Advances in the linear/nonlinear control of aeroelastic structural systems

Summary Active aeroelastic control is a recently emerging technology aimed at providing solutions to a large class of problems involving the aeronautical/aerospace flight vehicle structures that are prone to instability and catastrophic failures, and to oscillations that can yield structural failure by fatigue. In order to prevent such damaging phenomena to occur, the linear/nonlinear aeroelastic control technology should be applied. Its goals are among others: (i) to alleviate and even suppress the vibrations appearing in the subcritical flight speed range, (ii) to enlarge the flight envelope by increasing the flutter speed, and (iii) to enhance the post-flutter behavior by converting the unstable limit cycle oscillation to a stable one. A short review of the available control techniques and capabilities is presented first. Attention is focused on the open/closed-loop of 2D and 3D lifting surfaces as well as on panels exposed to supersonic flowfields. A number of concepts involving various control methodologies, such as proportional, velocity, linear quadratic regulator, modified bang-bang, sliding mode observer, time-delay control, fuzzy, etc., as well as results obtained with such controls are presented. Emphasis is placed on theoretical and numerical results obtained with the various control strategies that are considered in a comparative way. Finally, conclusions and directions for further work are presented.This paper represents an updated version of the General Lecture presented at the 3rd European Conference on Structural Control, July 12–15, 2004, Vienna, Austria.     

土-结构非线性相互作用混合约束模态实施方法

分析、探讨线性-非线性混合约束模态综合法用于土-结构相互作用的动力分析实施方法。据结构存在局部非线性特性,提出对整体结构划分为若干个线性、非线性子结构,对线性子结构只需一步提取特性矩阵并进行方程降阶,而对非线性子结构则逐步提取降阶后的等效特性矩阵。通过组装各子结构达到对整体结构特性矩阵的降阶处理。对非线性子结构凭借在小段时间 内利用分段等效线性化手段,通过ANSYS二次开发工具UPFs编辑写成可执行程序文件,并与MATLAB联立计算,实现特性矩阵提取。结果表明,该方法求得结构的各阶频率和动态响应时程曲线与ANSYS直接计算吻合良好。可逐步提取等效弹塑性特性矩阵,为采用变化的特性矩阵对结构进入塑性阶段深层次分析研究提供有效方法。     

Stochastic analysis of highly non‐linear structures

Instabilities can introduce highly non‐linear effects into structural problems. The instabilities, not clearly associated with a change in a parameter, result in a stochastic variation of the responses. This process variation can be distinguished from the effects of the parameter variation by mapping the response variation onto a predictable space and a residual space, where the predictable space contains the possible effects of the parameter variation, and the residual space contains the process effects. This study discusses the sources (mechanics) of the response variation in this class of problems, the use of response surfaces to distinguish between effects driven by design variable changes and bifurcations, and the visualization of unstable zones in the structure. Analytical problems, a headform impact problem, and an occupant safety study clarify the use of the proposed methodology. Copyright © 2005 John Wiley & Sons, Ltd.     

高层建筑气弹模型模态参数识别

以某-492m高的超高层建筑为工程背景,讨论了3种不同方法对风洞试验气动弹性模型的固有频率、固有振型和阻尼比等模态参数的识别效果.首先采用初始激励自由衰减振动试验和频域分析方法(简称"初激励频域法")对该气动弹性模型的模态参数进行了识别.然后以TJ-2风洞中人工紊流风场为环境随机激励源,对该气动弹性模型进行了环境风随机振动试验,并分别采用环境风随机振动频域分析方法和基于响应相关函数的环境风随机振动时域STD方法对其模态参数进行了识别.结果显示:对于固有频率和固有振型,上述3种方法的识别结果比较接近;对于模态阻尼比,初激励自由衰减频域法的识别结果与具有钢芯棒悬臂结构的气动弹性模型模态阻尼比的经验范围相符,但环境风随机振动法的识别结果由于模型振动响应幅值较小而明显偏大.     

