An Eigenvalue Method for Calculation of Stability and Limit Cycles in Nonlinear Systems

In the calculation of periodic oscillations of nonlinear systems –so-called limit cycles – approximative and systematic engineeringmethods of linear system analysis are known. The techniques, working inthe frequency domain, perform a quasi-linearization of the nonlinear system,replacing nonlinearities by amplitude-dependent describing functions.Frequently, the resulting equations for the amplitude and frequency ofpresumed limit cycles are solved directly by a graphical procedure in aNyquist plane or by solving the nonlinear equations or a parameteroptimization problem. In this paper, an indirect numerical approach isdescribed which shows that, for a system of nonlinear differentialequations, the eigenvalues of the quasi-linear system simply indicateall limit cycles and, additionally, yield stability regions for thelinearized case. The method is applicable to systems with multiplenonlinearities which may be static or dynamic. It is demonstrated foran example of aircraft nose gear shimmy dynamics in the presence ofdifferent nonlinearities and the results are compared with those fromsimulation.     

特征值摄动法在利用动响应结构小损伤检测中的应用

目前在使用遗传算法或神经网络方法进行结构动力学损伤检测,需要基于少量的在线测量损伤结构数据和大量的数值仿真数据来实现,其中通过有限元方法来获得仿真数据的巨大计算量是动力学结构损伤检测方法发展中所面临的一个重要问题。本文在建模方面应用近年来提出的调整单元刚度模拟损伤的先进方法,以保证在损伤前后结构自由度数目不变;在此基础上应用特征值摄动法来减少损伤检测中计算量,并通过对复合材料层合板响应信号的小波分析验证了使用一阶矩阵摄动在有效降低计算量的同时,可以获得对损伤检测而言足够准确的响应信号。     

Linear and Nonlinear Structural Mechanics

Book Review

Linear and Nonlinear Structural Mechanics     

Singular Nonlinear Eigenvalue Problem for a Second-Order Differential Equation with Energy Dissipation

We consider an eigenvalue problem for a singular nonlinear ordinary differential operator of the second order on the half line. We find sufficient conditions under which this problem has a solution that has a prescribed number of zeros and vanishes at infinity.     

On the Implementation of the Eigenvalue Method for Limit Cycle Determination in Nonlinear Systems

In many practical systems, limit cycles can be predicted with suitable precision by frequency domain methods using describing functions. Within such an approach, limit cycles can be predicted using the “eigenvalue method” [Somieski, G., Nonlinear Dynamics 26(1), 2001, 3–22]. This contribution presents a novel and advantageous implementation of this method, using singular value instead of eigenvalue calculations, and enhancing computational efficiency by avoiding a so called “frequency iteration”. An erratum to this article is available at .     

随机结构系统的一般实矩阵特征值问题的概率分析

由于工程实际结构的复杂性和所用材料在统计上的离散性以及测量、加工、制造误差的存在，必然导致具有随机参数的随机结构振动系统，按结构参数的性质来划分，随机振动问题包括两方面内容：(1)确定结构问题；(2)随机结构问题。本文以现代数学理论为依托，研究了随机结构系统的一般实矩阵的特征值问题。根据Kronecker代数、向量值和矩阵值函数的灵敏度分析、一般二阶矩法和概率摄动技术给出了计算随机结构系统的一般实矩阵的特征值和特征向量的数值方法，可以有效地得出随机结构系统的一般实矩阵的特征向量的统计量，发展了2D矩阵值函数的随机结构系统的特征值问题概率分析理论。     

A Simple Fast Method in Finding Particular Solutions of Some Nonlinear PDE

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结构非线性颤振半主动抑制

本文利用数字仿真技术，对结构非线性颤振半主动抑制－颤振驯化方案进行了探讨，仿真结果表明，对于一定的结构非线性类型和参数，利用非线性颤振的极限环特性，可使系统的颤振响应被抑制在幅值很小的稳定域内，从而达到减缓颤振的目的。     

A Nondifferentiable Optimization Algorithm for Structural Problems with Eigenvalue Inequality Constraints*

This paper presents a nondifferentiable optimization algorithm for solution of structural optimal design problems with eigenvalue inequality constraints. The algorithm is shown to be convergent both in the L _∞ and in the sequence space topologies.     

