An enhanced multi-term harmonic balance solution for nonlinear period-\varvec{\beta } dynamic motions in right-angle gear pairs

A nonlinear, time-varying dynamic model for right-angle gear pair systems is formulated to analyze the existence of sub-harmonics and chaotic motions. This pure torsional gear pair system is characterized by its time-varying excitation, clearance, and asymmetric nonlinearities as well. The period-1 dynamic motions of the same system were obtained by solving the dimensionless equation of gear motion using an enhanced multi-term harmonic balance method (HBM) with a modified discrete Fourier transform process and the numerical continuation method presented in another paper by the authors. Here, the sub-harmonics and chaotic motions are studied using the same solution technique. The accuracy of the enhanced multi-term HBM is verified by comparing its results to the solutions obtained using the more computational intensive direct numerical integration method. Due to its inherent features, the enhanced multi-term HBM cannot predict the chaotic motions. However, the frequency ranges where chaotic motions exist can be predicted using the stability analysis of the HBM solutions. Parametric studies reveal that the decrease in drive load or the increase of kinematic transmission error (TE) can result in more complex gear dynamic motions. Finally, the frequency ranges for sub-harmonics and chaotic motions, as a function of TE and drive load, are obtained for an example case.     

一种确定非线性裂纹转子解的形式的新方法

将小波变换与Ｐｏｉｎｃａｒｅ映射相结合，即用Ｐｏｉｎｃａｒｅ映射确定周期解，用谐波小波变换区分拟周期响应和混沌运动，提出了一种分析非线性裂纹转子系统解的形式随参数变化的新方法．结果表明这种方法是非常有效的，它比以前所用的计算Ｌｉａｐｕｎｏｖ指数的方法节约了计算时间，并且较易实施．     

Chaotic Dynamics of a Harmonically Excited Spring-Pendulum System with Internal Resonance

An investigation into chaotic responses of a weakly nonlinear multi-degree-of-freedom system is made. The specific system examined is a harmonically excited spring pendulum system, which is known to be a good model for a variety of engineering systems, including ship motions with nonlinear coupling between pitching and rolling motions. By the method of multiple scales the original nonautonomous system is reduced to an approximate autonomous system of amplitude and phase variables. The approximate system is shown to have Hopf bifurcation and a sequence of period-doubling bifurcations leading to chaotic motions. In order to examine what happens in the original system when the approximate system exhibits chaos, we compare the largest Lyapunov exponents for both systems.     

Sensitivity Resonance and Attractor Morphing Quantified by Sensitivity Vector Fields for Parameter Reconstruction

A novel method of parameter variation reconstruction for systems exhibiting chaotic dynamics is presented. The algorithm reconstructs variations of system parameters without the need for explicit system equations of motion, or knowledge of the nominal parameter values. The concept of a sensitivity vector field (SVF) is developed. This construct captures geometrical deformations to the dynamical attractor of the system in state space. These fields are collected by means of a proposed unique approach referred to as point cloud averaging (PCA). PCA is applied to discrete time series data from the system with nominal parameter values (healthy) and the system with changed parameters. Test variations are reconstructed from an optimal basis of SVF snapshots which is generated by means of proper orthogonal decomposition. The method is applied to two system models, a magneto-elastic oscillator (MEO) and an atomic force microscope (AFM). The method is shown to be highly accurate, and capable of identifying multiple simultaneous variations. The success of the method as applied to an AFM and a MEO indicates a potential for highly accurate readings by exploiting the geometric features of observed chaotic vibrations. An exciting new phenomenon referred to as sensitivity resonance was also observed, and some implications regarding its use in further improving algorithm performance are discussed.An earlier version of this work has been presented at the 20-th Biennial Conference on Mechanical Vibration and Noise, Long Beach, 2005.     

单自由度体系地震动力响应的混沌特性分析

引入非线性动力学理论和混沌时间序列分析方法考察地震动作用下单自由度体系动力响应的混沌特性。输入典型近断层地震动记录,定量计算了代表性周期的单自由度弹性和非弹性体系加速度响应时程的非线性特性参数。计算表明,这些加速度响应的关联维数为分数维,最大Lyapunov指数大于0;地震动激励下单自由度体系的地震动力响应具有混沌特性,不是完全的随机信号,为理解结构地震动力响应的不规则性与复杂性提供了新思路和新视角。     

Resonant chaotic motions of a buckled rectangular thin plate with parametrically and externally excitations

Resonant chaotic motions of a simply supported rectangular thin plate with parametrically and externally excitations are analyzed using exponential dichotomies and an averaging procedure for the first time. The formulas of the rectangular thin plate are derived by a von Karman type equation and the Galerkin’s approach. The critical condition to predict the onset of chaotic motions for the full system is obtained by developing a Melnikov function containing terms from the non-hyperbolic mode. We prove that the non-hyperbolic mode of the thin plate does not affect the critical condition for the occurrence of chaotic motions in the resonant case. Simulations also show that the chaotic motions of the hyperbolic subsystem are shadowed by the chaotic motions for the full system of the rectangular thin plate.     

