of the response between two frequencies, or the coexistence of several regions of VIV with different dynamics in the same structure, are discussed with the help of the proposed model.     

Local slamming impact of sandwich composite hulls

We develop a hydroelastic model based on a {3, 2}-order sandwich composite panel theory and Wagner’s water impact theory for investigating the fluid–structure interaction during the slamming process. The sandwich panel theory incorporates the transverse shear and the transverse normal deformations of the core, while the face sheets are modeled with the Kirchhoff plate theory. The structural model has been validated with the general purpose finite element code ABAQUS^®. The hydrodynamic model, based on Wagner’s theory, considers hull’s elastic deformations. A numerical procedure to solve the nonlinear system of governing equations, from which both the fluid’s and the structure’s deformations can be simultaneously computed, has been developed and verified. The hydroelastic effect on hull’s deformations and the unsteady slamming load have been delineated. This work advances the state of the art of analyzing hydroelastic deformations of composite hulls subjected to slamming impact.     

NECESSARY CONDITIONS FOR SEPARATION OF STRAIGHT BAR FROM HORIZONTAL PLANE WITHOUT SLIDING

该文研究的是起初竖直静置于水平面上的非均质细直杆的倾倒过程,利用质心运动定理、机械能守恒定律和定轴转动定理导出了直杆在下端不滑动情况下所受支持力的表达式,据此指出了直杆能够脱离水平面的必要条件同时包括两部分：对直杆质量分布的要求;对直杆与水平面之间的静摩擦因数的要求。     

Multi-mode interactions in vortex-induced vibrations of flexible curved/straight structures with geometric nonlinearities

A general low-order fluid–structure interaction model capable of evaluating the multi-mode interactions in vortex-induced vibrations of flexible curved/straight structures is presented. Cross-flow motions due to unsteady lift forces of inclined sagged cables and tensioned beams in uniform currents are investigated. In contrast to a linear equation governing the transverse motion of straight beams or cables typically considered in the literature, coupled horizontal/vertical (axial/transverse) displacements and geometric nonlinearities of curved cable (straight beam) are accounted for. A distributed nonlinear wake oscillator is considered in the approximation of space–time varying hydrodynamics. This semi-empirical fluid force model in general depends on the mass-damping parameter and has further been modified to capture both the effects of varying initial curvatures of the inclined cylinder and the Reynolds number. Numerical simulations are performed in the case of varying flow velocities and parametric results highlight several meaningful aspects of vortex-induced vibrations of long flexible cylinders. These comprise multi-mode lock-in, sharing, switching and interaction features in the space and time domains, the estimated maximum modal and total amplitudes, the resonant nonlinear modes of flexible cylinders and their space–time modifications, and the influence of fluid/structure parameters. A shortcoming of single-mode or linear structural model is underlined. Some quantitative and qualitative comparisons of numerical/experimental results are discussed to demonstrate the validity and required improvement of the proposed modelling and analysis predictions.     

A theoretical computational model of a plate in hypersonic flow

A theoretical model of an elastic panel in hypersonic flow is derived to be used for design and analysis. The nonlinear von Kármán plate equations are coupled with 1st order Piston Theory and linearized at the nonlinear steady-state deformation due to static pressure differential and thermal loads. Eigenvalue analysis is applied to determine the system’s stability, natural frequencies and mode shapes. Numerically time marching the equations provides transient response prediction which can be used to estimate limit cycle oscillation amplitude, frequency and time to onset. The model’s predictive capability is assessed by comparison to an experiment conducted at a free stream flow of Mach 6. Good agreement is shown between the theoretical and experimental natural frequencies and mode shapes of the fluid–structure system. Stability analysis is performed using linear and nonlinear methods to plot stability, flutter and buckling zones on a free stream static pressure vs temperature differential plane.     

