Buckling and initial post-buckling behavior of thin-walled I columns

The behavior of axially compressed thin-walled I columns after torsional buckling is discussed. At first the state of strain in a column subjected to a large angle of twist is outlined. The considerations are based on the following assumptions: (1) cross section is non-deformable in its own plane, (2) shear deformation is negligible, and (3) strains are small and elastic. The initial post-buckling equilibrium paths are determined by utilizing a perturbation approach. The finite-element and analytical procedure is presented. It has been shown that the point of bifurcation for simply supported, or clamped I column with constant cross section is symmetric and stable. Two examples of I columns of variable cross section are also considered. It is worthwhile noticing that in these examples the critical loads are beyond the limits described by the critical force for column with the minimum and the maximum cross section. The points of bifurcation in these cases are also symmetric and stable. This property of the bifurcation point is very important with regard to the sensitivity to the initial geometrical imperfections. In the case of the unstable point of bifurcation a drastic decrease of the value of the critical loads is possible, which does not hold for stable point.     

The nonlinear behaviour of a slender flexible cylinder pinned or clamped at both ends and subjected to axial flow

The nonlinear dynamics of a slender flexible cylinder subjected to axial flow is studied when both its ends are either pinned or clamped, particularly focusing on the post-critical behaviour. In both cases, the system is stable at low flow velocities until the critical flow for divergence, at which point the initial equilibrium position of the cylinder becomes unstable and a new stable buckled solution arises (together with its symmetric counterpart). The amplitude of the buckled solution increases with the flow velocity. At higher flow, the buckled stationary cylinder loses stability by a Hopf bifurcation, after which a periodic solution arises. The frequency of oscillation after this Hopf bifurcation, in the case of a pinned–pinned cylinder, is almost twice as high as that in the clamped–clamped case, due to the dynamic loss of stability in a higher mode in the former case. The periodic solution is followed by a period-doubling bifurcation, giving rise to period-2 oscillations. The system undergoes a torus bifurcation afterward, followed by quasiperiodic and chaotic oscillations at higher flow velocities. All the critical flow velocities for the pinned–pinned cylinder are smaller than those for the clamped–clamped one. In the case of a pinned–pinned cylinder, at still higher flow velocities, there exists a range of flow velocities in which chaotic and static solutions co-exist; this has not been observed in the clamped–clamped case.     

Level-ground walking for a bipedal robot with a torso via hip series elastic actuators and its gait bifurcation control

Recent studies have shown that a bipedal robot with a torso supported by springs on the hip can have a stable passive gait on a slope, while such a robot walking on level ground is a new challenge and has rarely been studied. This research adds actuators in series with the springs to form series elastic actuators on the hip and applies a state machine as controller to achieve stable walking on level ground. During walking, hip series elastic actuators support the torso from the legs as well as complement the energy to the system via elastic potential energy. The state machine uses the landing impact of the swing leg and the actuation durations as events to make the robot switch between successive active and passive walking processes. Because this simple scheme makes full use of the dynamics of the robot, it can lead to an efficient and natural gait. By means of numerical simulation, in addition to the stable period-1 gait, we found a variety of gait bifurcation phenomena, including the period-doubling bifurcation, the Neimark–Sacker bifurcation, the Neimark–Sacker-2 bifurcation, the period-X bifurcation, and the Neimark–Sacker-X bifurcation, among which many types have never been reported in previous studies. We also show that the unstable period-1 gait embedded in the bifurcation gait can be stabilized by applying the Ott–Grebogi–Yorke method. Not only can the gait bifurcation be suppressed, but also higher gait performance can be achieved.     

Buck-Boost DC-DC变换器中分叉与混沌问题的研究(Ⅱ)——数值分析与实验

研究了电流控制型Buck Boost变换器中的分叉与混沌问题. 在连续模式下的Buck Boost变换器的离散数学模型基础上, 从不动点及稳定性理论的角度对Buck Boost变换器的第一分叉点进行了严格的数值分析, 得到了三维参数空间曲面图. 此外, 建立了实验硬件电路, 实验结果验证了理论分析.     

