Nonlinear aeroelastic response of slender wings based on Wagner function

This paper presents a method for nonlinear aeroelastic analysis of Human Powered Aircraft (HPA) wings. In this type of aircraft there is a long, highly flexible wing. Wing flexibility, coupled with long wing span can lead to large deflections during normal flight operation; therefore, a wing in vertical and torsional motion using the second-order form of nonlinear general flexible Euler–Bernoulli beam equations is used for structural modeling. Unsteady linear aerodynamic theory based on Wagner function is used for determination of aerodynamic loading on the wing. Combining these two types of formulations yields the nonlinear integro-differentials aeroelastic equations. Using the Galerkin's method and modes summation technique, the governing equations will be solved by introducing an iterative numerical method to predict the aeroelastic response of the problem. The obtained results for a test case are compared with those of linear study which shows good agreement for speeds less than the flutter speed, but the nonlinear model shows limit cycle oscillations for the wing beyond the flutter boundary.     

Non-linear vibration of initially stressed plates with initial imperfections

Non-linear partial differential equations of vibration for isotropic plates having initial imperfection are derived. The derivation based on the classical plate theory aims to describe non-linear vibration of imperfect plates in a general state of arbitrary initial stresses. Galerkin method is used to reduce the non-linear partial differential equations to ordinary non-linear differential equations. Runge-Kutta method is used to obtain the non-linear and linear frequencies of vibration. A numerical example is presented to discuss the performances of perfect and imperfect plates. The initial stress is taken to be a combination of pure bending stress plus an extension stress in the plane of the plate. It is found that the existence of initial vibration amplitude, initial stress and geometric imperfect may result in a drastic change on the non-linear vibration behavior. The effects of various parameters on the non-linear free vibrations are discussed.     

A beam finite element for non-linear analyses of thin-walled elements

The aim of the present paper is to investigate a theoretical and numerical model which is able to study the behaviour of thin-walled beams with open cross section in presence of large torsion. The presented model takes into account for large torsion, linear and non-linear warping currently named shortening effects, pre-buckling deformation and flexural–torsional coupling. In numerical analysis, a 3D beam with two nodes and seven degrees of freedom per node is adopted. The equilibrium equations and the material behaviour are derived in discrete form without assumption on torsion angle amplitude. Due to large torsion context, all the equilibrium equations are non-linear and highly coupled. The linear behaviour is made possible by disregarding non-linear terms. For non-linear behaviour and stability, the tangent stiffness matrix is carried out. Due to large torsion context, new matrices are present. The element is incorporated in a homemade finite element code. Newton–Raphson iterative methods are used with different control parameters. In order to prove the efficiency of the model many examples are presented in linear and non-linear behaviour with presence of bifurcations.     

Elastic biaxial bending and torsion of thin-walled members

This paper presents a detailed treatment of the non-linear elastic biaxial bending and torsion of thin-walled open section members. The treatment is valid for uniform members of linear elastic material, and is limited to small strains and rotations, and moderate deflections. Shear straining of the mid-surface of the member wall is neglected, and it is assumed that the member does not distort or buckle locally. The effects of initial deformations, loads, stresses, and strains are incorporated.

The treatment is based on non-linear strain-displacement relationships, and these are used to derive the non-linear equilibrium and tangent stiffness equations in forms which are suitable for computer solution by the finite element method.

Approximate linear and non-linear differential equilibrium equations are derived, as are the differential equilibrium equations and the energy equation for neutral equilibrium at bifurcation buckling, and these are then related to the classical equations developed by Timoshenko, Vlasov, and others.     

Aerodynamic instability of a bridge deck section model: Linear and nonlinear approach to force modeling

The aerodynamic behavior of a bridge deck section model with a simple single-box shape was characterized in wind tunnel. At large nose-up mean angles of attack, a torsional instability arises, outlining a situation in which nonlinear aeroelastic effects may be critical. Such condition represents an interesting case to develop and validate nonlinear models for the aeroelastic problem. The experimental campaign allowed both to characterize the aerodynamic forces using forced motion tests and to study the aeroelastic behavior of the section model, when excited by actively generated turbulent wind. These aeroelastic tests are used to validate a numerical time-domain model for aerodynamic forces that takes into account the nonlinearities due to the reduced velocity and to the amplitude of the instantaneous angle of incidence. Results are critically analyzed and compared with those obtained with a linear model.     

Feasibility assessment of a leading-edge-flutter wind power generator

This study addresses the preliminary technical feasibility assessment of a mechanical apparatus for conversion of wind energy. The proposed device, designated as “leading-edge-flutter wind power generator”, employs aeroelastic dynamic instability of a blade airfoil, torsionally rotating about its leading edge. Although the exploitation of aeroelastic phenomena has been proposed by the research community for energy harvesting, this apparatus is compact, simple and marginally susceptible to turbulence and wake effects. The objective of this study was to identify a conceptual configuration that could become operational at low and medium wind speeds. Pre- and post-critical regimes are simulated; the prediction of mechanical efficiency in the absence of dissipation is also discussed.     