二维弹性结构入水冲击过程中的流固耦合效应

描述了一个研究弹性结构入水冲击过程中水弹性效应的数值方法,在弹性结构入水冲击过程中,流体域作用在结构上的水动力载荷由边界元法获得,而结构的弹性动力响应则由有限元方法求解,通过线性给离散Ｂｅｒｎｏｕｌｌｉ方程将有限元方程和边界元方程耦合到一起,从而获得了求解流场和结构动力响应的相互耦合的运动方程。在数值考虑了自由表面的非线性边界条件,通过引入射流单元以及最大射流厚度,较好地处理了冲击引起的射流问题。     

LOW-FREQUENCY RESPONSES OF NONLINEARLY MOORED VESSELS IN RANDOM WAVES: COUPLED SURGE,PITCH AND HEAVE MOTIONS

The responses of a multi-degree-of-freedom model of a moored vessel are analysed, accounting for the hydroelastic interaction between the nonlinear wave hydrodynamics and the nonlinear mooring stiffness. A two-scale perturbation method developed by Sarkar & Eatock Taylor to determine low-frequency hydrodynamic forces on a single-degree-of-freedom model of a nonlinearly moored vessel has been extended to analyse the nonlinear multi-degree-of-freedom dynamics of the system. Surge, heave and pitch motions are considered. The perturbation equations of successive orders are derived. To illustrate the approach, semi-analytical expressions for the higher-order hydrodynamic force components have been obtained for a truncated circular cylinder in finite water depth. In addition to conventional quadratic force transfer functions, a new type of higher-order force transfer function is introduced. This is used to characterize the hydrodynamic forces on the vessel which arise due to nonlinearity of the mooring stiffness. These are a type of radiation force, generated by the nonlinear interaction of the fluid–structure coupled system. Based on a Volterra series model, the power spectral densities of the new higher-order forces are then derived for the case of Gaussian random seas. It is shown that the additional response arising due to nonlinear dynamics of the mooring system can significantly contribute to low-frequency drift forces and responses of the vessel. Unlike conventional non-Gaussian second-order forces which are quadratic transformations of a Gaussian random process, the new higher-order forces arising due to the nonlinear mooring stiffness are polynomials of a Gaussian random process (up to fourth order for a Duffing oscillator model). This may significantly influence the extreme responses.     

A coupled BE–FE study for evaluation of seismically isolated cylindrical liquid storage tanks considering fluid–structure interaction

The effects of base-isolation on the seismic response of cylindrical vertical flexible liquid storage tanks subjected to horizontal seismic ground motion are presented in this paper. The whole system consists of two main parts: the base-isolation component, and the fluid–structure interaction subsystem. Dynamic analysis of liquid storage tank is achieved through the use of finite shell elements for the structure and internal boundary elements for the liquid region. The boundary element equations are employed to obtain an equivalent liquid mass matrix which is then coupled with the shell structure mass matrix, resulting in the coupled equations of motion. Finally, the coupled equations of motion are connected with the base-isolation system to observe the whole system behavior. A bilinear hysteretic element is used to illustrate the base-isolation system. The analysis is performed in the time domain in which, hydrodynamic interaction is taken into account. It is shown that the seismic response of the isolated tank could be significantly reduced compared with the fixed-foundation tanks. Parametric studies are carried out to study the effects of different system parameters on the effectiveness of the base-isolation system. These parameters are the tank geometry aspect ratio (height to radius), the flexibility of the isolation system, the liquid surface displacement variations, and the tank wall flexibility. It is observed that the seismic isolation is more effective in slender tanks in comparison with broad tanks. Furthermore, it is shown that the isolation efficiency is partially more significant in rigid tanks. It is also noticeable that flexible isolators considerably reduce the seismic response in comparison with stiff isolators. Despite the foregoing advantages, liquid surface displacement increases due to seismic isolation, especially in slender tanks.     

Study of static and dynamic stability of flexible rods in a geometrically nonlinear statement

We study static and dynamic stability problems for a thin flexible rod subjected to axial compression with the geometric nonlinearity explicitly taken into account. In the case of static action of a force, the critical load and the bending shapes of the rod were determined by Euler. Lavrent’ev and Ishlinsky discovered that, in the case of rod dynamic loading significantly greater than the Euler static critical load, there arise buckling modes with a large number of waves in the longitudinal direction. Lavrent’ev and Ishlinsky referred to the first loading threshold discovered by Euler as the static threshold, and the subsequent ones were called dynamic thresholds; they can be attained under impact loading if the pulse growth time is less than the system relaxation time. Later, the buckling mechanism in this case and the arising parametric resonance were studied in detail by Academician Morozov and his colleagues.In this paper, we complete and develop the approach to studying dynamic rod systems suggested by Morozov; in particular, we construct exact and approximate analytic solutions by using a system of special functions generalizing the Jacobi elliptic functions. We obtain approximate analytic solutions of the nonlinear dynamic problem of flexible rod deformation under longitudinal loading with regard to the boundary conditions and show that the analytic solution of static rod system stability problems in a geometrically nonlinear statement permits exactly determining all possible shapes of the bent rod and the complete system of buckling thresholds. The study of approximate analytic solutions of dynamic problems of nonlinear vibrations of rod systems loaded by lumped forces after buckling in the deformed state allows one to determine the vibration frequencies and then the parametric resonance thresholds.     

