Experimental analysis of a stochastic model for estimating wind-borne compact debris trajectory in turbulent winds

The impact of flying debris against building envelopes during high winds is a major source of structural damage. For example, damage produced by Hurricanes Katrina and Ike in the United States on the facades of tall buildings, located in urban areas, has been documented. It is therefore of relevance to analyze the vulnerability of tall buildings to debris-induced non-structural damage in the general context of performance-based wind engineering. In order to analyze the random trajectory of debris in highly turbulent winds, a numerical model combined with a probability-based algorithm was recently proposed by the authors (Moghim and Caracoglia, 2013). This model investigates the trajectory of “compact debris”, defined as point-mass objects of negligible mass moments of inertia and for which the aerodynamics is predominantly controlled by the drag force. The model replicates both the inherent randomness in debris properties and the effect of wind shear and atmospheric turbulence to estimate debris trajectory and the likelihood of impact against vertical building facades in a probabilistic setting.This paper describes the comparison between numerical model results and wind tunnel experiments. Tests were carried out in the Northeastern University׳s small scale wind tunnel in both smooth flow and grid-generated turbulent flow. The motion of spheres and cubes, simulating compact debris objects, was investigated in two dimensions (2D) on a vertical plane.The 2D motion of compact objects of various sizes was captured by a high-speed digital camera at different flow speeds. Experimental results showed to be consistent with numerical simulations. They also confirmed that not only mean flow speed but also turbulence features can have a non-negligible effect on the trajectory of compact objects.     

大跨悬索桥三维非平稳耦合抖振分析

基于Priestley(1967)演变功率谱模型，并采用Lin和Yang(1983)的建议，建立了脉动风速的非平稳功率谱模型。依据此模型，采用三维有限元法，建立了大跨桥梁非平稳耦合抖振运动方程。然后，将虚拟激励法和精细时程积分法相结合，建立了求解桥梁三维非平稳耦合抖振运动方程的快速算法。以某大跨悬索桥为例，分析了该桥的非平稳耦合抖振响应，并与平稳耦合抖振响应进行了比较。计算结果表明：随着脉动风速平稳部分持时的增大，非平稳抖振分析结果逐渐收敛于平稳抖振分析结果；但若脉动风速的平稳部分持时较短，非平稳抖振分析结果将低于平稳抖振分析结果。     

On the dynamics of tapping mode atomic force microscope probes

A?mathematical model is developed to investigate the grazing dynamics of tapping mode atomic force microscopes (AFM) subjected to a base harmonic excitation. A?multimode Galerkin approximation is utilized to discretize the nonlinear partial differential equation of motion governing the cantilever response and associated boundary conditions and obtain a set of nonlinearly coupled ordinary differential equations governing the time evolution of the system dynamics. A?comprehensive numerical analysis is performed for a wide range of the excitation amplitude and frequency. The tip oscillations are examined using nonlinear dynamic tools through several examples. The non-smoothness in the tip/sample interaction model is treated rigorously. A?higher-mode Galerkin analysis indicates that period doubling bifurcations and chaotic vibrations are possible in tapping mode microscopy for certain operating parameters. It is also found that a single-mode Galerkin approximation, which accurately predicts the tip nonlinear responses far from the sample, is not adequate for predicting all of the nonlinear phenomena exhibited by an AFM, such as grazing bifurcations, and leads to both quantitative and qualitative errors.     

考虑油膜力作用的船舶主推进系统冲击研究

舰船推进轴系是舰船动力装置的重要组成部分,它的稳定性是舰船生命力的基本保证.本文以舰船主推进轴系为研究对象,考虑了非线性油膜力的作用,在推进轴的非线性油膜力模型中,考虑了油膜涡动效应和挤压效应以及多个轴承油膜力的耦合影响,并考虑了轴承结构本身的刚度.文中采用弹性体动力学方法对推进轴建模,考虑了在外冲击作用下转速和轴承油膜力对于转轴冲击响应的影响,考虑横向运动与转动耦合作用,导出以非线性偏微分方程描述的舰船主推进轴的动力学方程,并用经典理论对其进行离散,最后采用龙格-库塔法进行数值仿真.实例分析指出转速对于转轴冲击响应的影响不可忽略.     