A modal aeroelastic finite element analysis method for advanced turbomachinery stages

This paper presents a formulation suitable for the flutter analysis of rotating bladed assemblies. The blades are modelled using the finite element method. The aeroelastic displacements, expressed in terms of travelling wave co-ordinates, are written as a linear combination of a number of undamped modes leading to a complex eigenvalues eigenvectors problem. The associated unsteady aerodynamic model is either two or three dimensional, leading for both cases to a consistent and flexible aeroelastic analysis method. Three numerical applications are presented which illustrate the theory and emphasize the possible effects of blade cross-section (chord) deformation on aeroelastic damping.     

非线性气动弹性系统的鲁棒稳定性分析

基于结构奇异值理论和特征多项式的值域方法,研究了带有结构和气动参数不确定性的非线性二元机翼的鲁棒稳定性问题.机翼模型包括非线性的扭转弹簧和能够反映失速效应的非线性气动模型.针对零摄动,使用肛方法分析线性化系统在平衡点的鲁棒稳定性.针对非零摄动,将平衡点位置看作关于不确定参数的函数并展开为泰勒级数,从而在μ方法的框架下考虑平衡点的不确定性.此外,计算不确定特征多项式的值域范围,并使用除零条件来判断其鲁棒稳定性.仿真数值结果给出了鲁棒颤振速度的上下界,表明了方法的有效性.     

Random vibrations: A spectral method for linear and nonlinear structures

The time-dependent power spectral density of any linear response to an input modulated stochastic process is calculated by a single numerical integration. Thus, computer costs solely depend on the efficient computation of the frequency response function. The spectral method can be extended to elasto-plastic structures with certain approximations on the nonlinear yielding process which allow the determination of the mean yielding-deformation increment. Time- and frequency dependent envelope functions of the power density of modal driving forces of the associated linear system are computed in a time-stepping procedure. Results compare favourably well with simulations in the stationary limit.     

基于矩阵分式多项式时变结构模态参数最小二乘辨识

基于时间相关矩阵分式多项式传递函数模型,给出线性时变结构时频域参数化模型。以时频域参数化模型为基础,将现有广泛用于时不变结构模态参数辨识的最小二乘复指数法拓展到时频域,提出基于矩阵分式多项式模型的时频域线性时变结构模态参数最小二乘辨识方法;针对时频域最小二乘对计算资源庞大需求问题,给出基于缩减正则方程的最小二乘问题求解方法。通过对两质量连续变化三自由度时变结构仿真算例,说明最小二乘中待估参数约束对模态参数辨识影响,阐述所提线性时变结构模态参数辨识方法特点,说明方法的有效性及潜在实用性。     

非线性双渐进法应用于水中结构瞬态运动的研究

为了解决线性双渐近法只能用于解决准静态条件下的流固耦合问题的局限性,提出了解决流固耦合问题的非线性双渐进法。并将其应用于瞬态载荷作用下具有初始航速结构的流固耦合计算当中。研究了水平初速度系数下流场的动压力时历曲线,分析非线性对结构运动前期、中期、后期周围流场动压力的影响,分析无量纲单位面积冲量随水平初速度系数的变化趋势。给出了非线性效应不能被忽略的初始速度条件,进而研究了非线性双渐近法的适用范围。     

Optimization of aeroelastic composite structures using evolutionary algorithms

The flutter/divergence speed of a simple rectangular composite wing is maximized through the use of different ply orientations. Four different biologically inspired optimization algorithms (binary genetic algorithm, continuous genetic algorithm, particle swarm optimization, and ant colony optimization) and a simple meta-modeling approach are employed statistically on the same problem set. In terms of the best flutter speed, it was found that similar results were obtained using all of the methods, although the continuous methods gave better answers than the discrete methods. When the results were considered in terms of the statistical variation between different solutions, ant colony optimization gave estimates with much less scatter.     

