Analytical period-3 motions to chaos in a hardening Duffing oscillator

In this paper, bifurcation trees of period-3 motions to chaos in the periodically forced, hardening Duffing oscillator are investigated analytically. Analytical solutions for period-3 and period-6 motions are used for the bifurcation trees of period-3 motions to chaos. Such bifurcation trees are based on the Hopf bifurcations of asymmetric period-3 motions. In addition, an independent symmetric period-3 motion without imbedding in chaos is discovered, and such a symmetric period-3 motion possesses saddle-node bifurcations only. The switching of symmetric to asymmetric period-3 motions is completed through saddle-node bifurcations, and the onset of asymmetric period-6 motions occurs at the Hopf bifurcations of asymmetric period-3 motions. Continuously, the onset of period-12 motions is at the Hopf bifurcation of asymmetric period-6 motions. With such bifurcation trees, the chaotic motions relative to asymmetric period-3 motions can be determined analytically. This investigation provides a systematic way to study analytical dynamics of chaos relative to period-m motions in nonlinear dynamical systems.     

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.     

Non-planar,non-linear oscillations of a beam—I. Forced motions

Large amplitude whirling motions of a simply supported beam constrained to have a fixed length are investigated. Equations of motion taking into account bending in two planes and longitudinal deformations are developed. Using the method of harmonic balance, response curves for certain planar and non-planar steady state, forced motions are obtained. Another approximate scheme is used to study the stability of these motions. Stable regions corresponding to non-planar motions are found, thus confirming the existence of whirling motions. Numerical results are presented and discussed for several specific cases.     

Stability of finite-amplitude and overstable convection of a conducting fluid through fixed porous bed

The stability of infinitestimal steady and oscillatory motions and finite amplitude steady motions of a conducting fluid through porous media with free boundaries which is heated from below and cooled from above is investigated in the presence of a uniform magnetic field. Infinitesimal steady motions are investigated using Liapunov method and its is shown that the principle of exchange of stability is valid only when P_m/P_r≤1 with a restricted value of the Hartmann number. It is shown that overstable motions are due to the zonal current induced by the magnetic field. Finite amplitude steady motions are investigated using Veronis [1] analysis and it is shown that for a restricted range of Hartmann numbers and porous parameter P_l, steady finite-amplitude motions can exist for values of the Rayleigh number smaller than that value corresponding to oscillatory motions. Since the Busse number is greater than the wave number the horizontal scale of the steady finite-amplitude motions is larger than that of the overstable motions.     

Forces and energetics of intermittent swimming

Experiments are reported on intermittent swim-ming motions. Water tunnel experiments on a nominally two-dimensional pitching foil show that the mean thrust and power scale linearly with the duty cycle, from a value of 0.2 all the way up to continuous motions, indicating that individ-ual bursts of activity in intermittent motions are independent of each other. This conclusion is corroborated by particle image velocimetry (PIV) flow visualizations, which show that the main vortical structures in the wake do not change with duty cycle. The experimental data also demonstrate that intermittent motions are generally energetically advan-tageous over continuous motions. When metabolic energy losses are taken into account, this conclusion is maintained for metabolic power fractions less than 1.     

Formation of intermittent region by coherent motions in the turbulent boundary layer: Coherent motions and intermittent region

Coherent motions in the turbulent boundary layer and their relation to the large-scale structures in the intermittent region are investigated experimentally. Using conditional sampling technique, we have shown that a coherent motion is a pair of counter-rotating fluid motions. These coherent motions of various sizes take place in the turbulent boundary layer. Most of them are short and are found only near the wall, but some are tall enough to reach the outer edge of the boundary layer. The turbulent bulges of the intermittent region are formed by these tall coherent motions.     

冲击渐进振动系统相邻基本振动的转迁规律

冲击振动现象广泛存在于动力机械系统中,使得系统表现出复杂的动力学响应.目前对冲击振动系统的p/1类基本振动的稳定性及分岔研究报道较少,而且已有的对冲击振动系统动力学的研究基本都是基于单参数分岔进行分析的.研究以小型振动冲击式打桩机为工程背景,建立了冲击渐进振动系统的力学模型.分析了激振器和缓冲垫发生碰撞的类型,以及滑块渐进运动的条件.给出了系统可能呈现的四种运动状态的判断条件和运动微分方程.通过二维参数分岔分析得到系统在(ω,l)参数平面内存在的各类周期振动的参数域和分布规律.详细分析了相邻p/1类基本振动的转迁规律.在5/1基本振动的参数域的右边区域,相邻p/1基本振动的参数域临界线上存在一个奇异点X_p,相邻p/1类基本振动的分岔特点以奇异点X_p为临界点.在l小于l_X_p的区域内,相邻p/1基本振动经实擦边分岔和鞍结分岔相互转迁,实擦边分岔线和鞍结分岔线之间存在迟滞域,迟滞域内,系统存在两个周期吸引子共存的现象.在l大于l_X_p的区域内,相邻p/l类基本振动的参数域之间存在一个中间过渡区域.中间过渡区域内,系统呈现(2p+2)/2和(2p+1)/2周期振动等.在5/1基本振动的参数域的左边区域,p/1基本振动经多重滑移分岔产生(P+1)/1基本振动.     

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.     

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.     

