缓坡方程的推广

为了描述水波和强烈的环境流在非平整海底上的相互作用,运用无旋运动的Lagrangian变分原理,对经典的Berkhoff缓坡方程进行了改进。假定水流沿水深方向基本上保持均匀性,这正如潮流运动的特征。海底地形由慢变、快变两个分量叠加构成;慢变分量满足缓坡逼近假定,快变分量的波长与表面波波长为同一量级,但其振幅小于表面波的振幅。在以上假定条件下,得到了适用于非平整海底的推广型浅水方程和应用性更加广泛的波－流－非平整海底相互作用的一般缓坡方程,并且从理论上证明一般缓坡方程包含了以下3种著名的缓坡型方程：经典的Berkhoff缓坡方程;波－流相互作用的Kirby缓坡方程、Dingemans关于沙纹海底的缓坡方程。最后,通过与Bragg反射实验数据的比较,表明该模型可以准确地反映快变海底的典型地貌特征。     

Evaluation of well‐balanced bore‐capturing schemes for 2D wetting and drying processes

We consider numerical solutions of the two‐dimensional non‐linear shallow water equations with a bed slope source term. These equations are well‐suited for the study of many geophysical phenomena, including coastal engineering where wetting and drying processes are commonly observed. To accurately describe the evolution of moving shorelines over strongly varying topography, we first investigate two well‐balanced methods of Godunov‐type, relying on the resolution of non‐homogeneous Riemann problems. But even if these schemes were previously proved to be efficient in many simulations involving occurrences of dry zones, they fail to compute accurately moving shorelines. From this, we investigate a new model, called SURF_WB, especially designed for the simulation of wave transformations over strongly varying topography. This model relies on a recent reconstruction method for the treatment of the bed‐slope source term and is able to handle strong variations of topography and to preserve the steady states at rest. In addition, the use of the recent VFRoe‐ncv Riemann solver leads to a robust treatment of wetting and drying phenomena. An adapted ‘second order’ reconstruction generates accurate bore‐capturing abilities.This scheme is validated against several analytical solutions, involving varying topography, time dependent moving shorelines and convergences toward steady states. This model should have an impact in the prediction of 2D moving shorelines over strongly irregular topography. Copyright © 2006 John Wiley & Sons, Ltd.     

Extended Mild-Slope Equation

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On the use of the theory of dynamical systems for transient problems

This paper is a preliminary work to address the problem of dynamical systems with parameters varying in time. An idea to predict their behavior is proposed. These systems are called transient systems, and are distinguished from steady systems in which parameters are constant. In particular, in steady systems the excitation is either constant (e.g., nought) or periodic with amplitude, frequency, and phase angle which do not vary in time. We apply our method to systems, which are subjected to a transient excitation that is neither constant nor periodic. The effect of switching-off and full-transient forces is investigated. The former can be representative of switching-off procedures in machines; the latter can represent earthquake vibrations, wind gusts, etc., acting on a mechanical system. This class of transient systems can be seen as the evolution of an ordinary steady system into another ordinary steady system, for both of which the classical theory of dynamical systems holds. The evolution from a steady system to the other is driven by a transient force, which is regarded as a map between the two steady systems.     

非平整运动海底上n层流动中波浪传播的Hamilton逼近

在海洋水域，界面波对大尺度变化流的作用是一种典型的分层流动现象．考虑一不可压缩、无黏的分层势流运动，建立了一个在非平整运动海底上的n层流体演化系统，并对其进行了Hamilton描述．每层流体具有各自的常密度、均匀流水平速度，其厚度由未扰动和扰动部分构成．相对于顶层流体的自由表面，刚性、运动的海底具有一般地形变化特征．在明确指出n层流体运动的控制方程和各层交界面上的运动学、动力学边界条件(包含各层交界面上张力效应)后，对该分层流动力系统进行了Hamilton构造，即给出其正则方程和其下述的正则变量：各交界面位移和各交界面上的动量势密度差。     

