An analytical prediction of sliding motions along discontinuous boundary in non-smooth dynamical systems

This paper presents a method for the analytical prediction of sliding motions along discontinuous boundaries in non-smooth dynamical systems. The methodology is demonstrated through investigation of a periodically forced linear oscillator with dry friction. The switching conditions for sliding motions in non-smooth dynamical systems are given. The generic mappings for the friction-induced oscillator are introduced. From the generic mappings, the corresponding criteria for the sliding motions are presented through the force product conditions. The analytical prediction of the onset and vanishing of the sliding motions is illustrated. Finally, numerical simulations of sliding motions are carried out to verify the analytical prediction. This analytical prediction provides an accurate prediction of sliding motions in non-smooth dynamical systems. The switching conditions developed in this paper are expressed by the total force of the oscillator, and the nonlinearity and linearity of the spring and viscous damping forces in the oscillator cannot change such switching conditions. Therefore, the achieved force criteria can be applied to the other dynamical systems with nonlinear friction forces processing a C ⁰-discontinuity.     

A time-domain nonlinear system identification method based on multiscale dynamic partitions

Based on a theoretical foundation for empirical mode decomposition, which dictates the correspondence between the analytical and empirical slow-flow analyses, we develop a time-domain nonlinear system identification (NSI) technique. This NSI method is based on multiscale dynamic partitions and direct analysis of measured time series, and makes no presumptions regarding the type and strength of the system nonlinearity. Hence, the method is expected to be applicable to broad classes of applications involving time-variant/time-invariant, linear/nonlinear, and smooth/non-smooth dynamical systems. The method leads to nonparametric reduced order models of simple form; i.e., in the form of coupled or uncoupled oscillators with time-varying or time-invariant coefficients forced by nonhomogeneous terms representing nonlinear modal interactions. Key to our method is a slow/fast partition of transient dynamics which leads to the identification of the basic fast frequencies of the dynamics, and the subsequent development of slow-flow models governing the essential dynamics of the system. We provide examples of application of the NSI method by analyzing strongly nonlinear modal interactions in two dynamical systems with essentially nonlinear attachments.     

Friction oscillator excited by moving base and colliding with a rigid or deformable obstacle

The dynamics of non-smooth oscillators has not yet sufficiently been investigated, when damping is simultaneously due to friction and impact. Because of the theoretical and practical interest of this type of systems, an effort is made in this paper to lighten the behaviour of a single-degree-of-freedom oscillator colliding with an obstacle and excited by a moving base, which transfers energy to the system via friction. The different nature of discontinuities arising in the combined problem of friction and impact has been recognized and discussed. Closed-form solutions are presented for both transient and steady-state response, assuming Coulomb's friction law and a rigid stop-limiting motion. Furthermore, a deformable (hysteretic) obstacle has been considered, and its influence on the response has been investigated.     

Nonlinear dynamic behavior of a bio-inspired embedded X-shaped vibration isolation system

To improve the vibration isolation performance and bandwidth, loading capacity and supporting stability of passive vibration isolation system by utilizing nonlinearity, a bio-inspired embedded X-shaped vibration isolation (BIE-XVI) structure is proposed considering muscle/tendon contractile functions, joint rotational friction and connecting rod mass simultaneously. Furthermore, the dynamic model with pure linear elements and geometric relationship are established and the nonlinear variation properties are investigated. The effects of the key parameters of the BIE-XVI structure on frequency response characteristics and vibration isolation range are analyzed thoroughly by incremental harmonic balance method in various working conditions. From the parametric investigations, it can be found that the sensitivities of the nonlinear resonance properties are markedly different with respect to the different structure parameters. For longer rod length, larger assembly angle and higher stiffnesses, the hardening nonlinearity is weakened, but the resonance peak does not necessarily decrease. Besides, the softening nonlinearity and hardening nonlinearity can be interconverted with changing isolated mass and excitation amplitude. The BIE-XVI structure can widen the isolation frequency range and reduce the resonance peak to improve the vibration isolation properties by adjusting/designing the structural parameters, which could realize quasi-zero-stiffness property for vibration isolation.

     

空间可展机构非光滑力学模型和动力学研究

空间可展机构广泛应用于展开和支撑柔性太阳能帆板和航天工程领域中的有效载荷, 包括抛物面天线、平面相控阵雷达和合成孔径雷达等. 非光滑特性及其相应的动力学现象在空间可展机构的设计中有着非常重要的作用. 该文系统地综述了空间可展机构非光滑力学建模与非线性动力学的研究进展. 首先详细描述了含间隙铰链的接触碰撞力和摩擦力等非光滑特点;然后系统地介绍了含间隙机构的动力学建模方法、分析方法和参数设计;进一步简单介绍了含间隙铰链空间可展机构的非线性动力学特性, 如谐波共振、周期运动的稳定性和各类分岔等;最后提出了空间可展机构非光滑动力系统动力学、稳定性与控制中亟待解决的若干问题.     

