通识课"混沌与非线性思维"建设

为探讨非线性动力学的普及化和整合科学新进展加强通识教育课程,本文总结和分析了通识教育课"混沌与非线性思维"的设计与实践。非线性动力学现在已经成为人类知识的一部分。"混沌与非线性思维"是为上海大学所有学生(包括理工、经营和人文三大类)开设的通识教育课。该课程聚焦于混沌的科学概念和文化影响,非线性不仅作为数学模型也作为思维模式。该课程帮助学生从非线性思维角度观察、分析和理解自然和社会中的不可预测和不确定现象。教学实践表明,非线性动力学可以恰当的方式普及而成为通识教育课程的实质性内容,并为各专业的本科生接受。     

核心通识课“无处不在的力学”的教学实践

力学通识教育对培养学生科学与工程素养具有重要意义。本文基于核心通识课“无处不在的力学”的多轮教学实践,对多学时条件下的力学通识课程的组织与实施进行了探索,建立形成了“学科规划、多人授课、专人负责”的通识课程管理模式、“价值引领、知识传授、能力培养”相融合的课程理念和“讲授式教学、研讨式教学、研究型教学”相结合的多元教学模式。课程开设以来,学生选课踊跃,教学团队投入度高,促进了学生科学与工程素养的提高,扩大了力学学科在师生中的影响。     

空气动力学研讨课建设的探索与实践

为了丰富基础课堂教学资源、满足本科生通识教育的需求,北京航空航天大学近期推出建设名师研讨课、新生研讨课、专业研讨课等,毫无疑问这些对学校本科教学的深化改革具有重要的促进作用.针对我校国家级精品课程空气动力学教学改革的实践,总结了在建设空气动力学名师研讨课过程中的构思和经验,以供参考.空气动力学是一门理论与实际紧密结合的基础课程,在研讨课建设中,提出以空气动力学典型问题为引导,通过启发、实验、讨论、分析等多种形式,引导学生积极主动思考,从鸟的飞行原理、实验现象和结果中获取灵感、凝练科学问题,对激发学习兴趣、提高创新实践能力等具有明显效果.     

准周期激励Duffing系统中奇异非混沌吸引子的判别方法

奇异非混沌动力学是非线性动力学领域中的新课题。本文以准周期激励Duffing振子为例,对其产生的奇异非混沌吸引子(strange nonchaotic attractors, SNAs)进行分析。通过三维庞加莱截面和定量方法如傅里叶变换、李雅普诺夫指数、李雅普诺夫维数、关联维数和盒维数检测SNAs是否存在。研究结果表明,傅里叶变换无法判断混沌与奇异非混沌行为。而李雅普诺夫指数、李雅普诺夫维数可以作为检测系统混沌与非混沌指标。关联维数和盒维数显著表明系统奇异与非奇异性,从而阐明适用于准周期驱动Duffing振子中存在SNAs的判别方法,并为其他类似系统检测SNAs提供指导。     

四自由度非线性系统的随机分岔混沌特性研究

以四自由度迟滞非线性随机振动模型为研究对象,以速度和位移立方的模型来模拟振动系统的迟滞非线性力,并以Monte Carlo法模拟随机位移激励,对迟滞非线性随机系统的动力学特性进行分析.通过系统的Poincare截面、分岔图及最大Lyapunov指数分析了系统迟滞非线性力各参数对系统混沌状态的影响.研究表明,非线性刚度系数对振动系统混沌状态的影响较小,线性阻尼项和线性刚度项次之,而非线性阻尼项的影响最为明显.不仅证明了非线性振动系统随机混沌振动现象的存在,更重要的是可以为非线性振动系统参数的合理取值提供理论依据.     

高斯色噪声和谐波激励共同作用下耦合SD振子的混沌研究

耦合SD振子作为一种典型的负刚度振子, 在工程设计中有广泛应用. 同时高斯色噪声广泛存在于外界环境中, 并可能诱发系统产生复杂的非线性动力学行为, 因此其随机动力学是非线性动力学研究的热点和难点问题. 本文研究了高斯色噪声和谐波激励共同作用下双稳态耦合SD振子的混沌动力学, 由于耦合SD振子的刚度项为超越函数形式, 无法直接给出系统同宿轨道的解析表达式, 给混沌阈值的分析造成了很大的困难. 为此, 本文首先采用分段线性近似拟合该振子的刚度项, 发展了高斯色噪声和谐波激励共同作用下的非光滑系统的随机梅尔尼科夫方法. 其次, 基于随机梅尔尼科夫过程, 利用均方准则和相流函数理论分别得到了弱噪声和强噪声情况下该振子混沌阈值的解析表达式, 讨论了噪声强度对混沌动力学的影响. 研究结果表明, 随着噪声强度的增大混沌区域增大, 即增大噪声强度更容易诱发耦合SD振子产生混沌. 当阻尼一定时, 弱噪声情况下混沌阈值随噪声强度的增加而减小; 但是强噪声情况下噪声强度对混沌阈值的影响正好相反. 最后, 数值结果表明, 利用文中的方法研究高斯色噪声和谐波激励共同作用下耦合SD振子的混沌是有效的.本文的结果为随机非光滑系统的混沌动力学研究提供了一定的理论指导.      