谐波齿轮系统的快慢振荡机制研究

谐波齿轮减速器是一种新型的传动装置, 因其具有诸多的优点, 因而得到了广泛应用. 谐波齿轮减速器涉及不同振荡尺度之间的耦合作用, 这通常会诱发复杂的快慢振荡, 严重影响了谐波齿轮系统的正常工作. 本文考虑涉及扭转刚度非线性因素的谐波齿轮系统, 旨在研究系统的快慢动力学, 揭示新型的快慢振荡机制. 首先, 构建了非线性扭转刚度下的谐波齿轮系统的快慢动力学模型. 然后, 通过改变扭转刚度系数, 得到了系统从常规振荡向快慢振荡的转迁过程. 接着, 简要地论述了有关快慢系统的基础理论. 在此基础上, 采用快慢分析法研究了快子系统的动力学特性, 揭示了快慢振荡的产生机制. 研究表明, 当系统参数改变时, 快子系统的平衡点曲线并未发生失稳或分岔; 然而, 在某一点附近, 平衡点曲线能够产生急剧量变, 其特征是平衡点在局部小范围内可以在正坐标值与负坐标值之间快速转迁. 在此基础上, 揭示了一种诱发快慢振荡的新型动力学机制, 比较了这种诱发机制与其他相关机制之间的区别. 本文丰富了系统通向快慢振荡的路径, 为实际谐波齿轮传动系统中的快慢振荡机理与控制研究提供参考.      

A coupled-mode model for water wave scattering by vertically sheared currents in variable bathymetry regions

A coupled-mode model is developed for treating the wave–current–seabed interaction problem, with application to wave scattering by non-homogeneous, sheared current with linear vertical velocity profile, over general bottom topography. The wave potential is represented by a series of local vertical modes containing the propagating and evanescent modes, plus additional terms accounting for the satisfaction of the boundary conditions. Using the above representation, in conjunction with a variational principle, a coupled system of differential equations on the horizontal plane is derived, with respect to the unknown modal amplitudes. In the case of small-amplitude waves, a linearized version of the above coupled-mode system is obtained, extending previous analysis by Belibassakis et al. (2011) to the propagation of water waves over variable bathymetry regions in the presence of vertically sheared currents. Keeping only the propagating mode in the vertical expansion of the wave potential, the present system reduces to a one-equation model, that is shown to extend known mild-slope mild vertical shear equation concerning wave–current interaction over slowly varying topography. After additional simplifications, the latter model is shown to be compatible with the extended mild-slope mild-shear equation by Touboul et al. (2016). Results are presented for various representative test cases demonstrating the usefulness of the present coupled mode system and the importance of various terms in the modal expansion, and compared against experimental data collected in wave flume validating the present method. The analytical structure of the present system facilitates extensions to model non-linear effects and applications concerning wave scattering by inhomogeneous currents in coastal regions with general 3D bottom topography.     

一类平面自治非线性时滞控制系统的多稳态和混沌

众所周知,平面自治系统即使具有光滑非线性存在,系统也不会出现复杂的动力学行为。本文研究这样的系统存在时滞时,时滞量对系统的动力学行为的影响。通过对一个平面自治非线性系统引入时滞反馈,得到数学模型。利用泛函分析和平均法建立系统平衡态随时滞量变化的失稳机理,研究表明：时滞量平面自治系统动力学行为的影响是本质的．时滞量不但可以使系统出现Hopf分岔,产生周期振动。而且还可以使系统出现多稳态的周期运动或周期吸引子,这些共存的吸引子相碰是导致系统复杂的动力学行为,包括概周期和混沌运动。     

求非线性动力系统周期解的切比雪夫多项式法

周期运动是一种在客观世界中普遍存在的运动形式,它与混沌运动之间存在十分密切的关系,因而具有很重要的研究价值。利用切比雪夫多项式的若干良好性质,对自治非线性动力系统进行分析,将状态矢量在主周期上展开为切比雪夫多项式的形式,从而将原问题转变为非线性代数方程组的求解问题,得出一种可以方便、迅速地获得周期轨道近似多项式表达式的方法。此方法不依赖于小参数假设,可以用于分析强非线性问题,而且对参数激励系统同样有效。在计算机条件允许时,对高维系统也能迅速、精确地得到其周期轨道的近似多项式表达式。以三维Rossler系统和五维非线性磁浮转子系统周期轨道的计算为例,通过与四阶Runge-Kutta数值积分结果比较,说明此方法的精确、高效性。     