A Degree Associated to Linear Eigenvalue Problems in Hilbert Spaces and Applications to Nonlinear Spectral Theory

We extend to the infinite dimensional context the link between two completely different topics recently highlighted by the authors: the classical eigenvalue problem for real square matrices and the Brouwer degree for maps between oriented finite dimensional real manifolds. Thanks to this extension, we solve a conjecture regarding global continuation in nonlinear spectral theory that we have formulated in a recent article. Our result (the ex conjecture) is applied to prove a Rabinowitz type global continuation property of the solutions to a perturbed motion equation containing an air resistance frictional force.

     

Component Mode Synthesis Using Nonlinear Normal Modes

This paper describes a methodology for developing reduced-order dynamic models of structural systems that are composed of an assembly of nonlinear component structures. The approach is a nonlinear extension of the fixed-interface component mode synthesis (CMS) technique developed for linear structures by Hurty and modified by Craig and Bampton. Specifically, the case of nonlinear substructures is handled by using fixed-interface nonlinear normal modes (NNMs). These normal modes are constructed for the various substructures using an invariant manifold approach, and are then coupled through the traditional linear constraint modes (i.e., the static deformation shapes produced by unit interface displacements). A class of systems is used to demonstrate the concept and show the effectiveness of the proposed procedure. Simulation results show that the reduced-order model (ROM) obtained from the proposed procedure outperforms the ROM obtained from the classical fixed-interface linear CMS approach as applied to a nonlinear structure. The proposed method is readily applicable to large-scale nonlinear structural systems that are based on finite-element models.     

结构损伤识别的非线性解析法

本文在直接解析法的基础上,进一步提出结构损伤识别的非线性解析法.它与直接解析法不同之处是:考虑了频率和振型对损伤参数的二阶偏导数项,以此构造损伤参数为未知量的超定非线性方程组.求解该方程组可以一次性得到全部损伤参数值.本方法不仅具有直接解析法的优点,而且由于构建方程组时增加了损伤参数的二次项,极大地提高了损伤识别的精度.最后,通过对一个五层框架结构进行数值模拟分析验证了该方法的可行性,同时经过比较计算结果表明:在损伤识别中,非线性解析法在精度及节省计算工作量上明显优于直接解析法.     

Approximation and Calibration of Nonlinear Structural Dynamics

In this paper, a methodology for the calibration of nonlinear structural dynamic models is presented. Calibration of nonlinear structural dynamics offers several additional challenges beyond that of linear dynamics. Even with advanced computational power, exact nonlinear finite element simulations often take several hours to complete on engineering workstations. Thus, the proposed model calibration method utilizes an approximate structural model. This approximate analysis is embedded in the outer loop, which utilizes an exact finite element analysis to verify the validity of the approximate model. If the approximate model is shown to be invalid at that point in parameter space, then the new exact analysis is used to develop an improved approximate model and the inner loop is executed again. Specifically, this paper will focus on the two key aspects of the inner loop, namely the development of an approximate model, and the parameter identification using the approximate model.     

估计区间特征值问题特征值界的一种新方法

在对称矩阵的性质和谱半径的单调性质等矩阵计算理论的基础上,给出了一种计算标准区间特征值问题特征值上下界的新方法.最后用两个数值例子与已有的方法进行了比较,进一步验证了本文方法的正确性和有效性.     

