Low speed flutter and limit cycle oscillations of a two-degree-of-freedom flat plate in a wind tunnel

This paper explores the dynamical response of a two-degree-of-freedom flat plate undergoing classical coupled-mode flutter in a wind tunnel. Tests are performed at low Reynolds number (Re~2.5×10⁴), using an aeroelastic set-up that enables high amplitude pitch–plunge motion. Starting from rest and increasing the flow velocity, an unstable behaviour is first observed at the merging of frequencies: after a transient growth period the system enters a low amplitude limit-cycle oscillation regime with slowly varying amplitude. For higher velocity the system transitions to higher-amplitude and stable limit cycle oscillations (LCO) with amplitude increasing with the flow velocity. Decreasing the velocity from this upper LCO branch the system remains in stable self-sustained oscillations down to 85% of the critical velocity. Starting from rest, the system can also move toward a stable LCO regime if a significant perturbation is imposed. Those results show that both the flutter boundary and post-critical behaviour are affected by nonlinear mechanisms. They also suggest that nonlinear aerodynamic effects play a significant role.     

Rich dynamics caused by diffusion

It is an awfully difficult task to design an efficient numerical method for bifurcation diagrams, the graphs of Lyapunov exponents, or the topological entropy about discrete dynamical systems by linear/nonlinear diffusion with the Direchlet/Neumann- boundary conditions. Until now there are less works concerned with such a problem. In this paper, we propose a scheme about bifurcating analysis in a series of discrete-time dynamical systems with linear/nonlinear diffusion terms under the periodic boundary conditions. The complexity of dynamical behaviors caused by the diffusion term are to be determined. Bifurcation diagrams are shown by numerical simulation and chaotic behavior (chaotic Turing patterns) is demonstrated by computing the largest Lyapunov exponent. Our theoretical model can give an interesting case study about the phenomenon: the individuals exhibit a very simple dynamics but the groups with linear/nonlinear coupling can own a complex dynamics including fluctuation, periodicity and even chaotic behavior. We find that diffusion can trigger chaotic behavior in the present system and there exist multiple Turing patterns. It is interesting as regular or chaotic patterns can be reported in this study. Chaotic orbits emerge when exploring further in the diffusion coefficient space, and such a behavior is entirely absent in the corresponding continuous time-space system. The method proposed in the present paper is innovative and the conclusion is novel.

     

Hopf bifurcation control for stochastic dynamical system with nonlinear random feedback method

Hopf bifurcation control in nonlinear stochastic dynamical system with nonlinear random feedback method is studied in this paper. Firstly, orthogonal polynomial approximation is applied to reduce the controlled stochastic nonlinear dynamical system with nonlinear random controller to the deterministic equivalent system, solvable by suitable numerical methods. Then, Hopf bifurcation control with nonlinear random feedback controller is discussed in detail. Numerical simulations show that the method provided in this paper is not only available to control the stochastic Hopf bifurcation in nonlinear stochastic dynamical system, but is also superior to the deterministic nonlinear feedback controller.     

Stability and Hamiltonian Hopf bifurcation for a nonlinear symmetric bladed rotor

The modal interaction which leads to Hamiltonian Hopf bifurcation is studied for a nonlinear rotating bladed-disk system. The model, which is discussed in the paper, is a Jeffcott rotor carrying a number of planar blades which bend in the plane of the motion. The rigid rotating disk is supported on nonlinear bearings. It is supposed that this dynamical system is a Hamiltonian system which is perturbed by small dissipative and nonlinear forces. Krein’s theorem is employed for obtaining a stability criterion. The nonlinear eigenvalue equations on the stability boundary are turned into ordinary differential equations (ODEs) by differentiating them over the rotating speed. By solving these ODEs, the eigenmodes and the eigenvalues on the stability boundary are obtained. The bifurcation analysis is performed by applying multiple scales method around the boundary. The rotor nonlinear behavior and damping effects are studied for different conditions on the rotating speed and nonlinearity type by the bifurcation equation. It is shown that the damping distribution between the blades and bearings may shift the unstable mode. Depending on the nonlinearity type, subcritical and supercritical Hopf bifurcation are possible.     

A free boundary problem modeling thermal instabilities: Stability and bifurcation

In this paper, we analyze a simple free boundary model associated with solid combustion and some phase transition processes. There is strong evidence that this one-phase model captures all major features of dynamical behavior of more realistic (and complicated) combustion and phase transition models. The principal results concern the dynamical behavior of the model as a bifurcation parameter (which is related to the activation energy in the case of combustion) varies. We prove that the basic uniform front propagation is asymptotically stable against perturbations for the bifurcation parameter above the instability threshold and that a Hopf bifurcation takes place at the threshold value. Results of numerical simulations are presented which confirm that both supercritical and subcritical Hofp bifurcation may occur for physically reasonable nonlinear kinetic functions.     

