Bifurcation and chaos analysis for aeroelastic airfoil with freeplay structural nonlinearity in pitch

The dynamics character of a two degree-of-freedom aeroelastic airfoil with combined freeplay and cubic stiffness nonlinearities in pitch submitted to supersonic and hypersonic flow has been gaining significant attention. The Poincare mapping method and Floquet theory are adopted to analyse the limit cycle oscillation flutter and chaotic motion of this system. The result shows that the limit cycle oscillation flutter can be accurately predicted by the Floquet multiplier. The phase trajectories of both the pitch and plunge motion are obtained and the results show that the plunge motion is much more complex than the pitch motion. It is also proved that initial conditions have important influences on the dynamics character of the airfoil system. In a certain range of airspeed and with the same system parameters, the stable limit cycle oscillation, chaotic and multi-periodic motions can be detected under different initial conditions. The figure of the Poincare section also approves the previous conclusion.

GLOBAL BIFURCATION AND CHAOS IN A VAN DER POL-DUFFING-MATHIUE SYSTEM WITH A SINGLE-WELL POTENTIAL OSCILLATOR

The global bifurcation and chaos are investigated in this paper for a van der Pol-Duffing-Mathieu system with a single-well potential oscillator by means of nonlinear dynamics. The autonomous system corresponding to the system under discussion is analytically studied to draw all global bifurcation diagrams in every parameter space. These diagrams are called basic bifurcation ones. Then fixing parameter in every space and taking the parametrically excited amplitude as a bifurcation parameter, we can observe how to evolve from a basic bifurcation diagram to a chaos pattern in terms of numerical methods. The results are sufficient to show that the system has distinct dynamic behavior. Finally, the properties of the basins of attraction are observed and the appearance of fractal basin boundaries heralding the onset of a loss of structural integrity is noted in order to consider how to control the extent and the rate of the erosion in the next paper.

APPLICATION OF WAVELET TRANSFORM TO BIFURCATION AND CHAOS STUDY

The response of a nonlinear vibration system may be of three types, namely, periodic, quasiperiodic or chaotic,,when the parameters of the system are changed. The periodic motions can be identified by Poincare map, and harmonic wavelet transform (HWT) can distinguish quasiperiod from chaos, so the existing domains of different types of motions of the system can be revealed in the parametric space with the method of HWT joining with Poincare map.

Study on Bifurcation and Chaos in Boost Converter Based on Energy Balance Model

Based on boost converter operating in discontinuous mode, this paper proposes an energy balance model (EBM) for analyzing bifurcation and chaos phenomena of capacitor energy and output voltage when the converter parameter is varying. It is found that the capacitor energy and output voltage dynamic behaviors exhibit the typical period-doubling route to chaos by increasing the feedback gain constant K of proportional controller. The accurate position of the first bifurcation point and the iterative diagram of the capacitor energy with every K can be derived from EBM. Finally, the underlying causes for bifurcations and chaos of a general class of nonlinear systems such as power converters are analyzed from the energy balance viewpoint. Com-paring with the discrete iterative model, EBM is simple and high accuracy. This model can be easily devel-oped on the nonlinear study of the other converters.

Bifurcation and chaos of airfoil with multiple strong nonlinearities

The bifurcation and chaos phenomena of two-dimensional airfoils with multiple strong nonlinearities are investigated. First, the strongly nonlinear square and cubic plunging and pitching stiffness terms are considered in the airfoil motion equations, and the fourth-order Runge-Kutta simulation method is used to obtain the numerical solutions to the equations. Then, a post-processing program is developed to calculate the physical parameters such as the amplitude and the frequency based on the discrete numerical solutions. With these parameters, the transition of the airfoil motion from balance, period, and period-doubling bifurcations to chaos is emphatically analyzed. Finally, the critical points of the period-doubling bifurcations and chaos are predicted using the Feigenbaum constant and the first two bifurcation critical values. It is shown that the numerical simulation method with post-processing and the prediction procedure are capable of simulating and predicting the bifurcation and chaos of airfoils with multiple strong nonlinearities.

Bifurcation and Chaos Analysis of Nonlinear Rotor System with Axial-grooved Gas-lubricated Journal Bearing Support

