Periodically forced delay limit cycle oscillator

This paper investigates the dynamics of a delay limit cycle oscillator under periodic external forcing. The system exhibits quasiperiodic motion outside of a resonance region where it has periodic motion at the frequency of the forcer for strong enough forcing. By perturbation methods and bifurcation theory, we show that this resonance region is asymmetric in the frequency detuning, and that there are regions where stable periodic and quasiperiodic motions coexist.     

Analytical solutions for asymmetric periodic motions to chaos in a hardening Duffing oscillator

In this paper, the analytical dynamics of asymmetric periodic motions in the periodically forced, hardening Duffing oscillator is investigated via the generalized harmonic balance method. For the hardening Duffing oscillator, the symmetric periodic motions were extensively investigated with the aim of a good understanding of solutions with jumping phenomena. However, the asymmetric periodic motions for the hardening Duffing oscillators have not been obtained yet, and such asymmetric periodic motions are very important to find routes of periodic motions to chaos in the hardening Duffing oscillator analytically. Thus, the bifurcation trees from asymmetric period-1 motions to chaos are presented. The corresponding unstable periodic motions in the hardening Duffing oscillator are presented, and numerical illustrations of stable and unstable periodic motions are carried out as well. This investigation provides a comprehensive understanding of chaos mechanism in the hardening Duffing oscillator.     

A parametrically excited pendulum with irrational nonlinearity

In this paper, we propose a parametrically excited pendulum with irrational nonlinearity which comprises a simple pendulum linked by a linear spring under base excitation. This parametric vibration system exhibits bistable state and discontinuous characteristics due to the geometry configuration. For small oscillations, this system can be described by Mathieu equation coupled with SD (Smooth and Discontinuous) oscillator whose dynamic response is examined analytically by using the averaging method in both smooth and discontinuous case. Numerical simulations are carried out to demonstrate the complicated dynamic behavior of multiple periodic motions and different types of chaotic motions.     

Experimental study of an impact oscillator with viscoelastic and Hertzian contact

Modeling an impact event is often related to the desired outcome of an impact oscillator study. If the only intent is to study the dynamic behavior of the system, numerous researchers have shown that simpler impact models will often suffice. However, when the geometric contours and material properties of the two colliding surfaces are known, it is often of interest to model the contact event at a greater level of complexity. This paper investigates the application of a finite time impact model to the study of a parametrically excited planar pendulum subjected to a motion-dependent discontinuity. Experimental and numerical studies demonstrate the presence of multiple periodic attractors, subharmonics, quasi-periodic motions, and chaotic oscillations.     

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.     

Chaos in a mapping describing elastoplastic oscillations

We study the local and global dynamical behavior of a two dimensional piecewise linear map which describes the asymptotic motions of a single degree of freedom, parametrically excited, elastoplastic oscillator after it has settled down to purely elastic oscillations. We give existence and stability conditions for periodic orbits and prove that chaos, in the form of a Smale horseshoe, exists at specific, but representative, parameter values. We interpret simulations of the elastoplastic oscillator itself in the light of these results.Partially supported by NSF grant number MSS-9016626.Partially supported by AFOSR 91-0329.     

有界噪声激励下单摆-谐振子系统的混沌运动

研究了具有同宿轨道和周期轨道的可积单摆-谐振子系统在弱Hamilton摄动(即弱耦合摄动)和弱非Hamilton摄动(即阻尼和有界噪声微扰)下的混沌运动．用Melnikov方程预测Hamilton系统中可能存在混沌运动的参数域，并用Poincare截面验证解析结果．用数值方法计算了有阻尼与有界噪声激励下系统的最大Lyapun0V指数和Poincare截面，结果表明有界噪声在频率上的扩散减小了引发系统产生混沌运动的效应。     

Modified Parker-Sochacki method for solving nonlinear oscillators

A new approach is presented for solving nonlinear oscillatory systems. Parker-Sochacki method (PSM) is combined with Laplace-Padé resummation method to obtain approximate periodic solutions for three nonlinear oscillators. The first one is Duffing oscillator with quintic nonlinearity which has odd nonlinearity. The second one is Helmholtz oscillator which has even nonlinearity. The last one is a strongly nonlinear oscillator, namely; relativistic harmonic oscillator which has a fractional order nonlinearity. Solutions are also obtained using Runge-Kutta numerical method (RKM) and Lindstedt-Poincare method (LPM). However, the LPM could not be used to solve the relativistic harmonic oscillator since it is a strongly nonlinear oscillator. The comparison between these solutions shows that the convergence zone for the Parker-Sochacki with Laplace-Padé method (PSLPM) is remarkably increased compared to PSM method. It also shows that the PSLPM solutions are in excellent agreement with LPM solutions for Duffing oscillator and are superior to LPM solutions in case of Helmholtz oscillator. The PSLPM succeeded to give an accurate periodic solution for the relativistic harmonic oscillator. For a wide range of solution domain, comparing PSLPM with RKM prove the correctness of the PSLPM method. Hence, the PSLPM method can be used with satisfied confidence to solve a broad class of nonlinear oscillators.     