一类平面自治非线性时滞控制系统的多稳态和混沌

众所周知,平面自治系统即使具有光滑非线性存在,系统也不会出现复杂的动力学行为。本文研究这样的系统存在时滞时,时滞量对系统的动力学行为的影响。通过对一个平面自治非线性系统引入时滞反馈,得到数学模型。利用泛函分析和平均法建立系统平衡态随时滞量变化的失稳机理,研究表明：时滞量平面自治系统动力学行为的影响是本质的．时滞量不但可以使系统出现Hopf分岔,产生周期振动。而且还可以使系统出现多稳态的周期运动或周期吸引子,这些共存的吸引子相碰是导致系统复杂的动力学行为,包括概周期和混沌运动。     

Stability and bifurcation for a coupled nonlinear relative rotation system with multi-time delay feedbacks

In this paper, we investigate the stability and bifurcation of a class of coupled nonlinear relative rotation system with multi-time delay feedbacks. Using dissipative system Lagrange equation, the dynamics equation of coupled nonlinear relative rotation system with three masses is established. The dynamical behaviors of the system under multi-time delay feedbacks, with two state variables, are discussed. First, characteristic roots and the stable regions of time delay are determined by direct method. The relation between two time delays ratio or time delay feedbacks gains and the stable regions of time delay is analyzed. Second, the direction and stability of Hopf bifurcation are decided by normal form theorem and center manifold argument. Finally, numerical simulation can confirm the validity of the conclusion.     

Bifurcation boundary analysis as a nonlinear damage detection feature: does it work?

Many systems in engineering and science are inherently nonlinear and require damage detection. For such systems, nonlinear damage detection methods may be useful. A bifurcation boundary analysis method as a new nonlinear damage detection tool was previously introduced in the literature to track bifurcation boundary changes due to damages over a small region of an aeroelastic panel model. Results of this method based upon a finite difference solution showed higher sensitivities to the small amount of damage than methods based upon linear models. In this paper, four methods including Finite Difference, Finite Element (FEM), Rayleigh-Ritz and Galerkin Approach are used to further investigate the sensitivity of the bifurcation boundary for damage detection. Results of the FEM and Rayleigh-Ritz method agree with each other and also show that the sensitivity of the bifurcation boundary to damage is much less than what previously reported when using a finite difference solution method.     

Complex dynamics and targeted energy transfer in linear oscillators coupled to multi-degree-of-freedom essentially nonlinear attachments

We study the dynamics of a system of coupled linear oscillators with a multi-DOF end attachment with essential (nonlinearizable) stiffness nonlinearities. We show numerically that the multi-DOF attachment can passively absorb broadband energy from the linear system in a one-way, irreversible fashion, acting in essence as nonlinear energy sink (NES). Strong passive targeted energy transfer from the linear to the nonlinear subsystem is possible over wide frequency and energy ranges. In an effort to study the dynamics of the coupled system of oscillators, we study numerically and analytically the periodic orbits of the corresponding undamped and unforced hamiltonian system with asymptotics and reduction. We prove the existence of a family of countable infinity of periodic orbits that result from combined parametric and external resonance interactions of the masses of the NES. We numerically demonstrate that the topological structure of the periodic orbits in the frequency–energy plane of the hamiltonian system greatly influences the strength of targeted energy transfer in the damped system and, to a great extent, governs the overall transient damped dynamics. This work may be regarded as a contribution towards proving the efficacy the utilizing essentially nonlinear attachments as passive broadband boundary controllers. PACS numbers: 05.45.Xt, 02.30.Hq     

Dynamical behavior analysis and bifurcation mechanism of a new 3-D nonlinear periodic switching system

This paper presents a new periodic switching chaotic system, which is topologically non-equivalent to the original sole chaotic systems. Of particular interest is that the periodic switching chaotic system can generate stable solution in a very wide parameter domain and has rich dynamic phenomena. The existence of a stable limit cycle with a suitable choice of the parameters is investigated. The complex dynamical evolutions of the switching system composed of the Rössler system and the Chua’s circuit are discussed, which is switched by equal period. Then the possible bifurcation behaviors of the system at the switching boundary are obtained. The mechanism of the different behaviors of the system is investigated. It is pointed out that the trajectories of the system have obvious switching points, which are decided by the periodic signal. Meanwhile, the system may be led to chaos via a period-doubling bifurcation, resulting in the switching collisions between the trajectories and the non-smooth boundary points. The complicated dynamics are studied by virtue of theoretical analysis and numerical simulation. Furthermore, the control methods of this periodic switching system are discussed. The results we have obtained clearly show that the nonlinear switching system includes different waveforms and frequencies and it deserves more detailed research.     