Coupled frequencies of a hydroelastic viscous liquid system

The coupled frequencies of a hydroelastic system consisting of an elastic shell and a viscous liquid layer with a free surface have been treated. The system exhibits no z-dependency and may be either an annular liquid layer around an elastic center shell or a liquid layer inside an elastic container. The first case has been evaluated numerically, where the influence of the liquid surface tension parameter, the elasticity parameter of the shell and the thickness of the layer have been determined. In contrast to the hydroelastic system with an ideal liquid, the system with viscous liquid exhibits instability of the liquid surface as well as the shell.     

Non-Newtonian flows in channels. Workable pressure gradient-volume flow rate formulae

Summary Some flows of a non-Newtonian incompressible liquid in stationary curved, and in rotating straight, channels under the influence of a uniform pressure gradient are investigated; the viscous and elastic properties of the liquid are defined in terms of a spectrum of relaxation times. By using a simple approximate secondary flow mechanism, pressure gradient-volume flow rate relations are derived in forms suitable for practical purposes.     

超细长弹性杆动力学仿真的曲面处理算法

超细长弹性杆动力学研究在DNA的平衡、稳定性等问题的研究中有重要的应用。为了便于超细长弹性杆动力学研究中数值结果图形后处理以及研究表面接触等问题的需要,需要建立弹性杆的表面模型和相应算法。本文利用Kirchhoff弹性杆模型的动力学比拟技巧,建立了描述超细长弹性杆曲面的常微分/积分方程组,利用Adames方法和递推方法设计了方程的数值解法,并给出了超长弹性杆的数值仿真结果的图形处理的计算实例。     

Influence of the connecting condition on the dynamic buckling of longitudinal impact for an elastic rod

The stress wave propagation law and dynamic buckling critical velocity are formulated and solved by considering a general axial connecting boundary for a slender elastic straight rod impacted by a rigid body. The influence of connecting stiffness on the critical velocity is investigated with varied impactor mass and buckling time. The influences of rod length and rod mass on the critical velocity are also discussed. It is found that greater connecting stiffness leads to larger stress amplitude, and further results in lower critical velocity. It is particularly noteworthy that when the connecting stiffness is less than a certain value,dynamic buckling only occurs before stress wave reflects off the connecting end. It is also shown that longer rod with larger slenderness ratio is easier to buckle, and the critical velocity for a larger-mass rod is higher than that for a lighter rod with the same geometry.     

Analytical effective width equations for limit state design of thin plates under non-homogeneous in-plane loading

The effective width concept has been widely used in engineering practice for the computation of ultimate strength of slender members. Many design codes employ this concept in order to compensate for the stiffness reduction in the post-buckling state. Extensive work was done to develop effective width equations for plates under uniform compression, while little attention has been given for plates under non-homogeneous in-plane loading. North American, British and European design codes provide only expressions for the computation of the elastic buckling loads for plates under this load combination, while the effective width calculation is based on the uniformly compressed plates. It will be shown that due to the non-uniformity of the applied load, the stress characteristics in the post-buckling state are different from the uniform compression case, thus requiring special treatments. The paper presents analytical closed form expressions for the computation of effective width of thin plates under non-homogeneous in-plane loading. The longitudinal edges are assumed to be straight and free to translate in the plane of the plate. The proposed expressions are very useful for limit state design of slender I-sections of beam columns or channel sections under this general type of loading. They enable the designers to compute the effective width of the section with the aid of simple formulas that, for design purposes, are suitable for hand-calculation and avoid the cost and effort that any numerical non-linear analysis may require.     