Growth patterns of the eight regions in the People's Republic of China (1980–1985)

The objective of this brief paper is very specific: to test with data from the People's Republic of China (PRC) an earlier model by the author on regional relative dynamics. Originally, that model was used to replicate the regional relative population and income growth of Nine Divisions in the United States for the period 1929–1979.Only one component of the original model is replicated in the case of the PRC, due to data limitations. Although the full regional dynamics model contains a system of two simultaneous differential equations, with a linear and a non-linear isocline respectively, here only the equation with the linear isocline is confirmed. This particular component of the model has also been independently verified for the case of Japan's regional growth.The author wishes to acknowledge the valuable contribution of Jian Zhang, his research assistant, for collecting and processing the data. Comments from T.R. Lakshmanan and two anonymous referees are also acknowledged, which helped improve the exposition of the paper's material     

Simulation of linear and non-linear propagation effects of a random turbulence field on bridge flutter instability

This paper presents the results of an analytical and numerical investigation on the implications associated with the propagation of an “uncertain” turbulence field on the aeroelastic stability of long-span bridges. Even though the influence of turbulence on flutter has received the attention of the wind engineering community in the recent past, an alternative formulation for simulating turbulence effects on flutter is proposed, in the general context of uncertainty and modeling error simulation. The coupled-mode flutter threshold, limited to the fundamental bending and torsional modes of a bridge, is estimated in the time domain by stochastic calculus techniques. A generalized modal correlation length is used to simulate turbulence modeling errors. Both linear and non-linear random propagation is investigated by means of stochastic stability. A numerical integration scheme is employed for the solution of the dynamic equations and to derive second-moment stability conditions for non-linear case. A set of simplified bridge examples is analyzed. Either a decrement or an increment in the critical velocity is observed for the non-linear random turbulence case, depending on the selected example. Nevertheless, this variation is usually small, of the order of a few percents in comparison with the “turbulence-free” scenario.     

Effect of relative amplitude on bridge deck flutter

Self-excited wind forces on a bridge deck can be non-linear even when the vibration amplitude of the body is small. This phenomenon is evaluated in this paper. Experiments detecting the nonlinearity are performed first, with the concept of “relative amplitude”, i.e. the amplitude of the externally triggered free vibration relative to the envelope of the ambient response of an elastically supported rigid sectional model. Two types of sectional model, a twin-deck bluff model (model A) and a partially streamlined box girder model (model B) are tested with two extreme cases of relative amplitude. Based on the flutter derivatives of model B, a flutter boundary prediction is subsequently carried out on a cable-supported bridge to manifest the changes of critical flutter wind velocity due to different relative amplitudes. The effect of relative amplitude on flutter derivatives and on the flutter boundary reveals, from the structural point of view, a complex relationship between the self-excited forces and the “structural vibration noise” due to turbulence that is inherent in the interaction of the ambient wind with the structure. Although the aeroelastic forces are linear when the body motion due to an external trigger is not affected significantly by this turbulence, they are postulated to be nonlinear when this “vibration noise” cannot be neglected.     

基于Wagner函数的细长机冀非线性空气弹性反应分析

介绍了人力飞机机翼的非线性空气弹性分析方法。这种飞机具有长而且高度弹性的机翼。弹性机翼和较长的机翼跨度会在正常飞行中导致机翼的大变形。因此,在机翼的竖向和扭转运动中采用二阶非线性弹性Euler-Bernoulli梁函数进行结构建模。采用基于Wagner函数的不稳定线性空气动力学理论确定机翼的空气动力荷载。将这两种公式结合起来就产生了非线性积分一微分空气弹性函数。采用Galerkin方法和模式求和方法,通过在空间弹性反应分析中引入一个迭代数值方法来求解控制方程。将分析结果与线性分析结果对比,发现在低于颤振速率的速度下,两者能很好地吻合,但是非线性模型显示出机翼的极限周期振幅超出了边界值。     

Investigation and optimization of nonlinear pendulum vibration absorber for horizontal vibration suppression of damped system

Nonlinear pendulum vibration absorber is investigated in this paper to control resonance peak of linear primary system with horizontal vibrations subjected to forced and motion excitations. Pendulum vibration absorbers have been utilized as tuned mass damper for many years, but by this knowledge, nonlinear analysis of this problem has not been investigated anywhere. Harmonic balance (HB) method is used to solve nonlinear differential equations and analyze the stability of their results. Optimum damping and natural frequency ratios are derived by minimizing the maximum steady‐state response of primary system with numerical optimization. The presented analysis shows that the pendulum absorber design based on linear model resulting in a small area around the original frequency for vibration absorption. For linear pendulum design, with a small deviation from the linear range, the amplitude of initial system greatly increased, and its performance will fall sharply. But if the design is based on a nonlinear pendulum with larger amplitude of motion, the resulting design will have a higher accuracy. In this paper, responses with inferior periods are inspected beside main period one. Different methods are used for optimized nonlinear pendulum. Finally, system robustness and chance of bifurcation will be predicted. Copyright © 2015 John Wiley & Sons, Ltd.     