TWO-DIMENSIONAL VORTEX-INDUCED VIBRATION OF CABLE SUSPENSIONS

Resonant responses of suspended elastic cables driven by a steady current are investigated. Phenomenological fluid force models for alternate vortex-shedding are coupled with the nonlinear partial differential equations of cable motion. Decoupled cross-flow and in-line vortex-induced vibrations (VIV) are examined first using linearized and nonlinear cable models. The linearized cable model predicts well the basic characteristics of VIV and the nonlinear cable model captures the hysteresis often observed in experiments. Next, coupled cross-flow and in-line vibrations are evaluated by considering two principal coupling mechanisms: (i) cable structural nonlinearities, and (ii) coupled fluid lift and drag. Attention is focused on a “worst-case” resonant response where the natural frequencies for cable modes in the cross-flow and in-line directions are in the same 1:2 ratio as the excitation frequencies associated with lift and drag. The inclusion of cable structural nonlinearities alone leads to coupled responses that differ qualitatively (i.e., in number and stability of periodic motions) when compared to those of the decoupled model. The inclusion of coupled fluid lift and drag produces non-planar “figure eight” motions of the cable cross-section that exhibit similar characteristics to those previously measured on spring supported cylinders.     

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.     

PROBABILITY MODEL AND SOLUTION ON EARTHQUAKE EFFECTS COMBINATION IN ALONG WIND RESISTANT DESIGN OF TALL-FLEXIBLE BUILDINGS

A model on the earthquake effects combination in wind resistant design of high-rise flexible structures is proposed in accordance with the probability method. Based on the Turkstra criteria, the stochastic characters of wind velocity, earthquake ground acceleration and excitations occurrence probability are taken into account and then the combination of the earthquake effects in structure wind resistant design is analyzed with the convolution approach. The results indicate that as for the tall flexible buildings whose lateral force is governed by wind loading, the maximum lateral loads verification with respect to the wind resistant design combined with earthquake effects may be more unfavorable compared with that in terms of the earthquake resistant design involving wind effects.     

Nonlinear dynamics of a resonant gas sensor

We develop a mathematical model for a resonant gas sensor made up of an microplate electrostatically actuated and attached to the end of a cantilever microbeam. The model considers the microbeam as a continuous medium, the plate as a rigid body, and the electrostatic force as a nonlinear function of the displacement and the voltage applied underneath the microplate. We derive closed-form solutions to the static and eigenvalue problems associated with the microsystem. The Galerkin method is used to discretize the distributed-parameter model and, thus, approximate it by a set of nonlinear ordinary-differential equations that describe the microsystem dynamics. By comparing the exact solution to that associated with the reduced-order model, we show that using the first mode shape alone is sufficient to approximate the static behavior. We employ the Finite Difference Method (FDM) to discretize the orbits of motion and solve the resulting nonlinear algebraic equations for the limit cycles. The stability of these cycles is determined by combining the FDM discretization with Floquet theory. We investigate the basin of attraction of bounded motion for two cases: unforced and damped, and forced and damped systems. In order to detect the lower limit of the forcing at which homoclinic points appear, we conduct a Melnikov analysis. We show the presence of a homoclinic point for a loading case and hence entanglement of the stable and unstable manifolds and non-smoothness of the boundary of the basin of attraction of bounded motion.     

随机风速下高速列车的运行安全可靠性

基于可靠性理论提出了随机风速作用下高速列车风致安全分析的新方法, 这种方法可以对随机风速作用下高速列车的运行安全可靠性进行有效评估.首先基于Cooper理论和谐波叠加法计算随车移动点的脉动风速, 建立随机风速作用下高速列车非定常气动载荷的计算方法, 并通过数值仿真得到气动载荷系数的标准差随侧偏角的变化规律. 然后建立高速列车车辆系统动力学模型, 并对计算模型的正确性进行验证.最后以随机风速、侧力系数、升力系数、侧滚力矩系数、摇头力矩系数和点头力矩系数为基本随机变量, 研究随机风速作用下高速列车的运行安全可靠性和可靠性灵敏度, 给出随机风速作用下高速列车的概率特征风速曲线.研究结果表明:随着车速和风速的增大, 系统的失效概率增大;通过可靠性灵敏度分析发现侧力系数和侧滚力矩系数对高速列车的运行安全影响最大, 应该特别注意这两个参数的变化对高速列车运行安全性的影响;传统确定性方法得到的高速列车的安全域曲线偏于保守, 基于可靠性的方法可得到更合理的安全域曲线.     