基于弹塑性模型的群桩-土-结构动力非线性数值分析

提出一种群桩-土弹塑性模型,结合动力文克尔理论,推导出了与桩(筏)-土属性及SSI体系频率相关的各项弹簧-阻尼单元动力阻抗,建立了三维框架土-结构相互作用有限元简化分析模型。针对不同地震激励,在不同桩-土条件下对模型进行了动力非线性时程分析,结果表明,在某些地震动和土-基础条件下,上部结构非线性作用效应结果可能大于固基假定情形,且桩-土弹塑性模型对上部结构柔弱层位置产生影响。应用本文简化方法可以快速、较准确有效地进行复杂的上下部结构动力时程分析及抗震评估。     

Investigation of the near-grazing behavior in hard-impact oscillators using model-based TS fuzzy approach

An impact oscillator is a non-smooth dynamical system with discontinuous state jumps whose dynamical behavior illustrates a variety of non-linear phenomena including a grazing bifurcation. This specific phenomenon is difficult to analyze because it coincides with an infinite stretching of the phase space in the neighborhood of the grazing orbit, resulting in the well-known problem of the square-root singularity of the Jacobian of the discrete-time map. A?novel Takagi?CSugeno fuzzy model-based approach is presented in this paper to model a hard impacting system as a non-smooth dynamical system including discontinuous jumps. Employing non-smooth Lyapunov theory, the structural stability of the system is analyzed to predict the onset of the destabilizing chaotic behavior. The proposed stability results, formulated as a Linear Matrix Inequality (LMI) problem, demonstrate how the new method can detect the loss of stability just before the grazing bifurcation.     

Example of a non-smooth Hopf bifurcation in an aero-elastic system

We investigate a typical aerofoil section under dynamic stall conditions, the structural model is linear and the aerodynamic loading is represented by the Leishman–Beddoes semi-empirical dynamic stall model. The loads given by this model are non-linear and non-smooth, therefore we have integrated the equation of motion using a Runge–Kutta–Fehlberg algorithm equipped with event detection. The main focus of the paper is on the interaction between the Hopf bifurcation typical of aero-elastic systems, which causes flutter oscillations, and the discontinuous definition of the stall model. The paper shows how the non-smooth definition of the dynamic stall model can generate a non-smooth Hopf bifurcation. The mechanisms for the appearance of limit cycle attractors are described by using standard tools of the theory of dynamical systems such as phase plots and bifurcation diagrams.     

基于功能的基础隔震结构一体化优化设计

“投资-效益”准则作为基于功能的抗震设计的一个基本原则，反映了现代结构抗震设计思想的一个重要转变，即从以往只注重结构安全，向全面注重结构的功能、安全及经济等诸多方面发展。本文根据“投资-效益”准则，建立了基于功能的基础隔震结构一体化优化设计模型，其中目标函数考虑了结构生命周期内的总造价，包括结构初始造价和不同发生概率的大震、中震、小震作用下的结构损失期望，约束条件包括隔震层水平位移、上部结构层间变形等，优化过程中同时对上部结构截面尺寸和隔震器参数进行优化。算例表明，本文建立的基于功能的隔震结构优化模型是合理可行的，可为实际隔震结构的优化设计提供参考。     

Dynamics of a linear system with time-dependent mass and a coupled light mass with non-smooth potential

Multi-scale dynamics of two coupled oscillators, the linear one with varying mass and a non-smooth light system is studied. The light system, namely nonlinear energy sink (NES) is implemented for passively controlling the linear system against external impulses and/or forces. Obtained invariant manifold of the system at fast time scale let us explain the system behavior during its transient and steady-state regime while predicted dynamics of the system at slow time scale let us detect the positions of fixed and fold singularities by explanation of its behavior.     