高机动飞行器非指令运动及其控制的研究进展

高机动飞行器往往都是通过大攻角飞行来实现高机动科目的, 在发展高机动飞行器的过程中, 其非指令运动是伴随着大攻角飞行而常常出现的运动形态. 为此, 应在飞行器设计的早期阶段, 充分研究所设计布局的大攻角流动性态及其相应的非指令运动的形态;揭示这类运动形态的主控流动;在此基础上形成和发展流动控制新技术, 以达到抑制非指令运动的目的. 由于大攻角前体非对称涡往往与非指令运动密切相关, 为此本文首先指出前体非对称涡流动对头部微扰动十分敏感, 以致长期以来让人们误认为这类流动具有不确定性. 研究表明, 通过设置人工微扰动可使前体非对称涡流动具有可重复性, 并揭示该流动随扰动周向角变化的响应、演化规律. 通过利用大、小后掠翼两类翼身组合体的典型布局形式, 研究它们所呈现的摇滚运动形态, 揭示其摇滚运动的不同主控流动机理, 在此基础上分别发展了抑制小、大后掠翼身组合体摇滚运动的流动控制技术: 快速旋转头部扰动和适当设置扰动位使翼、身的两对非对称涡处于反相. 在抑制非指令运动的研究中, 深入理解和揭示头部微扰动对非对称涡流动的响应、演化机理是至关重要的, 应予以特别关注.     

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。     

On steady motions of a rigid body bearing a two-degree-of-freedom control momentum gyroscope with precession axis arbitrarily oriented in the carrying body

Steady motions of a rigid body with a control momentum gyroscope are studied versus the gimbal axis direction relative to the body and the magnitude of the system angular momentum. The study is based on a formula that gives a parametric representation of the set of the system steady motions in terms of the rotation angle of the gimbal. It is shown that, depending on the values of the parameters, the system has 8, 12, or 16 steady motions and the number of stable motions is 2 or 4.     

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.     

Convective motions in a porous medium heated from below

Convective motions in a porous medium filling a horizontal cylinder with a cross section of arbitrary shape and heated from below are studied. The small-parameter method is used to obtain infinitely many stationary motions forming a one-parameter family. For small values of the parameter, all of these motions are stable with respect to small perturbations. The article also discusses the case of heating which is not strictly vertical. It is found that in this case only one stationary motion is stable.     

Steady motions of an axisymmetric satellite: an atlas of their bifurcations

The steady motions of an axially symmetric rigid satellite orbiting a fixed spherically symmetric rigid body are considered in this paper. Based on a model for the dynamics of the satellite which incorporates the classical approximation for the gravitational potential and includes the attitude-orbit coupling, the bifurcations and non-linear stability of the steady motions of the satellite are discussed. These results extend earlier works on this problem by Likins, Pringle, Rumyantsev, Stepanov, and Thomson, by showing how the motions found by these authors are interrelated, and how several of them are physically unrealistic because their orbital radii are too small.     

On limit motions of a system of control moment gyros with intrinsic dissipation in a homogeneous gravitational field

We study the limit motions of a free rigid body bearing n two-degree-of-freedom control moment gyros with dissipation in the gyro gimbal suspension axes. We show that, in the absence of dynamic symmetry, the limit motions of the system are only steady rotations at a constant angular velocity. In the case of dynamic symmetry, the gyros can be arranged so that, in addition to steady rotations, the system exhibits limit motions that are regular precessions.     

Balancing a Rotor with Two Coupled Perfectly Rigid Bodies

The steady motions of a vertical rotor balanced by two coupled perfectly rigid bodies (PRBs) are analyzed for stability. The rotor executes plane-parallel motions. The PRBs can rotate together about the rotor axis and about its transverse axes through equal angles but in opposite directions. It is established that when the mass of the PRBs is much less than that of the rotor and the rotor spins with a superresonant speed, only the basic steady motions, which are those of the best-balanced rotor, are asymptotically stable.     

Natural motion density of elastic shells under intensive dynamic loading

The question of the natural motion density in dynamics problems of elastic shells is considered. Motions are studied for which an exponential growth in amplitude with time occurs. The number of natural motions incident in a given range of variation of the exponent is computed by using an idea of R. Courant. The governing natural motions and condensation points at which the natural motion density tends to infinity are found. The condensation points and governing motions are compared in specific examples.     

Three-dimensional dynamics of supported pipes conveying fluid

This paper deals with the three-dimensional dynamics and postbuckling behavior of flexible supported pipes conveying fluid, considering flow velocities lower and higher than the critical value at which the buckling instability occurs. In the case of low flow velocity, the pipe is stable with a straight equilibrium position and the dynamics of the system can be examined using linear theory. When the flow velocity is beyond the critical value, any motions of the pipe could be around the postbuckling configuration (non-zero equilibrium position) rather than the straight equilibrium position, so nonlinear theory is required. The nonlinear equations of perturbed motions around the postbuckling configuration are derived and solved with the aid of Galerkin discretization. It is found, for a given flow velocity, that the first-mode frequency for in-plane motions is quite different from that for out-of-plane motions. However, the second- or third-mode frequencies for in-plane motions are approximately equal to their counterparts for out-of-plane motions, keeping almost constant values with increasing flow velocity. Moreover, the orientation angle of the postbuckling configuration plane for a buckled pipe can be significantly affected by initial conditions, displaying new features which have not been observed in the same pipe system factitiously supposed to deform in a single plane.