新工科背景下力学跨学科课程的建设与探讨

新一轮科技变革业已开始,我国正在大力推动“新工科”建设。力学专业作为工科的关键基础之一,需要结合新时代工程教育改革方向来推进专业教育与学科建设。本文以生物力学与仿生跨学科课程为例,分析了当前跨学科课程所面临的学生缺少了解、教师定位不清、教学方式单一等问题,提出了与低年级通识教育、基础力学课程、现代教育技术相结合的针对性措施,并对其实施效果进行总结,以期为现有工科中的跨学科课程建设提供借鉴和参考。     

非线性刚度不平衡转子径向碰摩动力学研究

以线性项和立方项之和来表示转轴材料的物理非线性因素,建立了考虑非线性油膜力和非线性刚度的轴转子系统的动力学模型,利用数值积分法对转子系统由于局部碰摩故障导致的非线性动力学行为进行了研究,发现此类非线性振动系统具有倍周期分岔、拟周期和混沌等复杂的动力学行为,为此类系统的安全运行和有效识别转子故障提供了理论参考。     

非线性动力学在心脏电活动研究中的应用

本文介绍了非线性动力学在心脏电活动研究中的应用。主要内容包括非线性分析指标、心率变异非线性分析、异常心率（心室振颤）的非线性分析、心脏细胞动作电位模型中的搏动节律的分叉与混沌现象及机理、混沌控制在心脏电活动研究中的应用,并对今后发展方向提出了展望。非线性动力学分析方法是研究心脏电活动的良好工具,使得解决心脏电活动中的难题成为可能。     

粘弹性矩形板的混沌和超混沌行为

从薄板Karman理论的基本假设出发；利用线性粘弹性理论中的Boltzman叠加原理，建立了粘弹性薄板非线性动力学分析的初边值问题，其运动方程是一组非线性积分──微分方程．在空间域上利用Galerkin平均化法之后，得到了变型的非线性积分──微分型的Duffing方程．综合利用动力系统中的多种方法，揭示了粘弹性矩形板在横向周期激励下的丰富的动力学行为，如不动点、极限环、混沌、奇怪吸引子、超混沌等，其中，混沌和超混沌是交替出现的．     

Leader-following consensus of nonlinear fractional-order multi-agent systems over directed networks

Nonlinear Dynamics - This paper investigates the leader-following consensus of nonlinear fractional-order multi-agent systems where the linear part is depicted by the general linear dynamics on...     

Estimation of the largest Lyapunov exponent-like (LLEL) stability measure parameter from the perturbation vector and its derivative dot product (part 2) experiment simulation

Controlling system dynamics with use of the Largest Lyapunov Exponent (LLE) is employed in many different areas of the scientific research. Thus, there is still need to elaborate fast and simple methods of LLE calculation. This article is the second part of the one presented in Dabrowski (Nonlinear Dyn 67:283–291, 2012). It develops method LLEDP of the LLE estimation and shows that from the time series of two identical systems, one can simply extract value of the stability parameter which value can be treated as largest LLE. Unlike the method presented in part, one developed method (LLEDPT) can be applied to the dynamical systems of any type, continuous, with discontinuities, with time delay and others. The theoretical improvement shows simplicity of the method and its obvious physical background. The proofs for the method effectiveness are based on results of the simulations of the experiments for Duffing and Van der Pole oscillators. These results were compared with ones obtained with use of the Stefanski method (Stefanski in Chaos Soliton Fract 11(15):2443–2451, 2000; Chaos Soliton Fract 15:233–244, 2003; Chaos Soliton Fract 23:1651–1659, 2005; J Theor Appl Mech 46(3):665–678, 2008) and LLEDP method. LLEDPT can be used also as the criterion of stability of the control system, where desired behavior of controlled system is explicitly known (Balcerzak et al. in Mech Mech Eng 17(4):325–339, 2013). The next step of development of the method can be considered in direction that allows estimation of LLE from the real time series, systems with discontinuities, with time delay and others.     

Complex dynamics of a harmonically excited structure coupled with a nonlinear energy sink

Nonlinear behaviors are investigated for a structure coupled with a nonlinear energy sink. The structure is linear and subject to a harmonic excitation, modeled as a forced single-degree-of-freedom oscillator. The nonlinear energy sink is modeled as an oscillator consisting of a mass,a nonlinear spring, and a linear damper. Based on the numerical solutions, global bifurcation diagrams are presented to reveal the coexistence of periodic and chaotic motions for varying nonlinear energy sink mass and stiffness. Chaos is numerically identified via phase trajectories, power spectra,and Poincaré maps. Amplitude-frequency response curves are predicted by the method of harmonic balance for periodic steady-state responses. Their stabilities are analyzed.The Hopf bifurcation and the saddle-node bifurcation are determined. The investigation demonstrates that a nonlinear energy sink may create dynamic complexity.     