On Loads Acting on a Structure with a Seismic Damper

The problem on amplification of the translation acceleration by a seismoisolated structure is considered with regard for the nonlinearity of friction and the multifrequency nature of the external perturbation. The equivalent coefficients of Coulomb and turbulent friction that provide a transient time equal to that of a system with viscous friction are calculated. Spectral curves that reflect the dependence of the ratio of the maximum absolute acceleration to the maximum translation acceleration on the natural part of a vibrating system are plotted for various equivalent parameters of nonlinear friction     

Grazing Bifurcation of a Cantilever Vibrating System with One-sided Impact

与光滑动力系统不同,擦边分岔是非光滑动力系统中的一种特殊分岔行为. 局部不连续映射 是研究非光滑动力系统擦边分岔的一种有力工具. 对一类单侧弹性碰撞悬臂振动系统进行了擦边分岔分析. 首先建立了系统对应的局部不连 续映射(ZDM)和全局Poincar\'{e}映射,进而在其他参数固定,碰撞间隙$g$为分 岔参数时利用数值仿真的方法分别对原系统和对应的Poincar\'{e} 映射进行擦边分岔分析,得到了该系统的两种不同类型的擦边分岔行为：周期1到周期2运 动和周期1到混沌,这两种擦边分岔与刚性碰撞系统的情况是不相同的. 由分析可知,对 于含高阶非线性项的非光滑动力系统的擦边分岔,同样可以利用局部不连续映射的方法进行 研究.     

Stationary mode of a nonlinear elastically hereditary oscillator

The steady-state forced oscillations of a single-mass system subject to an external pure harmonic force are considered. The role of restoring force is played by a nonlinear function, which takes account of hereditary effects as a sum of multiple integrals in accordance with the Volterra theory. The problem is solved by the method of equivalent linearization, the discussion being confined to a triple integral of the hereditary type. The influence of the hereditary nonlinearity on the system dynamic characteristics, namely, its amplitude, phase, dynamic rigidity, hysteresis loop area, and Q, is investigated. In particular, the reciprocal of the Q, which can be taken as a measure of the internal friction, is shown to be independent of the amplitude of oscillation and to be the same as that obtained by linear theory. The other dynamic characteristics prove sensitive to the nonlinear properties. The exponential rational fractions proposed by Yu. N. Rabotnov are used as concrete hereditary functions.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 11, No. 3, pp. 111–116, May–June, 1970.     

Seismic base isolation by nonlinear mode localization

Summary In this paper, the performance of a nonlinear base-isolation system, comprised of a nonlinearly sprung subfoundation tuned in a 1∶1 internal resonance to a flexible mode of the linear primary structure to be isolated, is examined. The application of nonlinear localization to seismic isolation distinguishes this study from other base-isolation studies in the literature. Under the condition of third-order smooth stiffness nonlinearity, it is shown that a localized nonlinear normal mode (NNM) is induced in the system, which confines energy to the subfoundation and away from the primary or main structure. This is followed by a numerical analysis wherein the smooth nonlinearity is replaced by clearance nonlinearity, and the system is excited by ground motions representing near-field seismic events. The performance of the nonlinear system is compared with that of the corresponding linear system through simulation, and the sensitivity of the isolation system to several design parameters is analyzed. These simulations confirm the existence of the localized NNM, and show that the introduction of simple clearance nonlinearity significantly reduces the seismic energy transmitted to the main structure, resulting in significant attenuation in the response. This work was supported in part by the National Science Foundation Grant CMS 00-00060. The authors are grateful for this support.     

几何非线性摩擦阻尼隔振系统动力学行为研究

非线性隔振系统由于具有较线性系统更优的隔振性能,因此在工程中应用广泛.本文通过配置与被隔振对象的运动方向相垂直的库伦摩擦阻尼器,构建了几何非线性摩擦阻尼模型.由于引入了几何非线性,因此其摩擦力与位移正相关,这与传统与位移无关摩擦力模型有显著不同.首先,建立了具有几何非线性摩擦阻尼的数学模型以及隔振系统的受迫振动方程;然后,使用谐波平衡法求解了动力学方程,并使用数值仿真方法验证了谐波平衡法求解的准确性;最后,研究了几何非线性摩擦阻尼隔振器的绝对位移传递率和相对位移传递率.研究结果表明,在库伦摩擦阻尼选择适当,非线性摩擦阻尼系统可以在保持高频振动衰减效果的前提下,显著降低系统共振峰,其性能优于传统的恒定摩擦阻尼隔振模型.同时,几何非线性摩擦阻尼系统能够避免传统摩擦阻尼系统中的“锁定”现象,从传递率角度来说,不利于共振峰控制;但从激励环境改变引发隔振系统失效的角度来看,几何非线性摩擦阻尼系统可以拓宽系统对激励幅值的适应范围,避免隔振系统失效.本文的研究结果对此类隔振系统的设计和摩擦阻尼参数的选择具有通用的指导意义.      