Axial-grooved gas-lubricated journal bearings have been widely applied to precision instrument due to their high accuracy, low friction, low noise and high stability. The rotor system with axial-grooved gas-lubricated journal bearing support is a typical nonlinear dynamic system. The nonlinear analysis measures have to be adopted to analyze the behaviors of the axial-grooved gas-lubricated journal bearing-rotor nonlinear system as the linear analysis measures fail. The bifurcation and chaos of nonlinear rotor system with three axial-grooved gas-lubricated journal bearing support are investigated by nonlinear dynamics theory. A time-dependent mathematical model is established to describe the pressure distribution in the axial-grooved compressible gas-lubricated journal bearing. The time-dependent compressible gas-lubricated Reynolds equation is solved by the differential transformation method. The gyroscopic effect of the rotor supported by gas-lubricated journal bearing with three axial grooves is taken into consideration in the model of the system, and the dynamic equation of motion is calculated by the modified Wilson-0-based method. To analyze the unbalanced responses of the rotor system supported by finite length gas-lubricated journal bearings, such as bifurcation and chaos, the bifurcation diagram, the orbit diagram, the Poincar6 map, the time series and the frequency spectrum are employed. The numerical results reveal that the nonlinear gas film forces have a significant influence on the stability of rotor system and there are the rich nonlinear phenomena, such as the periodic, period-doubling, quasi-periodic, period-4 and chaotic motion, and so on. The proposed models and numerical results can provide a theoretical direction to the design of axial-grooved gas-lubricated journal bearing-rotor system.

Bifurcation and chaos characteristics of hysteresis vibration system of giant magnetostrictive actuator

Chaotic motion and quasi-periodic motion are two common forms of instability in the giant magnetostrictive actuator(GMA).Therefore,in the present study we intend to investigate the influences of the system damping coefficient,system stiffness coefficient,disc spring cubic stiffness factor,and the excitation force and frequency on the output stability and the hysteresis vibration of the GMA.In this regard,the nonlinear piezomagnetic equation,Jiles-Atherton hysteresis model,quadratic domain rotation model,and the GMA structural dynamics are used to establish the mathematical model of the hysteresis vibration system of the GMA.Moreover,the multi-scale method and the singularity theory are used to determine the eo-dimensional two-bifurcation characteristics of the system.Then,the output response of the system is simulated to determine the variation range of each parameter when chaos is imposed.Finally,the fourth-order Runge-Kutta method is used to obtain the time domain waveform,phase portrait and Poincare mapping diagrams of the system.Subsequently,the obtained three graphs are analyzed.The obtained results show that when the system output is stable,the variation range of each parameter can be determined.Moreover,the stability interval of system damping coefficient,system stiffness coefficient,and the coefficient of the cubic stiffness term of the disc spring are obtained.Furthermore,the stability interval of the exciting force and the excitation frequency are determined.

Bifurcation and chaos in nonlinear rotor-bearing system

Bifurcation and chaos in rigid Jefccott rotor bearing system are studied, by following the multi variable Floquet theory. By calculating the largest Lyapunov exponent, the chaotic motion and ″periodic window″ phenomena are found for a certain bifurcation parameter. The results show that the motion of the rotor system features a complicated nonlinear dynamics phenomena, such as period doubling bifurcation, saddle node bifurcation, secondary Hopf bifurcation and chaotic motion.

BIFURCATION AND CHAOS OF THE CIRCULAR PLATES ON THE NONLINEAR ELASTIC FOUNDATION

According to the large amplitude equation of the circular plate on nonlinear elastic foundation , elastic resisting force has linear item , cubic nonlinear item and resisting bend elastic item. A nonlinear vibration equation is obtained with the method of Galerkin under the condition of fixed boundary. Floquet exponent at equilibrium point is obtained without external excitation. Its stability and condition of possible bifurcation is analysed. Possible chaotic vibration is analysed and studied with the method of Melnikov with external excitation . The critical curves of the chaotic region and phase figure under some foundation parameters are obtained with the method of digital artificial.

Complex Dynamics Caused by Torus Bifurcation in Power Systems

Torus bifurcation is a relatively complicated bifurcation caused by a pair of complex conjugate Floquet multipliers coming out of unit circle on the Poincare section. A three-bus system is employed to reveal the relationship between torus bifurcation and some complex dynamics. Based on theoretical analysis and simulation studies, it is found that torus bifurcation is a typical route to chaos in power system. Some complex dynamics usually occur after a torus bifurcation, such as self-organization, deep bifurcations, exquisite structure, coexistence of chaos and divergence. It is also found that chaos has close relationship with various instability scenarios of power systems. Studies of this paper are helpful to understand the mechanism of torus bifurcation in power system and relationship of chaos and power system instabilities.