Aerodynamic forces on a tall building in a turbulent boundary layer

The model studies were designed to obtain information concerning wind loads on a tall building by placing the model in a turbulent shear flow simulating expected atmospheric boundary-layer winds. Since current design codes are inadequate for predicting all possible motions of tall buildings, it is important that better knowledge of mean and fluctuating loadings and their distributions becomes available. Experiments were conducted to determine the mean and fluctuating forces and twisting moments at several levels over the surface of a model. By determining the effects at several levels simultaneously, it was possible to correlate forces and moments at five levels with one common level. A single model was tested at varying orientations. Tests were also conducted with an identical model placed upstream so that its wake influenced the flow around the instrumented model. Results are presented in terms of distributions of force and moment coefficients and correlations at different levels. The spectral character of the force and moment components is illustrated for one case. Paper was presented at 1977 SESA Spring Meeting held in Dallas, TX on May 15–20.     

具有基础隔振器结构在风载激励下的振动响应分析

具有分段线性特征的随机振动系统能借助于马尔柯夫扩散过程理论精确积分，并由此产生多维分布密度及统计量，本文应用这种协方差分析法确定具有基础隔振器的高层建筑在脉动风和平均风共同作用下结构响应统计量和多维分布密度。     

Deterministic and Stochastic Oscillations of a Flexible Cylinder in Arrays of Static Tubes

This study concentrates on vortex-induced vibrations of one flexiblecylinder in an array of fixed tubes. To describe approximately thedynamics of this system we generalize a previously developed linearsemi-empirical model that includes memory effects. We choose a cubicdamping term to model adequately vortex-induced oscillations and weobtain thus a nonlinear integro-differential equation governing thedisplacement of the cylinder. We use a two-variable expansion to derivecriteria for the appearance of stable, periodic, nonlinear oscillations.This approach predicts the appearance of a limit cycle and gives acriterion for the stability of the oscillations. In a stochasticoscillation model we apply additive white noise to the otherwisedeterministic oscillator model. For small values of the noise intensitywe can approximately solve the Fokker–Planck equation. A comparison ofthis approximation with numerical simulations shows a satisfactorydegree of agreement.     

结构非线性振动及其控制系统的等效力学模型研究

研究了结构非线性振动及其控制系统的等效力学模型建立问题,对结构进行等效线性化处理;对状态变量的观测,仅测量结构的前几个位移。将随机激励和观测噪声等维化处理后,导出了结构非线性振动及其控制系统的等效力学模型。通过等效力学模型的应用研究,得出了多高层结构在风与地震作用下的多维与一维等效力学模型、等效振型力学模型及其线性力学模型,给出了等效力学模型在多高层结构自适应抗风抗震控制中的应用。     

Response analysis of nonlinear vibro-impact system coupled with viscoelastic force under colored noise excitations

This paper is mainly dealing with the stochastic responses of nonlinear vibro-impact (VI) system coupled with viscoelastic force excited by colored noise. By the aid of approximate conversion for the viscoelastic force, the original stochastic VI system is transformed into an equivalent stochastic system without viscoelastic term. Then, the equations of the converted system are simplified by non-smooth transformation, and the stochastic averaging method is employed to solve the above simplified system. A Van der Pol VI oscillator coupled with viscoelastic force is worked out in detail to illustrate the application of the mentioned method, and therewith the analytical solutions fit the numerical simulation results based on the original system. Therefore, the present analytical means of investigating this system is proved to be feasible. Additionally, the exploration of stochastic P-bifurcation by two different ways is also demonstrated in this paper through varying the value of the certain system parameters. Besides, it shows a noteworthy fact that assigning zero or a positive value to the magnitude of viscoelastic force can also lead to the bimodal shape of different degrees in the process of stochastic bifurcations.     