Liquid sloshing in a horizontally forced vessel with bottom topography

This paper presents a numerical study of the free-surface evolution for inviscid, incompressible, irrotational, horizontally forced sloshing in a two-dimensional rectangular vessel with an inhomogeneous bottom topography. The numerical scheme uses a time-dependent conformal mapping to map the physical fluid domain to a rectangle in the computational domain with a time-dependent aspect ratio Q(t), known as the conformal modulus. The advantage of this approach over conventional potential flow solvers is the solution automatically satisfies Laplace's equation for all time, hence only the integration of the two free-surface boundary conditions is required. This makes the scheme computationally fast, and as grid points are required only along the free-surface, high resolution simulations can be performed which allows for simulations for mean fluid depths close to the shallow water water regime. The scheme is robust and can simulate both resonate and non-resonate cases, where in the former, the large amplitude waves are well predicted.Results of nonlinear simulations are presented in the case of non-breaking waves for both an asymmetrical ‘step’ and a symmetric ‘hump’ bottom topography. The natural free-sloshing mode frequencies are compared with the small topography asymptotic results of Faltinsen and Timokha (2009) (Sloshing, Cambridge University Press (Cambridge)), and are found to be lower than this asymptotic prediction for moderate and large topography magnitudes. For forced periodic oscillations it is shown that the hump profile is the most effective topography for minimizing the nonlinear response of the fluid, and hence this topography would reduce the stresses on the vessel walls generated by the fluid. Results also show that varying the width of the step or hump has a less significant effect than varying its magnitude.     

涡旋运动与混沌

本文综述了利用非线性动力学方法研究涡旋运动的最新进展。通过典型例子的分析,讨论了该领域的研究方法、主要困难和存在问题,并提出今后的可能研究方向。      

Numerical and experimental analysis of the bi-stable state for frictional continuous system

Unstable friction-induced vibrations are considered an annoying problem in several fields of engineering. Although several theoretical analyses have suggested that friction-excited dynamical systems may experience sub-critical bifurcations, and show multiple coexisting stable solutions, these phenomena need to be proved experimentally and on continuous systems. The present work aims to partially fill this gap. The dynamical response of a continuous system subjected to frictional excitation is investigated. The frictional system is constituted of a 3D printed oscillator, obtained by additive manufacturing that slides against a disc rotating at a prescribed velocity. Both a finite element model and an experimental setup has been developed. It is shown both numerically and experimentally that in a certain range of the imposed sliding velocity the oscillator has two stable states, i.e. steady sliding and stick–slip oscillations. Furthermore, it is possible to jump from one state to the other by introducing an external perturbation. A parametric analysis is also presented, with respect to the main parameters influencing the nonlinear dynamic response, to determine the interval of sliding velocity where the oscillator presents the two stable solutions, i.e. steady sliding and stick–slip limit cycle.

     

Optimization of heat‐transfer rate into time‐periodic two‐dimensional Stokes flows

The periodic boundary displacement protocol leading to the optimum wall‐to‐fluid heat‐transfer rate, or to the most efficient mixing rate, in 2‐D annular Stokes flows is determined by calculating the steady periodic velocity and temperature fields. To obtain the steady periodic state one usually solves the dynamical system obtained after the spatial coordinates have been discretized. Here, we calculate the steady periodic state using an implicit method based on the discretization of the time coordinate over a period and the asymptotic regime is enforced by the periodicity condition in the computed temperature field. The obtained system of equations is solved using a Newton‐type iterative algorithm with invariant Jacobian. At each iteration step, the sparse linearized system is solved using a multi‐grid algebraic technique of rapid convergence. From a computational point of view and for the problem considered here, this method is an order of magnitude faster than the one based on a spatial discretization. Copyright © 2006 John Wiley & Sons, Ltd.     

Analytic evaluation of periodic responses of a forced nonlinear oscillator

The periodic responses of a strongly nonlinear, single-degree-of-freedom forced oscillator with weak excitation and damping are examined. The presented methodology is based on a regular perturbation expansion, whose first term is the solution of the unforced, and undamped nonlinear problem. Higher order approximations are computed by explicitly solving linear differential equations possessing a periodically varying coefficient. The general theory is used for studying the periodic steady state motions of the periodically forced system. Moreover, it is shown that the presented analysis can be used to analytically study the orbital stability of the identified steady state motions. The proposed method can also be used for studying periodic responses due to nonperiodic transient forces, provided that these responses are close to the O(1) periodic generating solution.     

