Symmetry breaking bifurcations of a parametrically excited pendulum

This paper examines the bifurcation behavior of a planar pendulum subjected to high-frequency parametric excitation along a tilted angle. Parametric nonlinear identification is performed on the experimental system via an optimization approach that utilizes a developed approximate analytical solution. Experimental and theoretical efforts then consider the influence of a subtle tilt angle in the applied parametric excitation by contrasting the predicted and observed mean angle bifurcations with the bifurcations due to excitation applied in either the vertical or horizontal direction. Results show that small deviations from either a perfectly vertical or horizontal excitation will result in symmetry breaking bifurcations as opposed to pitchfork bifurcations.     

Model simulation of parametrically excited ship rolling

Two different models for simulating the ship motion in longitudinal or oblique seas are presented and studied in detail. Particular attention is devoted to the parametrically induced rolling which may be established by means of the nonlinear coupling between both heave-roll and/or pitch-roll motions. It is proved that the phenomenon is likely to occur with this mechanism when the roll frequency is subharmonic of the encounter wave frequency and when the vertical motions become resonant.     

Heteroclinic orbit and subharmonic bifurcations and chaos of nonlinear oscillator

Dynamical behavior of nonlinear oscillator under combined parametric and forcing excitation, which includes van der Pol damping, is very complex. In this paper, Melnikov'smethod is used to study the heteroclinic orbit bifurcations, subharmonic bifurcations and chaos in this system. Smale horseshoes and chaotic motions can occur from odd subharmonic bifurcation of infinite order in this system for various resonant cases.Finally the numerical computing method is used to study chaotic motions of this system. The results achieved reveal some new phenomena.     

Nonlinear response of a parametrically excited buckled beam

A nonlinear analysis of the response of a simply-supported buckled beam to a harmonic axial load is presented. The method of multiple scales is used to determine to second order the amplitude- and phase-modulation equations. Floquet theory is used to analyze the stability of periodic responses. The perturbation results are verified by integrating the governing equation using both digital and analog computers. For small excitation amplitudes, the analytical results are in good agreement with the numerical solutions. The large-amplitude responses are investigated by using a digital computer and are compared with those obtained via an analog-computer simulation. The complicated dynamic behaviors that were found include period-multiplying and period-demultiplying bifurcations, period-three and period-six motions, jump phenomena, and chaos. In some cases, multiple periodic attractors coexist, and a chaotic attractor coexists with a periodic attractor. Phase portraits, spectra of the responses, and a bifurcation set of the many solutions are presented.     

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.     

黏弹性传动带1:3内共振时的周期和混沌运动

研究了参数激励作用下黏弹性传动带在1:3内共振时的周期解分岔和混沌动力学. 同时考虑传动带的线性外阻尼因素和材料内阻尼因素. 首先建立了具有线性外阻尼情况下的黏弹性传动带平面运动时的非线性动力学方程, 黏弹性材料的本构关系用Kelvin模型描述. 然后考虑黏弹性传动带的横向振动问题, 利用多尺度法和Galerkin离散法得到黏弹性传动带系统在1:3内共振时的平均方程. 最后利用数值模拟方法研究了黏弹性传动带系统的周期振动和混沌动力学, 得到了系统在不同参数下的混沌运动. 数值模拟结果说明黏弹性传动带系统存在周期分岔, 概周期运动及混沌运动.     

Global transient dynamics of nonlinear parametrically excited systems

In this paper we examine the response of a typical nonlinear system that is subjected to parametric excitation. Particular attention is paid to how basins of attraction evolve such that the global transient stability of the system may be assessed. We show that at a forcing level that is considerably smaller than that at which the steady-state attractor loses its stability, there may exist a rapid erosion and stratification of the basin, signifying a global loss of engineering integrity of the system.We also show, for a system near its equilibrium state, that the boundaries in parameter space can become fractal. The significance of such an analysis is not only that it corresponds to a failure locus for a system subjected to a sudden pulse of excitation, but since the phase-space basin is often eroded throughout its central region, the determination of basin boundaries in control space can often reflect the characteristics of the phase-space basin structure, and hence on the macroscopic level they provide information regarding the global transient stability of the system.     