Nonlinear analysis of time-delay position feedback control of container cranes

Time-delay feedback control of container cranes is robustly stable and insensitive to initial conditions for most of the linearly stable region. To better understand this robustness and any limitations of the technique, we undertake a nonlinear analysis of the system. To this end, we develop a nonlinear model of the crane system by modeling the crane-hoist-payload assembly as a double pendulum. Then, we derive a linear approximation specific to this model. Finally, we derive a cubic model of the dynamics for nonlinear analysis. Using linear analysis, we determine the gain and time delay factors for stabilizing controllers. Also, we show that the controller undergoes a Hopf bifurcation at the linear stability boundary. Using the method of multiple scales on the cubic model, we determine the normal form of the Hopf bifurcation. We then show that for practical operating ranges, the controller undergoes a supercritical bifurcation that helps explain the robustness of the controller.     

Nonlinear dynamical stability analysis of the circular three-dimensional frame

The three-dimensional frame is simplified into flat plate by the method of quasiplate. The nonlinear relationships between the surface strain and the midst plane displacement are established. According to the thin plate nonlinear dynamical theory, the nonlinear dynamical equations of three-dimensional frame in the orthogonal coordinates system are obtained. Then the equations are translated into the axial symmetry nonlinear dynamical equations in the polar coordinates system. Some dimensionless quantities different from the plate of uniform thickness are introduced under the boundary conditions of fixed edges, then these fundamental equations are simplified with these dimensionless quantities. A cubic nonlinear vibration equation is obtained with the method of Galerkin. The stability and bifurcation of the circular three-dimensional frame are studied under the condition of without outer motivation. The contingent chaotic vibration of the three-dimensional frame is studied with the method of Melnikov. Some phase figures of contingent chaotic vibration are plotted with digital artificial method.     

Quasiperiodic oscillations in robot dynamics

Delayed robot systems, even of low degree of freedom, can produce phenomena which are well understood in the theory of nonlinear dynamical systems, but hardly ever occur in simple mechanical models. To illustrate this, we analyze the delayed positioning of a single degree of freedom robot arm which leads to an infinite dimensional dynamical system. Restricting the dynamics to a four dimensional center manifold, we show that the system undergoes a codimension two Hopf bifurcation for an infinite set of parameter values. This provides a mechanism for the creation of two-tori in the phase space and gives a theoretical explantion for self-excited quasiperiodic oscillations of force controlled robots. We also compare our results with experimental data.     

非线性参数激励系统的动力分叉研究

本文针对弹性梁动力曲屈分叉问题,建立了系统的非线性Mathiue方程,较全面地讨论了此类参数激励系统的1/2亚谐分叉特性,指出以往对此类问题的研究得到的只是一种退化情形下的分叉特性,阐述了分叉方程的截断对分叉结果的影响,得到了一些新的结果。文中还介绍了一个模型弹性梁系统分叉响应特性的实测结果,证实了理论分析的可靠性。     

双频1:2激励下修正蔡氏振子两尺度耦合行为

不同尺度耦合系统存在的复杂振荡及其分岔机理一直是当前国内外研究的热点课题之一. 目前相关工作大都是针对单频周期激励频域两尺度系统,而对于含有两个或两个以上周期激励系统尺度效应的研究则相对较少. 为深入揭示多频激励系统的不同尺度效应,本文以修正的四维蔡氏电路为例,通过引入两个频率不同的周期电流源,建立了双频1:2周期激励两尺度动力学模型. 当两激励频率之间存在严格共振关系,且周期激励频率远小于系统的固有频率时,可以将两周期激励项转换为单一周期激励项的函数形式. 将该单一周期激励项视为慢变参数,给出了不同激励幅值下快子系统随慢变参数变化的平衡曲线及其分岔行为的演化过程,重点考察了3种较为典型的不同外激励幅值下系统的簇发振荡行为. 结合转换相图,揭示了各种簇发振荡的产生机理. 系统的轨线会随慢变参数的变化,沿相应的稳定平衡曲线运动,而fold分岔会导致轨迹在不同稳定平衡曲线上的跳跃,产生相应的激发态. 激发态可以用从分岔点向相应稳定平衡曲线的暂态过程来近似,其振荡幅值的变化和振荡频率也可用相应平衡点特征值的实部和虚部来描述,并进一步指出随着外激励幅值的改变,导致系统参与簇发振荡的平衡曲线分岔点越多,其相应簇发振荡吸引子的结构也越复杂.      