基于随机子空间的递推在线模态识别算法

摘要：基于随机子空间算法,根据矩阵空间的性质,利用QR分解将行空间到过去行空间的投影展开为一种用于跟踪的修改递推模式。通过子空间跟踪算法,不断跟踪计算投影的左奇异值向量,再利用最小二乘法求出系统的模态参数,实现了单独利用响应数据,在线识别模态参数的方法。最后通过仿真算例,利用算法跟踪不断改变的模态空间和时变模态参数,验证方法的有效性及稳定性。     

Fast modal response method for structures

Regardless of their simplicity, all structures have an infinite number of degrees-of-freedom (d.o.f.) when subjected to dynamic loading. The usual finite element method reduces the infinite d.o.f. system to a model with a limited d.o.f. while capturing the significant physical behaviour. The modal analysis reduces the number of d.o.f. further to a limited number of modal co-ordinates. However, accurate results comparable to the original finite element model may not be possible unless higher modes are included. The present paper is to recommend a response analysis which makes use of both the natural modes and the mass and stiffness matrices of the system to improve the convergence with respect to the number of modes. While the effects of lower modes are analysed similar to the modal analysis, the effects of higher modes are included in the system matrices and the information for higher modes is not needed.     

Application of the proper orthogonal decomposition for linear and nonlinear structures under transient excitations

Model reduction has become very important in order to save calculation time. In particular, in structural dynamics, computations become very time-consuming when the critical time step of explicit integrators becomes very small. The main focus of this paper is on the application of the Proper Orthogonal Decomposition (POD) method to a structure subjected to transient earthquake loading. It is shown that based on the information of only a small portion of the transient excitation and the structure (“snapshots”), it is possible to assemble a reduced-order model, which yields a very accurate and time-saving approximation of the response to the entire earthquake. The POD reduction method is applied not only to linear, but also to nonlinear structures under earthquake loading. In the linear case, the POD results can be compared to those obtained by the classical method of modal truncation. In the nonlinear case, base isolation systems (friction pendulum systems) are integrated in the structure. Error estimations are applied in order to assess the solution of the POD-reduced system of the linear and the nonlinear systems. The POD can be applied successfully if the snapshots within the chosen time interval describe the main behavior of the system well. In both the linear and nonlinear cases, the approximation of the system as reduced by the POD is very accurate even if only a few POD modes are used. The advantage over the method of Modal Truncation is not only the optimality of the POD modes concerning their associated energy, but also its applicability to nonlinear systems.     

New conceptual design of aeroelastic wing structures by multi-objective optimization

Internal structural layouts and component sizes of aircraft wing structures have a significant impact on aircraft performance such as aeroelastic characteristics and mass. This work presents an approach to achieve simultaneous partial topology and sizing optimization of a three-dimensional wing-box structure. A multi-objective optimization problem is assigned to optimize lift effectiveness, buckling factor and mass of a structure. Design constraints include divergence and flutter speeds, buckling factor and stresses. The topology and sizing design variables for wing internal components are based on a ground element approach. The design problem is solved by multi-objective population-based incremental learning (MOPBIL). The Pareto optimum results lead to unconventional wing structures that are superior to their conventional counterparts.     

Substructuring in linear and nonlinear analysis

The procedures for utilizing substructuring and static condensation in structural analysis are well known. However, until recently there have been no general-purpose structural mechanics computer systems that offer multi-level substructuring combined with a convenient method for defining the structural model. Now that such systems are available, engineers must decide when substructuring techniques are useful. Substructuring, with and without condensation, has proved to be highly efficient in the analysis of certain classes of structure. It can reduce computer costs by a factor of from 2 to 100. Yet, indiscriminate use of condensation may result in unnecessary and expensive computations. This paper examines the advantages and disadvantages of substructuring relative to data entry and computational efficiency. Guidelines are proposed for engineers to follow when using substructuring in the analysis of linear and nonlinear structures. The FINITE system is used to illustrate actual implementation of substructuring features in a general purpose finite element system.     