Chaotic attitude motion of a magnetic rigid spacecraft in an elliptic orbit and its control

This paper deals with the chaotic attitude motion of a magnetic rigid spacecraft with internal damping in an elliptic orbit. The dynamical model of the spacecraft is established. The Melnikov analysis is carried out to prove the existence of a complicated nonwandering Cantor set. The dynamical behaviors are numerically investigated by means of time history, Poincaré map, Lyapunov exponents and power spectrum. Numerical simulations demonstrate the chaotic motion of the system. The input-output feedback linearization method and its modified version are applied, respectively, to control the chaotic attitude motions to the given fixed point or periodic motion. The project supported by the National Natural Science Foundation of Chine (10082003)     

混沌时间序列预测的局域法在边坡变形分析中的应用

边坡作为一个复杂系统,其本身的各种参量是不确定的和随机的,在其演化过程中,表现出复杂的非线性行为,发生一系列的混沌现象。本文运用现代混沌理论,对边坡变形的预测问题进行探索性研究,把混沌时间序列理论引入到边坡工程研究中,对该理论的建立及预测方法进行系统地讨论,为该领域的研究提供完整的技术方法。通过对新滩滑坡的研究结果表明,混沌时间序列方法对混沌序列的预测较线性时间序列具有较高的精度。     

STUDY ON THE PREDICTION METHOD OF LOW_DIMENSION TIME SERIES THAT ARISE FROM THE INTRINSIC NONLINEAR DYNAMIC

ＩｎｔｒｏｄｕｃｔｉｏｎＬｏｔｓｏｆｔｉｍｅｓｅｒｉｅｓｆｒｏｍｐｒａｃｔｉｃａｌｐｒｏｂｌｅｍｓｂｅｌｏｎｇｔｏｎｏｎｌｉｎｅａｒｃｈａｏｔｉｃｔｉｍｅｓｅｒｉｅｓ.Ｉｔｈａｓｂｅｅｎｐｒｏｖｅｄｉｎｐｒａｃｔｉｃｅｔｈａｔｔｈｅｌｉｎｅａｒｍｏｄｅｌｓｏｆｅｉｔｈｅｒｌｏｗｏｒｄｅｒｓｏｒｈｉｇｈｏｒｄｅｒｓｃａｎｎｏｔｂｅｕｓｅｄｔｏｄｅｓｃｒｉｂｅｎｏｎｌｉｎｅａｒｃｈａｏｔｉｃｔｉｍｅｓｅｒｉｅｓ.Ｈｅｎｃｅｉｔｉｓｖｅｒｙｉｍｐｏｒｔａｎｔｔｏｉｎｖｅｓｔｉｇａｔｅｃｈａｏｔｉｃｔｉｍ…     

Bifurcation,periodic and chaotic motions of the modified equal width-Burgers (MEW-Burgers) equation with external periodic perturbation

The modified equal width-Burgers (MEW-Burgers) equation is introduced for the first time. The bifurcation behavior of the MEW-Burgers equation is studied. Considering an external periodic perturbation, the periodic and chaotic motions of the perturbed MEW-Burgers equation are investigated by using phase projection analysis, time series analysis, Poincaré section and bifurcation diagram. The strength (\(f_0\)) of the external periodic perturbation plays a crucial role in the periodic and chaotic motions of the perturbed MEW-Burgers equation.     