Dynamic Processes in a Spheroidal Seismic-Damping Mechanism

A mathematical model of a variable-structure system of solids with rolling friction and unilateral sliding friction bonds is described. The model can be applied to seismic isolation mechanisms. Conditions for transition between the possible structures of the equations of state are formulated. The behavior of the system with kinematically defined motion of the base is analyzed as an example__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 1, pp. 119–127, January 2005.     

带有摩擦耗能元件的框架结构动力分析

本文提出粘性屈服模型来模拟摩擦耗能元件的力-速度关系,该模型是连续变化的,克服了库仑摩擦模型不连续导致数值计算复杂的缺点,在进行摩擦耗能体系的动力分析中,采用缩减自由度技术,并作适当的变换,则带有摩擦耗能元件体系的动力分析归结为求解微分代数方程,本文采用增量型Rosenbrock二级三阶半隐式Runge-Kutta法求解该方程,以考虑框架和支撑的材料和几何非线性。对带有摩擦耗能元件的钢框架进行了弹塑性动力分析,研究了支撑刚度与结构层刚度的比值、摩擦力的大小以及地震波类型等参数对体系的影响。     

Nonlinear aeroelastic behavior of an oscillating airfoil during stall-induced vibration

In this paper, nonlinear aeroelastic behavior of a two-dimensional symmetric rotor blade in the dynamic stall regime is investigated. Two different oscillation models have been considered here: pitching oscillation and flap–edgewise oscillation. Stall aeroelastic instability in such systems can potentially lead to structural damage. Hence it is an important design concern, especially for wind turbines and helicopter rotors, where such modes of oscillation are likely to take place. Most previous analyses of such dynamical systems are not exhaustive. System parameters like structural nonlinearity or initial conditions have not been studied which could play a significant role on the overall dynamics. In the present paper, a parametric study on the aeroelastic instability and the nonlinear dynamical behavior of the system has been performed. Emphasis is given on the effect of structural nonlinearity and initial conditions. The aerodynamic loads in the dynamic stall regime have been computed using the Onera model. The qualitative influence of the system parameters is different in the two systems studied. The effect of structural nonlinearity on the bifurcation pattern of the system response is significant in the case of pitching oscillation. The initial condition plays an important role on the aeroelastic stability as well as on the bifurcation pattern in both the systems. In the forced response study, interesting dynamical behavior, like period-3 response, has been observed in the pitching oscillation case. On the other hand, for the flap–edgewise oscillation case, super-harmonic and quasi-harmonic response have been found.     

Semi-analytical solution for a system with clearance nonlinearity and periodic excitation

Many dynamical systems such as gears, tire-pavement, automotive brakes, and cam-follower have clearance nonlinearity and excitation, which are periodic in nature. It is essential to accurately predict the steady-state response of these systems using contact-mechanics-based model for understanding their nonlinear dynamic behavior. Among the methods available to theoretically solve the system’s nonlinear governing equation(s), a semi-analytical technique such as the harmonic balance method (HBM) is preferred over numerical approaches for various reasons, including accuracy. An HBM formulation that can predict the fundamental, sub-, and super-harmonic solutions is presented here. As multiple variants of HBM exist in the literature, this work focuses on comparatively evaluating the most appropriate variant for the system under consideration. Since the system has multiple discontinuities in terms of contact stiffness and damping forces, these have to be smoothed precisely to be utilized in the HBM. Hence, a novel smoothing function was proposed and evaluated against other existing smoothing functions in literature based on various criteria. Next, the most applicable HBM variant was selected with reference to steady-state solutions from numerical methods. The predictions from the selected HBM variant were validated against the results furnished in the literature for a similar system. Finally, the nonlinear frequency response of the system with multiple discontinuities was estimated using the selected HBM and found to be in good agreement with numerical results.

     

地下爆炸冲击波引起的基底滑移隔震建筑的动力响应

对埋置在土体中的圆形衬砌内发生意外爆炸引起的附近地表的滑移隔震建筑 (考虑了土 结构相互作用 )的动力响应进行了研究。在土 结构相互作用分析中 ,分别采用有限元、无穷元和集中质点体系建立了衬砌周围的有限土区、四周的无限土区以及基底滑移隔震建筑的计算模型 ;在基底滑移隔震分析中 ,提出了用连续摩擦力模型取代传统的库仑摩擦力模型 ,避免跟踪啮合滑移的过渡边界并可减小计算误差。通过数值模拟 6层基底滑移隔震建筑在遭受地下爆炸冲击波作用且考虑土 结构相互作用所产生的动力响应 ,分析了基底滑移隔震效果以及不同隔震参数对隔震效果的影响 ,为相关的理论研究和工程实践提供了依据。