Studies of Store-Induced Limit-Cycle Oscillations Using a Model with Full System Nonlinearities

The authors investigate limit-cycle oscillations of a wing/store configuration. Unlike typical aeroelastic studies that are based upon a linearized form of the governing equations, herein full system nonlinearities are retained, and include transonic flow effects, coupled responses from the structure, and store-related kinematics and dynamics. Unsteady aerodynamic loads are modeled with the equations from transonic small disturbance theory. The structural dynamics for the cantilevered wing are modeled by the nonlinear equations of motion for a beam. The effects of general store-placement are modeled by the nonlinear equations of motion related to the position-induced nonlinear kinematics. Chordwise deformations of the wing surface, as well as pylon and store flexibility, are assumed negligible. Nonlinear responses are studied by examining bifurcation and related response characteristics using direct simulation. Particular attention is given to cases for which large-time, time-dependent behavior is dependent on initial conditions, as observed for some configurations in flight test. Comparisons of results in which selective nonlinearities are excluded indicate that the accurate prediction of nonlinear responses such as limit cycle oscillations (LCOs) may depend upon consideration of all nonlinearities related to the full system.     

The Effect of Downsizing on Hierarchical Work Flow Dynamics in Organizations

The objective was to determine the impact of the downsizing of management personnel on the flow of work throughout the organization, given that the quantity of workers (or technicians) must remain the same. In particular the objective of study was to assess the impact of downsizing on performance dynamics, where social factors such as power objectives of the management groups, job insecurity, and perceptions of fairness were not active variables in the process. Three groups of university students (19-33 in a group) were formed to play the Chaos Exercise in a three-level hierarchy. In the three experimental conditions, management groups were downsized during the game from 3 to 2 players, from 2 players to 1, or from 3 players to 1. Nonlinear dynamical structures were explored for each condition and at all three levels of hierarchy using production records generated by each group. Results showed chaos for two out of three conditions at the lowest level of the hierarchy, and usually a tendency toward asymptotic stabilization in middle management, although the functions themselves were markedly varied. The 2-to-1 condition, however, showed a tendency toward performance stabilization at the worker and middle management levels, but chaos at the top management level.     

Nonlinear pedagogy: a constraints-led framework for understanding emergence of game play and movement skills

Team sport competition can be characterized as a complex adaptive system in which concepts from nonlinear dynamics can provide a sound theoretical framework to understand emergent behavior such as movement coordination and decision making in game play. Nonlinear Pedagogy is presented as a methodology for games teaching, capturing how phenomena such as movement variability, self-organization, emergent decision making, and symmetry-breaking occur as a consequence of interactions between agent-agent and agent-environment constraints. Empirical data from studies of basketball free-throw shooting and dribbling are used as task vehicles to exemplify how nonlinear phenomena characterize game play in sport. In this paper we survey the implications of these data for Nonlinear Pedagogy, focusing particularly on the manipulation of constraints in team game settings. The data and theoretical modeling presented in this paper provide a rationale in nonlinear dynamics for the efficacy of a prominent model of game play teaching, Teaching Games for Understanding approach.     

Mutual information matrix based on Rényi entropy and application

Nonlinear Dynamics - Quantifying information dynamics in a nonlinear system is crucial in complex dynamics. The Mutual Information Matrix (MIM) method was developed to study nonlinear interactions...     

On the dynamics of a nonlinear energy harvester with multiple resonant zones

Nonlinear Dynamics - The dynamics of a nonlinear vibration energy harvester for rotating systems is investigated analytically through harmonic balance, as well as by numerical analysis. The...     

Nonlinear damping in a micromechanical oscillator

Nonlinear elastic effects play an important role in the dynamics of microelectromechanical systems (MEMS). A Duffing oscillator is widely used as an archetypical model of mechanical resonators with nonlinear elastic behavior. In contrast, nonlinear dissipation effects in micromechanical oscillators are often overlooked. In this work, we consider a doubly clamped micromechanical beam oscillator, which exhibits nonlinearity in both elastic and dissipative properties. The dynamics of the oscillator is measured in both frequency and time domains and compared to theoretical predictions based on a Duffing-like model with nonlinear dissipation. We especially focus on the behavior of the system near bifurcation points. The results show that nonlinear dissipation can have a significant impact on the dynamics of micromechanical systems. To account for the results, we have developed a continuous model of a geometrically nonlinear beam-string with a linear Voigt–Kelvin viscoelastic constitutive law, which shows a relation between linear and nonlinear damping. However, the experimental results suggest that this model alone cannot fully account for all the experimentally observed nonlinear dissipation, and that additional nonlinear dissipative processes exist in our devices.     

Nonlinear dynamics of dry friction oscillator subjected to combined harmonic and random excitations

Nonlinear Dynamics - The dynamics of a nonlinear single degree freedom oscillator on a moving belt subjected to combined harmonic and random excitations is numerically investigated. The dynamics is...