圆柱涡激振动的结构-尾流振子耦合模型研究

建立了一个新的结构-尾流振子耦合模型. 流场近尾迹动力学特征被模化为非线性阻尼振子,采用van der Pol方程描述. 以控制体中结构与近尾迹流体间受力互为反作用关系来实现流固耦合. 采用该模型进行了二维结构涡激振动计算,得到了合理的振幅随来流流速的变化规律和共振幅值,并正确地预计了共振振幅值$A_{\max}^\ast$随着质量阻尼参数$\left( {m^\ast + C_A } \right)\zeta$的变化规律,给出了预测$A_{\max }^\ast$值的拟合公式. 采用该模型计算了三维柔性结构在均匀来流和简谐波形来流作用下的VIV响应. 结构在均匀来流作用下振动呈现由驻波向行波的变化过程, 并最后稳定为行波振动形态.在简谐波形来流作用下,结构呈现混合振动形态,幅值随时间呈周期变化.      

Geometric nonlinear formulation for curved beams with varying curvature

Based on exact Green strain of spatial curved beam, the nonlinear strain-displacement relation for plane curved beam with varying curvature is derived. Instead of using the previous straight beam elements, curved beam elements are used to approximate the curved beam with varying curvature. Based on virtual work principle, rigid-flexible coupling dynamic equations are obtained. Physical experiments were carried out to capture the large overall motion and the strain of curved beam to verify the present rigid-flexible coupling formulation for curved beam based on curved beam element. Numerical results obtained from simulations were compared with those results from the physical experiments. In order to illustrate the effectiveness of the curved beam element methodology, the simulation results of present curved beam elements are compared with those obtained by previous straight beam elements. The dynamic behavior of a slider-crank mechanism with an initially curved elastic connecting rod is investigated. The advantage of employing generalized-α method is pointed out and the special nonlinear dynamic characteristics of the curved beam are concluded.     

Numerical investigation of the effect of boundary conditions on hydroelastic stability of two parallel plates interacting with a layer of ideal flowing fluid

The paper studies the hydroelastic stability of two parallel identical rectangular plates interacting with a flowing fluid confined between them. General equations describing the behavior of ideal compressible liquid in the case of small perturbations are written in terms of the perturbation velocity potential and transformed using the Bubnov–Galerkin method. The small deformations of elastic plates are defined using the first-order shear deformation plate theory. A mathematical formulation of the dynamic problem for elastic structures is developed using the variational principle of virtual displacements, which takes into account the work done by the inertial forces and hydrodynamic pressure. The numerical solution of the problem is carried out in three-dimensional formulation by means of the finite element method. A stability criterion is based on the analysis of complex eigenvalues of the coupled system of equations obtained for different values of flow velocity. The existence of different types of instability has been shown depending on the combinations of the kinematic boundary conditions defined at the edges of both plates. We considered both the symmetric and asymmetric types of clamping. It has been found that the dependence of the lowest eigenfrequency of two parallel plates on the height of quiescent fluid is nonmonotonic with a pronounced peak. At the same time, critical velocities of instability change insignificantly if the distance between plates is greater than half of the maximum linear dimensions of the structure. It should be noted that the critical velocities of divergence increase monotonically with growth of the height of the fluid layer, but critical velocities for the onset of flutter instability have sharp jumps. The cause of these jumps is a change in the mode shapes at which the system loses stability.     

A coupled WC-TL SPH method for simulation of hydroelastic problems

A coupled weakly compressible (WC) and total Lagrangian (TL) smoothed particle hydrodynamics (SPH) method is developed for simulating hydroelastic problems. The fluid phase is simulated using WCSPH method, while the structural dynamics are solved using TLSPH method. Fluid and solid components of the method are validated separately. A sloshing water tank problem is solved to test the WCSPH method while oscillation of a thin plate and large deformation of a cantilever beam are simulated to test the TLSPH method. After validating each component, the coupled WC-TL SPH scheme is used to simulate two benchmark hydroelastic problems. The first test case shows the evolution of water column with an elastic boundary gate, and the second one investigates the breaking water column impact on elastic structures. The agreement between WC-TL SPH results and literature data shows the ability of the proposed method in simulating hydroelastic phenomena.