Analysis of global dynamics in a parametrically excited thin plate

The global bifurcations and chaos of a simply supported rectangular thin plate with parametric excitation are analyzed. The formulas of the thin plate are derived by von Karman type equation and Galerkin's approach. The method of multiple scales is used to obtain the averaged equations. Based on the averaged equations, the theory of the normal form is used to give the explicit expressions of the normal form associated with a double zero and a pair of pure imaginary eigenvalues by Maple program. On the basis of the normal form, a global bifurcation analysis of the parametrically excited rectangular thin plate is given by the global perturbation method developed by Kovacic and Wiggins. The chaotic motion of thin plate is also found by numerical simulation. The project supported by the National Natural Science Foundation of China (10072004) and by the Natural Science Foundation of Beijing (3992004)     

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.     

Influence of concurrent use of springs with characteristics of different types on the equilibrium of an inverted pendulum

The effect of the concurrent use of springs with characteristics of different types (hard, soft, or linear) on the equilibrium of an inverted simple pendulum is studied __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 7, pp. 107–113, July 2007.     

参-强激励联合作用下输流管的分岔和混沌行为研究

研究输送脉动流的两端固定输流管道在其基础简谐运动激励下的分岔和混沌行为,考虑管道变形的几何非线性和管道材料的非线性因素,推导了系统的非线性运动方程,并应用Galerkin方法对其进行了离散化处理。通过采用数值模拟方法,对系统的运动响应进行仿真,重点探讨了流体平均流速、流速脉动振幅以及基础简谐运动激励振幅对系统动态特性的影响。结果表明,系统在不同的参数下会发生围绕不同平衡点的周期和混沌等运动,并在系统中发现了两条通向混沌运动的途径:倍周期分岔和阵发混沌运动。     

Stochastic stability of parametrically excited random systems

Multidegree-of-freedom dynamic systems subjected to parametric excitation are analyzed for stochastic stability. The variation of excitation intensity with time is described by the sum of a harmonic function and a stationary random process. The stability boundaries are determined by the stochastic averaging method. The effect of random parametric excitation on the stability of trivial solutions of systems of differential equations for the moments of phase variables is studied. It is assumed that the frequency of harmonic component falls within the region of combination resonances. Stability conditions for the first and second moments are obtained. It turns out that additional parametric excitation may have a stabilizing or destabilizing effect, depending on the values of certain parameters of random excitation. As an example, the stability of a beam in plane bending is analyzed.Published in Prikladnaya Mekhanika, Vol. 40, No. 10, pp. 135–144, October 2004.     

Rotating a pendulum with an electromechanical excitation

The objective of this paper is showing investigation of pendulum rotations via vertical, non-linear electromechanical excitation generated using a RLC-circuit-powered solenoid, which is originally built for an electro-vibro-impact mechanism. Various non-linear phenomena of pendulum dynamics, namely period-1 rotation, period-1 oscillation and period-2 oscillation, have been observed experimentally from the proposed apparatus. A mathematical model has been developed for the experimental rig and the system parameters have also been identified for the mathematical model. Finally, numerical results have been generated using the developed mathematical model and identified parameters, and their correlations with experimental observations have been discussed.     

Periodic Response and Chaos in Nonlinear Systems with Parametric Excitation and Time Delay

In this paper, the periodic motions of a nonlinear system with quadratic,cubic, and parametrically excited stiffness terms and with time-delayterms are obtained by the incremental harmonic balance (IHB) method. Theelements of the Jacobian matrix and residue vector arising in the IHBformulation are derived in closed form. A mechanism model representingthe one-mode oscillation of beams and plates is considered as anexample. A path-following algorithm with an arc-length parametriccontinuation procedure is used to obtain the response diagrams. Thesystem also exhibits chaotic motion through a cascade of period-doublingbifurcations, which is characterized by phase planes, Poincaré sectionsand Lyapunov exponents. The interpolated cell mapping (ICM) procedure isused to obtain the initial condition map corresponding to multiplesteady-state solutions.     

Local and global bifurcations of valve mechanism

In this paper we study in detail problems of nonlinear oscillations of valve mechanism at internal combustion engine. The practical measurement indicates that stiffness of valve mechanism is not constant but is a function of the rotational angle of the cam. For simplicity of analysis we replace valve mechanism of internal combustion engine with a nonlinear oscillator of single degree of freedom under combined parametric and forcing excitation. We use the method of multiple scales and normal form theory to study local and global bifurcations of valve mechanism at internal combustion engine.     