Study on the Prediction Method of Low-Dimension Time Series that Arise from the Intrinsic Nonlinear Dynamics

The prediction methods and its applications of the nonlinear dynamic systems determined from chaotic time series of low-dimension are discussed mainly. Based on the work of the foreign researchers, the chaotic time series in the phase space adopting one kind of nonlinear chaotic model were reconstructed. At first, the model parameters were estimated by using the improved least square method. Then as the precision was satisfied, the optimization method was used to estimate these parameters. At the end by using the obtained chaotic model, the future data of the chaotic time series in the phase space was predicted. Some representative experimental examples were analyzed to testify the models and the algorithms developed in this paper. The results show that if the algorithms developed here are adopted, the parameters of the corresponding chaotic model will be easily calculated well and true. Predictions of chaotic series in phase space make the traditional methods change from outer iteration to interpolations. And if the optimal model rank is chosen, the prediction precision will increase notably. Long term superior predictability of nonlinear chaotic models is proved to be irrational and unreasonable. Paper from Chen Yu-shu, Member of Editorial of Committee, AMM Foundation item: the National Natural Science Foundation of China (19990510); the National Key Basic Research Special Fund(G1998020316) Biography: Ma Jun-hai(1965-), Professor, Doctor     

Noise-induced chaotic motions in Hamiltonian systems with slow-varying parameters

This paper studies chaotic motions in quasi-integrable Hamiltonian systems with slow-varying parameters under both harmonic and noise excitations. Based on the dynamic theory and some assumptions of excited noises, an extended form of the stochastic Melnikov method is presented. Using this extended method, the homoclinic bifurcations and chaotic behavior of a nonlinear Hamiltonian system with weak feed-back control under both harmonic and Gaussian white noise excitations are analyzed in detail. It is shown that the addition of stochastic excitations can make the parameter threshold value for the occurrence of chaotic motions vary in a wider region. Therefore, chaotic motions may arise easily in the system. By the Monte-Carlo method, the numerical results for the time-history and the maximum Lyapunov exponents of an example system are finally given to illustrate that the presented method is effective.     

CHAOTIC MOTIONS AND LIMIT CYCLE FLUTTER OF TWO-DIMENSIONAL WING IN SUPERSONIC FLOW

Based on the piston theory of supersonic flow and the energy method, the flutter motion equations of a two-dimensional wing with cubic stiffness in the pitching direction are established. The aeroelastic system contains both structural and aerodynamic nonlinearities. Hopf bifurcation theory is used to analyze the flutter speed of the system. The effects of system parameters on the flutter speed are studied. The 4th order Runge-Kutta method is used to calculate the stable limit cycle responses and chaotic motions of the aeroelastic system. Results show that the number and the stability of equilibrium points of the system vary with the increase of flow speed. Besides the simple limit cycle response of period 1, there are also period-doubling responses and chaotic motions in the flutter system. The route leading to chaos in the aeroelastic model used here is the period-doubling bifurcation. The chaotic motions in the system occur only when the flow speed is higher than the linear divergent speed and the initial condition is very small. Moreover, the flow speed regions in which the system behaves chaos axe very narrow.     

随机激励对软弹簧杜芬振子动力学的分散作用

讨论了有界噪声激励对软弹簧杜芬振子的倍周期分岔至混沌运动的影响。利用蒙特卡罗方法,通过对系统受侵蚀安全盆的变化状况进行了观察,并由此对后继动力学分析的初始点进行了选取。系统的相图、倍周期分岔图以及庞加莱映射图等方面的数值结果表明,外加随机激励的作用往往掩盖原确定性系统内在的规则运动,对原确定性系统的运动具有较典型的分散作用,可延缓系统的倍周期分岔,也可使得系统内在随机行为提前发生,即可使得系统更容易出现混沌运动。     

Nonlinear nonplanar dynamics of a parametrically excited inextensional elastic beam

The nonlinear dynamics of a clamped-clamped/sliding inextensional elastic beam subject to a harmonic axial load is investigated. The Galerkin method is used on the coupled bending-bending-torsional nonlinear equations with inertial and geometric nonlinearities and the resulting two second order ordinary differential equations are studied by the method of multiple time seales and by direct numerical integration. The amplitude equations are analyzed for steady and Hopf bifurcations. Depending on the amplitude of excitation, the damping and the ratio of principal flexural rigidities, various qualitatively distinct frequency response diagrams are uncovered and limit cycles and chaotic motions are found. In the truncated two-degree-of-freedom system the transition from periodic to chaotic amplitude-modulated motions is via the process of torus doubling and subsequent destruction of the torus.     

Dither signals with particular application to the control of windscreen wiper blades

This study verifies chaotic motion of an automotive wiper system, which consists of two blades driven by a DC motor via the two connected four-bar linkages and then elucidates a system for chaotic control. A bifurcation diagram reveals complex nonlinear behaviors over a range of parameter values. Next, the largest Lyapunov exponent is estimated to identify periodic and chaotic motions. Finally, a method for controlling a chaotic automotive wiper system will be proposed. The method involves applying another external input, called a dither signal, to the system. Some simulation results are presented to demonstrate the feasibility of the proposed method.     