苏通大桥气弹模型气动失稳分析

大跨柔性桥梁气弹模型在风荷载作用下会产生明显的竖向、侧向位移和附加攻角。推导气弹模型实测位移修正表达式,对苏通大桥气弹模型高风速时实测位移进行修正,对比分析修正前后差异。绘制了苏通大桥主跨跨中断面、1/4断面和边跨跨中3个典型断面扭转中心的运动轨迹,分析其特点,对苏通大桥气弹模型气动失稳现象进行解释。基于随机搜索方法和随机子空间方法识别得到的模态参数,对苏通大桥气弹模型进行复模态颤振分析。分析结果表明:苏通大桥气弹模型可视为一种非常规索网复合系统,其气动失稳振动表现为保持平衡状态的竖向、侧向和扭转耦合滚动,扭转频率成分在竖弯和侧弯振动中参与很多,而竖弯和侧弯频率成分在其他两种振动中参与很少。     

Aeroelastic effects in a traffic sign panel induced by a passing vehicle

Here, a simple theoretical model of the vehicle induced flow and its effects on traffic sign panels is presented. The model is a continuation of a previous one by Sanz-Andrés and coworkers, now including the flexibility of the panel (and, therefore, the flow effects associated to the motion of the panel). Through the paper an aeroelastic one-degree-of-freedom model is developed and the flow effects are computed from unsteady potential theory. The influence of panel's mechanical properties (mass, damping ratio, and stiffness) in the motion induced forces are numerically analyzed.     

Mechanics of thin-walled curved beams made of composite materials, allowing for shear deformability

In this paper, a new theoretical model is developed for the generalized linear analysis of composite thin-walled curved beams with open and closed cross-sections. In the present model two important concepts concerning to composite thin-walled curved beams are addressed. The first one is the incorporation in the model of what is called full shear deformability, i.e. shear flexibility due to both bending and non-uniform warping is considered. The second feature is connected with the constitutive aspects, and it contemplates the use of different hypotheses that can be adopted in the formulation. These topics are treated in a straightforward way by means of the Linearized Principle of Virtual Work. In order to obtain the motion equations of the model a non-linear displacement field, whose rotations are formulated by means of the rule of semitangential transformation, is employed. This model allows the study of many problems of statics, free and forced vibrations with arbitrary initial stresses and linear stability of composite thin-walled curved beams with general cross-sections. A discussion about the constitutive equations is performed in order to explain characteristic features of the effects included in the theory. This paper presents the theoretical formulation together with finite element procedures that are developed to obtain the numerical approximations to the general equations of thin-walled shear-deformable composite curved beams. For this kind of structural member, iso-parametric finite elements are introduced. Numerical examples are carried out in several topics of statics, dynamics and buckling problems, focusing attention in the validation of the theory with respect to experimental data and with 2D and 3D computational approaches. Also, new parametric studies are performed in order to show the influence of shear deformability on the mechanics of the thin-walled composite curved-beams with open and closed cross-sections as well as to illustrate the utility of the model.     

Interference excitation mechanisms on a 3DOF aeroelastic CAARC building model

A comprehensive wind tunnel test program was conducted to investigate interference excitation mechanisms on translational and torsional responses of an identical pair of tall buildings. Motion responses of a three-degree-of-freedom aeroelastic building model were measured. Both upstream and downstream interference effects were studied in this research. The experimental results showed that with an open terrain wind model, both dynamic translational and dynamic torsional responses generally increased under interference effects for an operating reduced wind velocity of 6. Measured response spectra indicated that amplified along-wind, cross-wind and torsional responses were largely induced by the wake of an upstream interfering building. The significance of interference effects and the dominant interference mechanisms depended upon the location of the aeroelastic model in the wake region. Furthermore, coupled translational–torsional motion of the aeroelastic building model tested was found to cause only small increases in the resultant motions at the building corner.     