Two-parameter bifurcation analysis of limit cycles of a simplified railway wheelset model

The effect of the nonlinear terms on bifurcation behaviors of limit cycles of a simplified railway wheelset model is investigated. At first, the stable equilibrium state loses its stability via a Hopf bifurcation. The bifurcation curve is divided into a supercritical branch and a subcritical one by a generalized Hopf point, which plays a key role in determining the occurrence of flange contact and derailment of high-speed railway vehicles, and the occurrence of this critical situation is an important decision-making criteria for design parameters. Secondly, bifurcations of limit cycles are discussed by comparing the bifurcation behavior of cycles for two different nonlinear parameters. Unlike local Hopf bifurcation analysis based on a single bifurcation parameter in most papers, global bifurcation analysis of limit cycles based on two bifurcation parameters is investigated, simultaneously. It is shown that changing nonlinear parameter terms can affect bifurcation types of cycles and division of parameter domains. In particular, near the branch points of cycles, two symmetrical limit cycles are created by a pitchfork bifurcation and then two symmetrical cycles both undergo a period-doubling bifurcation to form two stable period-two cycles. Around the resonant points, period orbits can make several turns, whose number of turns corresponds to the ratio of resonance. Thirdly, near the Neimark–Sacker bifurcation of cycles, a stable torus is created by a supercritical Neimark–Sacker bifurcation, which shows that the orbit of the model exhibits modulated oscillations with two frequencies near the limit cycle. These results demonstrate that nonlinear parameter terms can produce very complex global bifurcation phenomena and make obvious effects on possible hunting motions even though a simple railway wheelset model is concerned.

     

Adjoint operator method and normal forms of higher order for nonlinear dynamical system

I.IntroductionInordertostudybifurcationsofnonlineardynamicalsystemsinthedegeneratecasesofhighercodimensionnumber(>3),wemustcomputenormalformsofhigherorderfornonlineardynamicalsystems.Inrecenttwentyyearsmanyscientistsmadeveryimportantcontributionstodevelop…     

Nonlinear dynamics of self-, parametric,and externally excited oscillator with time delay: van der Pol versus Rayleigh models

The regular and chaotic vibrations of a nonlinear structure subjected to self-, parametric, and external excitations acting simultaneously are analysed in this study. Moreover, a time delay input is added to the model to control the system response. The frequency-locking phenomenon and transition to quasi-periodic oscillations via Hopf bifurcation of the second kind (Neimark–Sacker bifurcation) are determined analytically by the multiple time scales method up to the second-order perturbation. Approximate solutions of the quasi-periodic motion are determined by a second application of the multiple time scales method for the slow flow, and then, slow–slow motion is obtained. The similarities and differences between the van der Pol and Rayleigh models are demonstrated for regular, periodic, and quasi-periodic oscillations, as well as for chaotic oscillations. The control of the structural response, and modifications of the resonance curves and bifurcation points by the time delay signal are presented for selected cases.

     

Dynamic modeling and extended bifurcation analysis of flexible-link manipulator

Abstract

In this article, the nonlinear dynamic analysis of a flexible-link manipulator is presented. Especially, the possibility of chaos occurrence in the system dynamic model is investigated. Upon the occurrence of chaos, the system dynamical behavior becomes unpredictable which in turn brings about uncertainty and irregularity in the system motion. The importance of this investigation is pronounced in similar systems such as double pendulum and single-link flexible manipulator. What makes this study distinct from previous ones is the increase in the number of links as well as the changing the bifurcation parameters from system mechanical parameters to force and torque inputs. To this aim, the motion equations of the N-link robot, which are derived with the aid of the recursive Gibbs-Appell formulation and the assumed modes method, are used. In the end, the equations of motion are developed for a two-link flexible manipulator, and its nonlinear dynamical behavior is analyzed via numerical integration of discrete equations. The results are presented in the form of bifurcation diagrams (for variation of torque amplitude), time histories, phase-plane portraits, Poincaré sections, and fast Fourier transforms. The outcomes indicate that when there is no offset, the decrease in damping results in chaotic generalized modal coordinates. In addition, as the excitation frequency decreases from 2π to π, a limiting amplitude is created at 0.35 before which the behavior of generalized rigid and modal coordinates is different, while this behavior has more similarity after this point. An experimental setup is also used to check the torques as the system input.     

Nonlinear dynamics of a fluid-conveying curved pipe subjected to motion-limiting constraints and a harmonic excitation

This paper investigates the dynamical behaviour of a fluid-conveying curved pipe subjected to motion-limiting constraints and a harmonic excitation. Based on a Newtonian method, the in-plane equation of motion of this curved pipe is derived. Then a set of discrete equations in spatial space obtained by the differential quadrature method (DQM) is solved numerically. Emphasis is placed on the possible dynamical behaviour of the curved pipe conveying fluid. The numerical results show that the pipe without motion-limiting constraints but with a harmonic force behaves as an ordinary linear system. If, however, the pipe is subjected to cubic motion-limiting constraints, nonlinear dynamic phenomena of the system will occur. Calculations of bifurcation diagrams, phase-plane portraits, time responses, power spectrum diagrams, and Poincaré maps of the oscillations clearly demonstrate the existence of chaotic and quasiperiodic motions. Moreover, it is shown that the route to chaos is via a sequence of period-doubling bifurcations.