Torus doublings and chaotic amplitude modulations in a two degree-of-freedom resonantly forced mechanical system

This work studies the response of a weakly non-linear vibratory system with two degrees-of-freedom when the system is excited near resonance. The two linear modes are in 1:3 internal resonance. The asymptotic method of averaging and direct numerical integration are used to obtain the response of the system. Over a range of excitation frequencies and modal damping, the averaged equations in slow time are found to possess limit cycle solutions. These solutions undergo period doubling bifurcations to chaotic solutions. The averaging theory then implies the existence of amplitude modulated motions, the exact nature of modulations not being well defined. Numerical simulation of the original vibratory two degree-of-freedom system shows that the system does undergo amplitude modulated motions. For sufficiently large damping, only periodic modulations arise in the form of a 2-torus. For lower damping, the 2-torus can undergo doubling and ultimate destruction to result in a chaotic attractor. Poincare sections of steady state solutions are used to characterize the various types of amplitude modulated motions.     

On Modal Interaction,Stability and Nonlinear Dynamics of a Model Two D.O.F. Mechanical System Performing Snap-Through Motion

Dynamics of a simple two degrees of freedom (d.o.f.) mechanical system is considered, to illustrate the phenomena of modal interaction. The system has a natural symmetry of shape and is subjected to symmetric loading. Two stable equilibrium configurations are separated by an unstable one, so that the model system can perform cross-well oscillations. Nonlinear statics and dynamics are considered, with the emphasis on detecting conditions for instability of symmetric configurations and analysis of bi-modal non-symmetric motions. Nonlinear local dynamics is analyzed by multiple scales method. Direct numerical integration of original equations of motions is carried out to validate analysis of modulation equations. In global dynamics (analysis of cross-well oscillations) Lyapunov exponents are used to estimate qualitatively a type of motion exhibited by the mechanical system. Modal interactions are demonstrated both in the local dynamics and for snap-through oscillations, including chaotic motions. This mechanical system may be looked upon as a lumped parameters model of continuous elastic structures (spherical segments, cylindrical panels, buckled plates, etc.). Analyses performed in the paper qualitatively describe complicated phenomena in local and global dynamics of original structures.     

Chaos in a single equilibrium point system: Finite deformations

The purpose of this paper is to examine a highly nonlinear model of a slender beam which yields chaotic solutions for some forcing amplitudes. The study is unique in that the governing partial differential equations are solved directly, and that the model lends itself to a more physical analysis of the beam than traditional chaotic models. In addition, the analysis will provide proof that a beam experiencing moderate deformations without stops or an initial axial force can exhibit chaotic motion. The model represents a simply-supported. Euler-Bernoulli beam subjected to a transverse load. The forcing function is sinusoidally distributed in space with an amplitude which also varies sinusoidally in time and is assumed to reach a maximum sufficient to allow nonlinearities associated with finite deformations to become important. During motion, even though displacements are large, the beam is assumed to attain only small strain levels and thus is assumed to be linearly elastic. The results indicate that for most levels of the forcing function the response of the beam is periodic. However, the steady state motion is not sinusoidal in time and in fact exhibits some bifurcated motions. At a certain level of the forcing amplitude, an asymmetry is observed and the periodicity of the motion breaks down as the beam experiences a period doubling cascade which culminates in a chaotic motion. The progression from periodic to chaotic motion is presented through a series of phase plane and Poincané plots, and physical variables such as bending moment are examined.     

Non-linear dynamic phenomena in the behaviour of a railway wheelset model

A dicone moving on a pair of cylindrical rails can be considered as a simplified model of a railway wheelset. Taking into account the non-linear friction laws of rolling contact, the equations of motion for this non-linear mechanical system result in a set of differential-algebraic equations. Previous simulations performed with the differential-algebraic solver DASSL, [2], and experiments, [7], indicated non-linear phenomena such as limit-cycles, bifurcations as well as chaotic behaviour. In this paper the non-linear phenomena are investigated in more detail with the aid of special in-house software and the path-following algorithm PATH [10]. We apply Poincaré sections and Poincaré maps to describe the structure of periodic, quasiperiodic and chaotic motions. The analyses show that part of the chaotic behaviour of the non-linear system can be fully understood as a non-linear iterative process. The resulting stretching and folding processes are illustrated by series of Poincaré sections.