Mode Localization Induced by a Nonlinear Control Loop

We present the analysis of a nonlinear controlsystem that is used to excite and maintain a specified amplitude ofoscillation in the Jet Propulsion Laboratory vibratory gyroscope.This experimental application shows that nonlinear localizationthrough active means can be implemented in a practical system when itis desirable to confine the response to a favorable mode. Theclosed-loop system response predicted by the model shows very closeagreement with the experimental results for a significant range ofcontroller parameters. We also experimentally demonstrate that theactively localized motion is eliminated through bifurcation, similarto what was observed in previous passive localization studies appliedto extended flexible oscillators.     

Dynamics of Nonlinear Cyclic Systems with Structural Irregularity

The dynamics of weakly coupled, nonlinear cyclic assemblies are investigated in the presence of weak structural mistuning. The method of multiple scales is used to obtain a set of nonlinear algebraic equations which govern the steady-state, synchronous (modal-like) motions for the structures. Considering a degenerate assembly of uncoupled oscillators, spatially localized modes are computed corresponding to motions during which vibrational energy is spatially confined to one oscillator (strong localization) or a subset of oscillators (weak localization). In the limit of weak substructural coupling, asymptotic solutions are obtained which correspond to (i) spatially extended, (ii) strongly localized, and (iii) weakly localized modes for fully coupled systems. Throughout the analysis, the influence of structural mistunings on the resulting solutions are discussed. Additionally, numerical solutions (including linearized stability characteristics) are obtained for prototypical two- and three-degree-of-freedom (DoF) systems with various structural mistunings. The numerical results provide insight into the strong influence of structural irregularities on the dynamical behavior of nonlinear cyclic systems, and demonstrate that the combined influences of structural mistunings and nonlinearities do not lead to uniform improvement of motion confinement characteristics.     

Milling Bifurcations from Structural Asymmetry and Nonlinear Regeneration

This paper investigates multiple modeling choices for analyzing the rich and complex dynamics of high-speed milling processes. Various models are introduced to capture the effects of asymmetric structural modes and the influence of nonlinear regeneration in a discontinuous cutting force model. Stability is determined from the development of a dynamic map for the resulting variational system. The general case of asymmetric structural elements is investigated with a fixed frame and rotating frame model to show differences in the predicted unstable regions due to parametric excitation. Analytical and numerical investigations are confirmed through a series of experimental cutting tests. The principal results are additional unstable regions, hysteresis in the bifurcation diagrams, and the presence of coexisting periodic and quasiperiodic attractors which is confirmed through experimentation.     

SVD-Unscented卡尔曼滤波的非线性结构系统识别

提出一种基于奇异值分解的unscented卡尔曼滤波(SVD-UKF)非线性滞回结构系统识别方法.SVD-UKF可被看成改进的unscented卡尔曼滤波(UKF)方法,相对UKF而言,SVD-UKF具有更好的鲁棒性和灵活性.此方法不仅避免象扩展卡尔曼滤波(EKF)为了计算Jacobians矩阵的所需的导数运算,并且可以克服常规UKF方法在计算协方差时经常遇到的病态条件的缺点.对非线性系统参数的识别和突然变化的识别的数值模拟结果显示了所提出方法的鲁棒性和灵活性.     

On Mode Interactions for a Weakly Nonlinear Beam Equation

In this paper an initial-boundary value problem for a weakly nonlinear beam equation with a Rayleigh perturbation will be studied. It will be shown that the calculations to find internal resonances in this case are much more complicated than and differ substantially from the calculations for the weakly nonlinear wave equation with a Rayleigh perturbation as for instance presented in [3] or [7]. The initial-boundary value problem can be regarded as a simple model describing wind-induced oscillations of flexible structures like suspension bridges or iced overhead transmission lines. Using a two-timescales perturbation method approximations for solutions of this initial-boundary value problem will be constructed.     

非线性结构动力学瞬态分析的摄动法

在用直接积分法求解非线性结构的动力响应时,常常需要做迭代运算。本文引入摄动方法后,加快了收敛速度,提高了计算效益。