Laminar-Turbulent Bifurcation Scenario in 3D Rayleigh-Benard Convection Problem

We are considering two initial-boundary value problems for Rayleigh-Benard convection in Oberbeck-Boussinesq approximation for incompressible fluid in 3D-rectangular domain with 4:4:1 geometric ratio with periodicity in two directions and cubic domain with 1:1:1 ratio and zero velocity and temperature gradient boundary conditions. For this purpose, we use two numerical method: one is a Pseudo-Spectral-Galerkin method with trigonometric-Chebyshev polynomials and the other is finite element/volume method with WENO interpolation for advection term. Numerical methods are presented shortly and are benchmarked against known DNS data and against one another (for quasi-periodic domain problem). Then we perform stability analysis using analytical expression for main stationary solutions and eigenvalue numerical analysis by applying Implicitly Restarted Arnoldi (IRA) method. The IRA is used to perform linear stability analysis, find bifurcations of stationary points and analyze eigenvalues of monodromy matrices. Thus characteristic exponents of the system for time periodic solutions (limited cycles of various periods and resonance invariant tori) are computed. We show, numerically, the existence of multistable rotes to chaos through chaotic fractal attractors, full Feigenbaum-Sharkovski cascades and multidimensional torus attractors (Landau-Hopf scenario). The existence of these attractors is shown through analysis of phase subspaces projections, Poincare sections and eigenvalue analysis of numerically computed DNS data. These attractors burst into chaos with the increase of Rayleigh number either through resonance and phase-locking or through emergence of singular chaotic attractors.

Cavity optomechanical chaos

Cavity optomechanics provides a powerful platform for observing many interesting classical and quantum nonlinear phenomena due to the radiation-pressure coupling between its optical and mechanical modes.In particular,the chaos induced by optomechanical nonlinearity has been of great concern because of its importance both in fundamental physics and potential applications ranging from secret information processing to optical communications.This review focuses on the chaotic dynamics in optomechanical systems.The basic theory of general nonlinear dynamics and the fundamental properties of chaos are introduced.Several nonlinear dynamical effects in optomechanical systems are demonstrated.Moreover,recent remarkable theoretical and experimental efforts in manipulating optomechanical chaotic motions are addressed.Future perspectives of chaos in hybrid systems are also discussed.

Bifurcation and chaos of the bladed overhang rotor system with squeeze film dampers

To study the nonlinear dynamic behavior of the bladed overhang rotor system with squeeze film damper (SFD), a blade-overhang rotor-SFD model is formulated using the lumped mass method and the Lagrange approach. The cavitated short bearing model is employed to describe the nonlinear oil force of the SFD. To reduce the scale of the nonlinear coupling system, a set of orthogonal transformations is employed to decouple the one nodal diameter equations of blades, which are coupled with the dy- namical equations of the rotor, with other equations of blades. In this way, the original system with 16+4n (n≥3) degrees of freedom (DoF) is reduced to a system with 24 DoF only. Then the parametric excitation terms in the blade-overhang rotor-SFD model are simplified in terms of periodic transforma- tions. The coupling equations are numerically solved and the solutions are used to analyze the dy- namic behavior of the system in terms of the bifurcation diagram, whirl orbit, Poincaré map and spec- trum plot. A variety of motion types are found such as multi-periodic, quasi-periodic, and chaotic mo- tions. Moreover, the typical nonlinear dynamic evolutions including the periodic-doubling bifurcation and reverse bifurcation are noted. It is noticed that there exist apparent differences in the dynamic behavior between the blade-overhang rotor-SFD models without and with considering the effect of blades.

Snap-through behaviors and nonlinear vibrations of a bistable composite laminated cantilever shell:an experimental and numerical study

The snap-through behaviors and nonlinear vibrations are investigated for a bistable composite laminated cantilever shell subjected to transversal foundation excitation based on experimental and theoretical approaches.An improved experimental specimen is designed in order to satisfy the cantilever support boundary condition,which is composed of an asymmetric region and a symmetric region.The symmetric region of the experimental specimen is entirely clamped,which is rigidly connected to an electromagnetic shaker,while the asymmetric region remains free of constraint.Different motion paths are realized for the bistable cantilever shell by changing the input signal levels of the electromagnetic shaker,and the displacement responses of the shell are collected by the laser displacement sensors.The numerical simulation is conducted based on the established theoretical model of the bistable composite laminated cantilever shell,and an off-axis three-dimensional dynamic snap-through domain is obtained.The numerical solutions are in good agreement with the experimental results.The nonlinear stiffness characteristics,dynamic snap-through domain,and chaos and bifurcation behaviors of the shell are quantitatively analyzed.Due to the asymmetry of the boundary condition and the shell,the upper stable-state of the shell exhibits an obvious soft spring stiffness characteristic,and the lower stable-state shows a linear stiffness characteristic of the shell.