A two-body continuous-discrete filter

In the theory of classical mechanics, the two-body central forcing problem is formulated as a system of the coupled nonlinear second-order deterministic differential equations. The uncertainty introduced by the small, unmodeled stochastic acceleration is not assumed in the particle dynamics. The small, unmodeled stochastic acceleration produces an additional random force on a particle. Estimation algorithms for a two-body dynamics, without introducing the stochastic perturbation, may cause inaccurate estimation of a particle trajectory. Specifically, this paper examines the effect of the stochastic acceleration on the motion of the orbiting particle, and subsequently, the stochastic estimation algorithm is developed by deriving the evolutions of conditional means and conditional variances for estimating the states of the particle-earth system. The theory of the nonlinear filter of this paper is developed using the Kolmogorov forward equation “between the observations" and a functional difference equation for the conditional probability density “at the observation." The effectiveness of the nonlinear filter is examined on the basis of its ability to preserve perturbation effect felt by the orbiting particle and the signal-to-noise ratio. The Kolmogorov forward equation, however, is not appropriate for the numerical simulations, since it is the equation for the evolution of “the conditional probability density." Instead of the Kolmogorov equation, one derives the evolutions for the moments of the state vector, which in our case consists of positions and velocities of the orbiting body. Even these equations are not appropriate for the numerical implementations, since they are not closed in the sense that computing the evolution of a given moment involves the knowledge of higher order moments. Hence, we consider the approximations to these moment evolution equations. This paper makes a connection between classical mechanics, statistical mechanics and the theory of the nonlinear stochastic filtering. The results of this paper will be of use to astrophysicists, engineers and applied mathematicians, who are interested in applications of the nonlinear filtering theory to the problems of celestial and satellite mechanics. Simulation results are introduced to demonstrate the usefulness of an analytic theory developed, in this paper.     

PATH INTEGRAL SOLUTION OF NONLINEAR DYNAMIC BEHAVIOR OF STRUCTURE UNDER WIND EXCITATION

IntroductionThe dynamic behavior of the nonlinear structure under wind excitation has beenobserved very complicated.Taking guyed masts as an example,only a few collapsingaccidents occurred under extreme atmospheric conditions[1],many took place under mild…     

Dynamic flight stability of a model dronefly in vertical flight

The dynamic flight stability of a model dronefly in hovering and upward flight is studied.The method of computational fluid dynamics is used to compute the stability derivatives and the techniques of eigenvalue and eigenvector used to solve the equations of motion.The major finding is as following.Hovering flight of the model dronefly is unstable because of the existence of an unstable longitudinal and an unstable lateral natural mode of motion.Upward flight of the insect is also unstable,and the instability increases as the upward flight speed increases.Inertial force generated by the upward flight velocity coupled with the disturbance in pitching angular velocity is responsible for the enhancement of the instability.     

非平稳非高斯湿下击暴流风雨荷载的模拟

为模拟湿下击暴流风雨荷载,使用时变平均风速和调制的kaimal谱数值模拟具有非平稳非高斯特征的下击暴流脉动风速。从模拟结果可以看出,模拟的脉动风速相关函数明显具有时变的特征,而且瞬时相关函数均与目标函数吻合,因此模拟的脉动风速具有非平稳的目标特性;与此同时,在任意时刻,目标函数也与模拟的脉动风速概率密度函数吻合,表明模拟的脉动风速也具有非高斯的目标特性。因此,所采用的非平稳非高斯过程模拟方法有效可行。针对雨滴冲击荷载,首先对单个雨滴运动分析,推导出单个雨滴的冲击力,进而运用单位体积雨滴谱将单个雨滴冲击扩展到大量雨滴的冲击状态。结果表明,风速对雨滴冲击荷载的影响大,雨荷载变化趋势基本与风速一致,说明湿下击暴流的雨滴冲击作用主要由风速拖拽所致;此外,通过改变降雨强度发现,随着降雨强度增大,雨滴冲击力的变化趋势几乎不变,但峰值增大。     

Nonlinear dynamics of oriented elastic solids

Based on a continuum model for oriented elastic solids the set of nonlinear dispersive equations derived in Part I of this work allows one to investigate the nonlinear wave propagation of the soliton type. The equations govern the coupled rotation-displacement motions in connection with the linear elastic behavior and large-amplitude rotations of the director field. In the one-dimensional version of the equations and for two simple configurations an exhaustive study of solitons is presented. We show that the transverse and/or longitudinal elastic displacements are coupled to the rotational motion so that solitons, jointly in the rotation of the director and the elastic deformations, are exhibited. These solitons are solutions of a system of linear wave equations for the elastic displacements which are nonlinearly coupled to a sine-Gordon equation for the rotational motion. For each configuration, the solutions are numerically illustrated and the energy of the solitions is calculated. Finally, some applications of the continuum model and the related nonlinear dynamics to several physical situations are given and additional more complex problems are also evoked by way of conclusion.