结构动力学模型中阻尼特性参数的修正

将结构分析计算的稳态位移响应和相应实测数据的差异映射为具有能量意义的误差范数，通过最小化该误差范数来修正与结构系统有关的阻尼特性参数．通过计算的或实测的频响函数获得稳态响应数据，使用的幅值在理论上完全包含了所有的复模态信息，参数值正过程采用迭代的方式．给出仿真算例，并对实验数据有噪声和无噪声两种情况进行了研究，结果表明方法合理且可行．     

迭代图胞映射方法的快速生成实现

在迭代图胞映射方法的框架下,基于摄动微分多项式的思想讨论了常微分方程的快速求解,将所得结果与迭代图胞映射方法有机结合,有效地解决了迭代图胞映射动力系统的快速生成问题,克服了微分方程动力系统生成迭代图胞映射系统过程中耗时较多、效率低下的不足,大大提高了计算效率。通过对典型非线性系统——杜芬方程的应用分析,证实了该方法的有...     

A NUMERICAL PROCEDURE FOR PREDICTING MULTIPLE SOLUTIONS OF A SPHERICAL TAYLOR–COUETTE FLOW

A new numerical procedure for predicting multiple solutions of Taylor vortices in a spherical gap is presented. The steady incompressible Navier–Stokes equations in primitive variables are solved by a finite- difference method using a matrix preconditioning technique. Routes leading to multiple flow states are designed heuristically by imposing symmetric properties. Both symmetric and asymmetric solutions can be predicted in a deterministic way. The current procedure gives very fast convergence rate to the desired flow modes. This procedure provides an alternative way of finding all possible stable steady axisymmetric flow modes.     

Steady State Passive Nonlinear Energy Pumping in Coupled Oscillators: Theoretical and Experimental Results

We study theoretically and experimentally the effect that anonlinear energy sink (NES) has on the steady state dynamics of a weaklycoupled system. The NES possesses essentially nonlinear(nonlinearizable) stiffness nonlinearity of the third degree. We findthat, in contrast to the classical linear vibration absorber, the NES iscapable of absorbing steady state vibration energy from the linearoscillator over a relatively broad frequency range. This results inlocalization of the steady state vibration in the NES, away from thedirectly forced subsystem. For a forward frequency sweep the localizedbranch of steady state motions is suddenly eliminated by a jump to alinearized low-amplitude motion, whereas, for a backward frequency sweepa reverse jump occurs. The difference in the frequencies of the twojumps introduces a nonlinear hysteresis loop. This work extends to thesteady state case of earlier transient passive energy pumping results.The notion of passively transferring vibration energy to an a prioridetermined NES, weakly attached to a main structure, is novel. The useof nonlinear energy sinks for passively absorbing energy from a linearmain structure can form the basis of relatively simple and modularvibration and shock isolation designs of mechanical systems.     

破碎岩体非等温渗流的非线性动力学研究

分别从固体及流体导热的能量方程出发,导出破碎岩体非等温渗流的能量本构方程, 结合渗流的连续性方程、运动方程、状态方程等建立了破碎岩体非等温渗流的一维非线性动力学方程组;结合Mathcad软件计算得到了系统的无量纲化平衡态, 利用逐次亚松弛迭代法分析了对应于不同参数时平衡态的稳定性;指出非等温渗流系统存在鞍结分岔及折叠突变, 与等温渗流相比, 考虑温度场的破碎岩体渗流动力系统更容易发生渗流突变.      

A robust and well‐balanced scheme for the 2D Saint‐Venant system on unstructured meshes with friction source term

In the following lines, we propose a numerical scheme for the shallow‐water system supplemented by topography and friction source terms, in a 2D unstructured context. This work proposes an improved version of the well‐balanced and robust numerical model recently introduced by Duran et al. (J. Comp. Phys., 235 , 565–586, 2013) for the pre‐balanced shallow‐water equations, accounting for varying topography. The present work aims at relaxing the robustness condition and includes a friction term. To this purpose, the scheme is modified using a recent method, entirely based on a modified Riemann solver. This approach preserves the robustness and well‐balanced properties of the original scheme and prevents unstable computations in the presence of low water depths. A series of numerical experiments are devoted to highlighting the performances of the resulting scheme. Simulations involving dry areas, complex geometry and topography are proposed to validate the stability of the numerical model in the neighbourhood of wet/dry transitions. Copyright © 2015 John Wiley & Sons, Ltd.