转子-机匣耦合系统碰摩非线性特征研究

转子-机匣碰摩问题是航空发动机中最常见故障之一。在临界转速附近,转子振动幅值会急剧增大,超过转子-机匣初始间隙就会引起碰摩。为了进一步探究碰摩机理,建立了考虑阻尼的非线性碰摩力模型并将其耦合在转子-机匣系统模型中,采用四阶Runge-Kutta法求解,给出了不同转速下转子-机匣系统的响应分岔图,研究了转子-机匣系统的振动响应。同时,分别讨论了碰撞恢复系数、碰摩间隙对转子运动和碰摩力的影响。结果表明:碰撞恢复系数减小会导致转子-机匣嵌入深度增大,从而导致碰摩力增大。碰摩间隙会改变碰摩的频率,且随着间隙的减小,碰摩的频率增大。

非圆齿轮行星轮系非线性动力学特性分析

为解决非圆齿轮行星轮系动力学模型不完善、非线性动力学特性难以获取等问题,提出了一种非圆齿轮行星轮系动力学建模方法,并聚焦于时变参数激励下系统的动态响应机制,对非圆齿轮行星轮系非线性动力学特性进行研究。为精确分析系统非线性振动,对非圆齿轮齿侧间隙函数进行了拟合,在综合考虑齿面摩擦、时变啮合刚度、黏弹性阻尼、静态传递误差的基础上,通过引入相对位移坐标消除非线性方程的变量耦合。在此基础上,建立非圆齿轮行星轮系传动系统的动力学模型,并利用四阶变步长Runge-Kutta数值方法对系统非线性动力学方程组进行求解。获取了分岔图、时域图、相轨迹以及Poincaré映射,得到阻尼、齿面摩擦、时变啮合刚度等控制参数激励下系统的动态响应分布规律。结果表明:随各激励参数取值不同,系统呈现出混沌和周期运动相互过渡状态;合理选取激励参数,可减小系统在混沌与周期运动之间的时间间隔,快速进入稳定运动状态。研究成果可为抑制非圆齿轮行星轮系传动系统非线性振动、预测系统的动力学行为提供理论依据。

Global bifurcations of strongly nonlinear oscillator induced by parametric and external excitation

The global bifurcation of strongly nonlinear oscillator induced by parametric and external excitation is researched. It is known that the parametric and external excitation may induce additional saddle states, and result in chaos in the phase space, which cannot be detected by applying the Melnikov method directly. A feasible solution for this problem is the combination of the averaged equations and Melnikov method. Therefore, we consider the averaged equations of the system subject to Duffing-Van der Pol strong nonlinearity by introducing the undetermined fundamental frequency. Then the bifurcation values of homoclinic structure formation are detected through the combined application of the new averaged equations with Melnikov integration. Finally, the explicit application shows the analytical conditions coincide with the results of numerical simulation even disturbing parameter is of arbitrary magnitude.

磁液双悬浮轴承-单盘转子系统动力学行为研究

随着轴承系统高转速、高效率的发展需求,磁液双悬浮轴承转子与静子的装配间隙不断减小,导致碰摩事故经常发生。综合考虑转子偏心比、转速比、磁液双悬浮轴承与静子碰摩等多种耦合故障,建立磁液双悬浮轴承转子系统“间隙-碰摩”动力学方程,数值模拟转子运行规律。研究结果表明:随着偏心比的增加,转子由周期1运行演化出周期2、周期3、拟周期、混沌等多种运行规律;当偏心比ρ∈(0.28~0.4)及转速比w∈(1.2~1.7)时,转子位移波动剧烈,在此区域内转子极易发生分岔甚至混沌,且在碰摩区间内,转速比在1.5、1.67附近时,转子轴承处与转盘处碰撞力分别出现最大值。

一类附加斜弹簧支撑的悬臂梁碰撞系统的全局动力学

本文研究了双侧非对称刚性约束下附加斜弹簧支撑的悬臂梁碰撞系统的次谐分岔和混沌的全局动力学。由于斜弹簧支撑结构的刚度项为超越函数,给解析研究系统混沌和次谐分岔造成很大的困难。本文近似拟合了该系统的刚度项,并对比分析了近似系统和原系统的同宿轨道及其内部的次谐轨道。将Melnikov方法发展应用于非光滑的碰撞悬臂梁系统,给出了发生同宿混沌和次谐分岔的阀值条件。利用光滑流形的特征乘子结合碰撞函数分析了碰撞次谐轨道的稳定性,并分析了次谐分岔与混沌的关系。基于阀值条件研究了阻尼、激励频率、激励幅值以及碰撞恢复系数对混沌和次谐分岔的影响,进一步验证了理论分析的正确性。

横向止挡对车辆动力学行为的影响研究

近年来,城轨列车的快速发展使转向架的研发越来越重要。而二系横向止挡是缓和转向架中枕梁与构架冲击的重要构件之一。为避免城轨列车转向架与车体有过大的偏移量,在设计转向架时通常设置有横向止挡,并使横向止挡与车体之间留有一定的间隙。该间隙属于强非线性因素,会对车辆的横向动力学行为产生重要的影响,同时间接影响城轨车辆的运行品质。因此,有必要对城轨地铁车辆中的二系横向止挡悬挂系统的非线性动力学特征进行更深入、更细微的探索性研究。其次,全面分析含横向止挡的城轨车辆转向架系统的非线性动力学行为,为横向止挡转向架的平稳性设计可以提供一定的理论依据和参数值。