Non-linear vibrations of a beam with a pair of piezoceramic actuators

In this study, the non-linear transverse vibrations of a system consisting of a beam with two piezoelectric actuators symmetrically bonded to its top and bottom surfaces are investigated. Due to the immovable end conditions the vibrating system is stretched. This stretching is modified by the residual in-plane stresses generated by the actuators, which are under the same in-phase voltages. A version of the Lindstedt–Poincare method is used to acquire the approximate solutions to the problem. The natural frequencies are computed from the solution to the linear problem, which is described by the equations of the first power of the small parameter. The orthogonality condition is applied to find the first correction of the linear frequency. The non-linear natural frequency of the system is derived depending on the vibration amplitude. Finally, the obtained solutions are applied to a steel beam with two piezoceramic layers. As the mass and stiffness of the piezoelectric elements are factors which affect the structure of the system, their influence on the natural linear frequency and the amplitude–frequency relationship is discussed. The main results concern the effect of the residual piezoforce on the non-linear vibration of the system.     

Phytoplankton modeling involving random rate constants

The deterministic framework is presented (part one) for the stochastic analysis (part two) of a model for phytoplankton production in a marine ecosystem. A system of nonlinear differential equations modeling the flow of nutrient through three components of an ecological system is considered. Modeling parameters and assumptions are discussed and numerical solutions are obtained. The dampened oscillatory characteristic of many predator prey systems is observed. A linear deterministic system is also derived which approximates the nonlinear and for which is presented a closed form solution. The linear solution is seen to accurately reflect the behavior of the nonlinear system. Discussion of the stochastic aspects of biological modeling are also discussed in preparation for the stochastic analysis of part two (to be published in a future issue of this Journal).     

Random analysis of geometrically non-linear FE modelled structures under seismic actions

In the framework of the finite element (FE) method, by using the “total Lagrangian approach”, the stochastic analysis of geometrically non-linear structures subjected to seismic inputs is performed. For this purpose the equations of motion are written with the non-linear contribution in an explicit representation, as pseudo-forces, and with the ground motion modelled as a filtered non-stationary white noise Gaussian process, using a Tajimi-Kanai-like filter. Then equations for the moments of the response are obtained by extending the classical Itô's rule to vectors of random processes. The equations of motion, and the equations for moments, obtained here, show a perfect formal similarity. By using this similarity a very effective computational procedure for evaluating response moments of any order is proposed.

Within the framework of non-Gaussian closure schemes, a technique is here presented based on a truncated Gram-Charlie expansion. To achieve this the Hermite coefficients are evaluated for multi-degree-of-freedom systems, once the multi-dimensional Hermite polynomials have been obtained in compact form.     

Nonlinear Analysis of Structural Frame Systems by the State-Space Approach

This article presents the formulation and solution of the equations of motion for distributed parameter nonlinear structural systems in state space. The essence of the state-space approach (SSA) is to formulate the behavior of nonlinear structural elements by differential equations involving a set of variables that describe the state of each element and to solve them in time simultaneously with the global equations of motion. The global second-order differential equations of dynamic equilibrium are reduced to first-order systems by using the generalized displacements and velocities of nodal degrees of freedom as global state variables. In this framework, the existence of a global stiffness matrix and its update in nonlinear behavior, a cornerstone of the conventional analysis procedures, become unnecessary as means of representing the nodal restoring forces. The proposed formulation overcomes the limitations on the use of state-space models for both static and dynamic systems with quasi-static degrees of freedom. The differential algebraic equations (DAE) of the system are integrated by special methods that have become available in recent years. The nonlinear behavior of structural elements is formulated using a flexibility-based beam macro element with spread plasticity developed in the framework of state-space solutions. The macro-element formulation is based on force-interpolation functions and an intrinsic time constitutive macro model. The integrated system including multiple elements is assembled, and a numerical example is used to illustrate the response of a simple structure subjected to quasi-static and dynamic-type excitations. The results offer convincing evidence of the potential of performing nonlinear frame analyses using the state-space approach as an alternative to conventional methods.     

基于参变量变分原理的地基梁位移非线性分析

提出一种分析地基梁非线性位移的新方法。首先采用分段线性函数对非线性的基底压力(p) -基底沉降(s)关系曲线进行拟合,通过引入控制变量,得到p-s曲线统一表达式。根据地基梁模型能量泛函,结合参变量变分原理和分段线性地基模型中的互补条件,导得一个标准的线性互补模型。该模型可用较为成熟的规划算法进行求解,使地基梁位移非线性求解问题转化为一个标准的数学问题。在对该法的合理性进行验证后,详细推导了集中荷载作用下地基梁位移的非线性求解方程,并对其进行求解。在此基础上,对线性与非线性计算的差异及影响非线性计算结果的因数进行分析,得到如下主要结论：考虑非线性影响时,地基梁位移曲线不均匀沉降增大,地基梁内力增大;随着p-s曲线非线性程度增加以及p-s曲线上进入非线性段临界压力值的减小,非线性影响越明显;荷载大小及梁与地基的相对刚度均会影响地基梁位移分布形式。