Bifurcation Analysis of Parametrically Excited Rayleigh–Liénard Oscillators

A parametrically excited Rayleigh–Liénard oscillator is investigatedby an asymptotic perturbation method based on Fourier expansion and timerescaling. Two coupled equations for the amplitude and the phase ofsolutions are derived and the stability of steady-state periodic solutionsas well as parametric excitation-response and frequency-response curvesare determined. Comparison with the parametrically excited Liénardoscillator is performed and analytic approximate solutions are checkedusing numerical integration. Dulac's criterion, thePoincaré–Bendixson theorem, and energy considerations are used in order to study the existence and characteristics of limit cycles of the twocoupled equations. A limit cycle corresponds to a modulated motion forthe Rayleigh–Liénard oscillator. Modulated motion can be also obtainedfor very low values of the parametric excitation, and in this case, anapproximate analytic solution is easily constructed. If the parametricexcitation is increased, an infinite-period bifurcation is observed because the modulation period lengthens and becomes infinite, while themodulation amplitude remains finite and suddenly the attractor settlesdown into a periodic motion. Floquet's theory is used to evaluatethe stability of the periodic solutions, and in certain cases,symmetry-breaking bifurcations are predicted. Numerical simulationsconfirm this scenario and detect chaos and unbounded motions in theinstability regions of the periodic solutions.     

Geometric aspects of the parametrically driven pendulum

The behaviour of the parametrically driven pendulum is very complex. Therefore, a global study is carried out to cover all possible situations. The study is mainly numeric, though primary bifurcations of subharmonic motions, as well as the homoclinic intersection of the hilltop saddle, are evaluated according to the Melnikov theory. Extended use is made of the cell-to-cell mapping algorithm to evaluate attracting basins of the various periodic motions. Heteroclinic intersections are always present, independently of the excitation intensity, so that the boundaries of attracting basins are always very complicated, even below the homoclinic tangency of the hilltop saddle. The oscillator exhibits various kinds of rotating and oscillating motions. All these motions lead to chaos after a period doubling cascade. It is shown that chaos usually occurs at a much greater excitation level than at that which produces homoclinic tangency of the hilltop saddle; the greater the damping, the greater the difference. The oscillatory chaotic motion is associated with the first change in the period two Birkhoff signature.     

Determination of limit cycles for a modified van der Pol oscillator

The determination of amplitude and period of limit cycles is a crucial question in non-linear mechanics. In this paper, a van der Pol oscillator containing a periodic potential is considered. Amplitude and period of limit cycles are calculated by He’s variational method and Krylov–Bogoliubov–Mitropolsky (KBM) method.     

两自由度塑性碰撞振动系统的动力学研究

用三维映射表示具有单侧刚性约束的两自由度振动系统在塑性碰撞时的动力学方程。借助理论分析与数值方法研究了系统周期n-1振动的存在性与稳定性,描述了系统周期n-1振动的特点,讨论了碰撞振子与约束擦边引起的Poincare映射奇异性对系统全局分岔的影响。     

Dynamics of a plastic-impact system with oscillatory and progressive motions

A mathematical model is developed to describe oscillatory and progressive motions in dynamics of a plastic impact oscillator with a frictional slider. Dynamics of the impact oscillator is analyzed by a five-dimensional map, which describes free flight and sticking solutions of two masses of the system, between impacts, supplemented by transition conditions at the instants of impacts. Piecewise property and singularity are found to exist in the Poincaré map. The piecewise property is caused by the transitions of free flight and sticking motions of impacting masses immediately after the impact, and the singularity of the map is generated via the grazing contact of impacting masses immediately before the impact. These properties of the map have been shown to exhibit particular types of sliding and grazing bifurcations of periodic-impact motions under parameter variation. The influence of piecewise property, grazing singularities and various parameters on dynamics of the vibro-impact system is analyzed. The global bifurcation diagrams for before-impact velocity versus forcing frequency are plotted to predict much of the qualitative behavior of the system. The global bifurcations of period-n single-impact motions of the plastic-impact oscillator are found to exhibit extensive and systematic characteristics.     