悬臂输流管道的运动分岔现象和混沌运动

研究受约束悬臂输流管道的稳定性和运动分岔问题.在静态与动态失稳区域边界上的一个交叉点附近,用理论分析的方法详细研究了该系统可能发生的复杂运动和运动分岔现象.在动态失稳区域内发现了输流管道的概周期运动和由于概周期运动环面破裂而导致混沌的现象.理论分析结果与数值模拟结果相吻合.     

Global Bifurcations and Chaotic Dynamics in Nonlinear Nonplanar Oscillations of a Parametrically Excited Cantilever Beam

This paper presents the analysis of the global bifurcations and chaotic dynamics for the nonlinear nonplanar oscillations of a cantilever beam subjected to a harmonic axial excitation and transverse excitations at the free end. The governing nonlinear equations of nonplanar motion with parametric and external excitations are obtained. The Galerkin procedure is applied to the partial differential governing equation to obtain a two-degree-of-freedom nonlinear system with parametric and forcing excitations. The resonant case considered here is 2:1 internal resonance, principal parametric resonance-1/2 subharmonic resonance for the in-plane mode and fundamental parametric resonance–primary resonance for the out-of-plane mode. The parametrically and externally excited system is transformed to the averaged equations by using the method of multiple scales. From the averaged equation obtained here, the theory of normal form is applied to find the explicit formulas of normal forms associated with a double zero and a pair of pure imaginary eigenvalues. Based on the normal form obtained above, a global perturbation method is utilized to analyze the global bifurcations and chaotic dynamics in the nonlinear nonplanar oscillations of the cantilever beam. The global bifurcation analysis indicates that there exist the heteroclinic bifurcations and the Silnikov type single-pulse homoclinic orbit in the averaged equation for the nonlinear nonplanar oscillations of the cantilever beam. These results show that the chaotic motions can occur in the nonlinear nonplanar oscillations of the cantilever beam. Numerical simulations verify the analytical predictions.     

Electromagnetically driven parametric instability in an annular layer

Local measurements of the free surface amplitude of an annular layer of mercury submitted to a low frequency vertical magnetic filed are performed. The results confirm the linear analysis which predicts two types of classical transitions (harmonic and sub-harmonic) and a specific electromagnetic transition occurring with a combination of frequences.     

Milling Bifurcations from Structural Asymmetry and Nonlinear Regeneration

This paper investigates multiple modeling choices for analyzing the rich and complex dynamics of high-speed milling processes. Various models are introduced to capture the effects of asymmetric structural modes and the influence of nonlinear regeneration in a discontinuous cutting force model. Stability is determined from the development of a dynamic map for the resulting variational system. The general case of asymmetric structural elements is investigated with a fixed frame and rotating frame model to show differences in the predicted unstable regions due to parametric excitation. Analytical and numerical investigations are confirmed through a series of experimental cutting tests. The principal results are additional unstable regions, hysteresis in the bifurcation diagrams, and the presence of coexisting periodic and quasiperiodic attractors which is confirmed through experimentation.     

A new numerical technique for the analysis of parametrically excited nonlinear systems

A new computational scheme using Chebyshev polynomials is proposed for the numerical solution of parametrically excited nonlinear systems. The state vector and the periodic coefficients are expanded in Chebyshev polynomials and an integral equation suitable for a Picard-type iteration is formulated. A Chebyshev collocation is applied to the integral with the nonlinearities reducing the problem to the solution of a set of linear algebraic equations in each iteration. The method is equally applicable for nonlinear systems which are represented in state-space form or by a set of second-order differential equations. The proposed technique is found to duplicate the periodic, multi-periodic and chaotic solutions of a parametrically excited system obtained previously using the conventional numerical integration schemes with comparable CPU times. The technique does not require the inversion of the mass matrix in the case of multi degree-of-freedom systems. The present method is also shown to offer significant computational conveniences over the conventional numerical integration routines when used in a scheme for the direct determination of periodic solutions. Of course, the technique is also applicable to non-parametrically excited nonlinear systems as well.