Effectiveness of linear bifurcation analysis for predicting the nonlinear stability limits of structures

As an effort to predict effectively the actual collapse load of a structure, a series of numerical studies on the stability of shell structures are made. The difference in formulation between the two types of linear buckling loads, the classical and the fully linearized, is first demonstrated. Their correlations with respect to the actual stability limit of the structure are compared, and finally the two types of critical load approximations are obtained at various stages of a nonlinear analysis to study the pattern of convergence to the actual collapse load. It is found that the fully linearized buckling analysis, when combined with nonlinear analysis, can serve as a useful tool for prediction of the stability limit of a structure. While for most types of structures the approximation is within engineering accuracy, the rate of convergence of the extrapolated critical load also gives some insight to the accuracy of the approximation.     

Size-dependent internal resonances and modal interactions in nonlinear dynamics of microcantilevers

The size-dependent internal energy transfer in the nonlinear dynamical behaviour of a microcantilever with an intermediate spring-support is investigated. A geometric size-dependent nonlinearity due to large changes in the curvature is taken into account in the longitudinal and transverse motions. Based on the modified couple stress theory, the potential energy of the system is developed; the kinetic energy is also constructed in term of the displacement field. The energy terms are balanced with the potential energy stored in the intermediate spring-support. The centreline-inextensibility assumption is applied leading to the continuous model of the system involving nonlinear inertial components as well as size-dependent nonlinear curvature components. Based on a weighted-residual technique, the continuous model is reduced and the resultant truncated model is solved via use of a continuation technique. The linear component of the truncated model is solved through an eigenvalue extraction method in order to verify the occurrence of internal energy transfer and modal interaction mechanisms. For the system tuned to internal resonances, the highly nonlinear dynamical response is obtained, taking into account both inertial and geometric (due to large rotations) nonlinearities. It is shown that taking into account the length-scale parameter changes the internal energy transfer mechanisms significantly.     

Optimization of complex structures to satisfy static,dynamic and aeroelastic requirements

A structural optimization problem is considered in which the design requirements include restrictions on the strength, stability, frequency and flutter characteristics of the structure. One of the central concerns of this phase of the work has been to overcome the problems inherent in analysing the dynamic and aeroelastic behaviour of structures with many degrees of freedom. The multiweb delta wing structure under supersonic flight conditions is the model upon which this exploratory study is based. The finite element idealization, with three different kinds of elements, is used to model the wing structure. The constant stress triangular plate elements, the rectangular shear panels and the pin-jointed bar elements are used to represent, respectively, the cover skins, webs and stringers of the wing structure. The design problem is formulated as a minimum weight optimization problem and is solved by using non-linear programming techniques. Computationally efficient schemes are developed for the necessary derivatives of the behaviour constraints. Numerical examples are presented to illustrate the feasibility and the computational effectiveness of the method.     

Probabilistic response of linear structures equipped with nonlinear damper devices (PIS method)

Passive control introducing energy absorbing devices into the structure has received considerable attention in recent years. Unfortunately the constitutive law of viscous fluid dampers is highly nonlinear, and even supposing that the structure behaves linearly, the whole system has inherent nonlinear properties. Usually the analysis is performed by a stochastic linearization technique (SLT) determining a linear system equivalent to the nonlinear one, in a statistical sense. In this paper the effect of the non-Gaussianity of the response due to the inherent nonlinearity of the damper device will be studied in detail via the Path Integral Solution (PIS) method. A systematic study is conducted showing that for a very wide range of parameters the SLT gives satisfactory results in terms of variance of displacement and velocity but not in terms of joint Probability Density Function (PDF). It has also been shown that at steady state the two processes, displacement and velocity, may be considered as independent ones.     

Stochastic stability of linear viscoelastic systems

The stochastic almost-sure stability of a single degree-of-freedom linear viscoelastic system subjected to random fluctuation in the stiffness parameter is investigated. For small damping and weak random fluctuation, asymptotic expressions are derived for the Lyapunov exponent and the rotation number using the method of stochastic averaging. From the sign of the Lyapunov exponent, the condition for asymptotic stability with probability 1 of the trivial equilibrium state is obtained.