The Response of an Oscillator Moving along a Parabola to an External Excitation

The behavior of a mass point moving along a parabola under theeffect of an external periodic excitation in resonance with the naturalfrequency of the oscillator is studied. The asymptotic perturbationmethod based on temporal rescaling and balancing of the harmonic termswith a simple iteration is used in order to determine the nonlinearmodulation equations for the amplitude and the phase of the oscillation.External force-response curves are shown and moreover jump phenomena arealso observed. In certain cases a second low frequency appears inaddition to the forcing frequency and then stable two-periodquasi-periodic motions are present with amplitudes depending on theinitial conditions. The value of the low frequency depends on theamplitude of the external excitation. A higher order perturbationanalysis is developed and the validity of the method is highlighted bycomparing the leading order and the higher order approximate analyticsolutions to numerical results.     

Subharmonic and grazing bifurcations for a simple bilinear oscillator

In this paper, subharmonic and grazing bifurcations for a simple bilinear oscillator, namely the limit discontinuous case of the smooth and discontinuous (SD) oscillator are studied. This system is an important model that can be used to investigate the transition from smooth to discontinuous dynamics. A combination of analytical and numerical methods is used to investigate the existence, stability and bifurcations of symmetric and asymmetric subharmonic orbits. Grazing bifurcations for a particular periodic orbit are also discussed and numerical results suggest that the bifurcations are discontinuous. We show via concrete numerical experiments that the dynamics of the system for the case of large dissipation is quite different from that for the case of small dissipation.     

Analytic evaluation of periodic responses of a forced nonlinear oscillator

The periodic responses of a strongly nonlinear, single-degree-of-freedom forced oscillator with weak excitation and damping are examined. The presented methodology is based on a regular perturbation expansion, whose first term is the solution of the unforced, and undamped nonlinear problem. Higher order approximations are computed by explicitly solving linear differential equations possessing a periodically varying coefficient. The general theory is used for studying the periodic steady state motions of the periodically forced system. Moreover, it is shown that the presented analysis can be used to analytically study the orbital stability of the identified steady state motions. The proposed method can also be used for studying periodic responses due to nonperiodic transient forces, provided that these responses are close to the O(1) periodic generating solution.     

The Dissipative Nonlocal Oscillator

The most important characteristics of a nonlocal and nonlinear oscillator subject to dissipative forces are extensively studied by means of an asymptotic perturbation method, based upon temporal rescaling and harmonic balance. The conditions under which bifurcations and limit cycles appear are determined. If the parameters satisfy particular conditions, a quasi-periodic motion is predicted, because a second small frequency adds to the natural frequency of the oscillator. The analytical results are validated by numerically solving the original system.     

SD振子的设计及非线性特性实验研究初探

具有光滑与不连续转迁特征的SD振子发现和提出以来, 引起了广泛关注. 基于双稳系统大位移特征的测量法困难, SD振子的实验研究还未见报道. 该文提出并设计了具有SD振子系统光滑特征的非线性实验装置, 用实验的方法揭示由几何关系产生的强非线性系统的非线性动力学行为. 设计的非线性实验装置基本振动参数均有良好的可调性和可测量性, 对SD振子在不同频率及幅值的简谐激励作用下的非线性动力学响应进行了实验研究. 为克服大位移测量难题, 研究采用高速摄像机采集振子振动视频信号并进行分析. 结果表明, SD振子系统在一定的参数条件下会产生周期振动、周期5振动及混沌运动等复杂非线性动力学现象, 在相同实验参数条件下进行了数值仿真, 仿真结果与实验结果一致.      

Modelling of Ground Moling Dynamics by an Impact Oscillator with a Frictional Slider

This paper describes current research into the mathematical modelling of a vibro-impact ground moling system. Due to the structural complexity of such systems, in the first instance the dynamic response of an idealised impact oscillator is investigated. The model is comprised of an harmonically excited mass simulating the penetrating part of the mole and a visco-elastic slider, which represents the soil resistance. The model has been mathematically formulated and the equations of motion have been developed. A typical nonlinear dynamic analysis reveals a complex behaviour ranging from periodic to chaotic motion. It was found out that the maximum progression coincides with the end of the periodic regime.     

Jump and bifurcation of Duffing oscillator under narrow-band excitation

The jump and bifurcation of Duffing oscillator with hardening spring subject to narrow-band random excitation are systematically and comprehensively examined. It is shown that, in a certain domain of the space of the oscillator and excitation parameters, there are two types of more probable motions in the stationary response of the Duffing oscillator and jumps may occur. The jump is a transition of the response from one more probable motion to another or vise versa. Outside the domain the stationary response is either nearly Gaussian or like a diffused limit cycle. As the parameters change across the boundary of the domain the qualitative behavior of the stationary response changes and it is a special kind of bifurcation. It is also shown that, for a set of specified parameters, the statistics are unique and they are independent of initial condition. It is pointed out that some previous results and interpretations on this problem are incorrect. The project supported by National Natural Science Foundation of China