Bifurcation mechanism underlying echelon-mode formation

This paper presents a theory on the underlying mathematical mechanism of the echelon mode (a series of parallel short wrinkles that looks like a flight of stairs or wild geese arranged in formation) which has been observed ubiquitously with uniform materials, but which has long denied successful numerical simulations. It is shown by means of the group-theoretic bifurcation theory that the echelon mode formation can be explained as a recursive (secondary, tertiary, …) symmetry-breaking bifurcation if O(2) × O(2) is chosen as the underlying symmetry to model the local uniformity of materials. This implies, for example, that the use of periodic boundaries is essential to successfully realize the oblique stripe patterns and the subsequent echelon mode formation in numerical simulations. In fact, a recursive bifurcation analysis of a rectangular domain with periodic boundaries subject to uniform uniaxial compression yields various kinds of patterns, such as diamond, stripe and echelon modes, which are often observed for materials under shear.     

A numerical algorithm coupling a bifurcating indicator and a direct method for the computation of Hopf bifurcation points in fluid mechanics

This paper deals with the computation of Hopf bifurcation points in fluid mechanics. This computation is done by coupling a bifurcation indicator proposed recently (Cadou et al., 2006) [1] and a direct method (Jackson, 1987; Jepson, 1981) [2] and [3] which consists in solving an augmented system whose solutions are Hopf bifurcation points. The bifurcation indicator gives initial critical values (Reynolds number, Strouhal frequency) for the direct method iterations. Some classical numerical examples from fluid mechanics, in two dimensions, are studied to demonstrate the efficiency and the reliability of such an algorithm.     

Bifurcation and post-buckling analysis of laminated composite plates via reduced basis technique

A reduced basis technique and a problem-adaptive computational algorithm are presented for the bifurcation and post-buckling analysis of laminated anisotropic plates. The computational algorithm can be conveniently divided into three distinct stages. The first stage is that of determining the bifurcation point. The plate is discretized by using displacement finite element (or finite difference) models. The special symmetries exhibited by the response of the anisotropic plate are used to reduce the size of the analysis region. The vector of unknown nodal parameters is expressed as a linear combination of a small number of basis vectors, and a Rayleigh-Ritz technique is used to approximate the finite element equations by a small system of algebraic equations. The reduced equations are used to determine the bifurcation point and the associated eigen mode of the panel.In the second stage of the bifurcation buckling mode is used to obtain a nonlinear solution in the vicinity of the bifurcation point and new (updated) sets of basis vectors and reduced equations are generated. In the third stage the reduced equations are used to trace the post-buckling paths.The effectiveness of the proposed technique for predicting the bifurcation and post-buckling behavior of plates is demonstrated by means of numerical examples for plates loaded by means of prescribed edge displacements.     

利用谐波平衡法对Buck变换器中倍周期分岔的仿真研究

利用谐波平衡法对Buck变换器中的倍周期分岔进行了仿真研究，首先给出了连续控制模式下电压控制型Buck变换器的动力学模型，然后采用谐波平衡法进行分析，获得了产生倍周期分岔的充要条件，同时也得到了分岔的准确位置。基于这个分岔条件，可以设计一个前馈控制来避免倍周期分岔的发生。此控制法有利于输入电压工作范围的大幅度扩大，以及较好的输出电压校准。     

Buck-Boost DC-DC变换器中分叉与混沌问题的研究(Ⅰ)——建模与仿真

研究了电流控制型Buck-Boost变换器中的分叉与混沌问题. 首先, 对连续模式下的Buck-Boost变换器建立了离散数学模型, 在此基础上, 在输入电压E、参考电流Iref、电阻R、电感L和电容C等分叉参数作用下, 通过数值方法对Buck-Boost变换器中的分叉与混沌进行了详细的仿真研究. 仿真研究结果表明: Buck-Boost变换器具有丰富的非线性行为———分叉与混沌, 随着各个分叉参数的变化系统会遵循倍周期分叉的规律走向混沌.     

二维Logistic映射的动力学分析

对二维logistic映射的动力学研究有助于认识和预测更复杂的高维非线性系统的性态.利用解析计算和实验分析相结合的方法揭示出:(1) 参数空间中二维logistic映射发生第一次分岔的边界方程;(2) 二维logistic映射可按倍周期分岔和Hopf分岔走向混沌;(3) 二维logistic映射的吸引盆中周期和非周期区域之间的边界是分形的,这意味着无法预测相平面上点运动的归宿;(4) Mandelbrot-Julia集的结构由控制参数决定,且它们的边界是分形的.     

Bandwidth and gain improvements of low‐profile H‐shaped microstrip patch antenna under quadruple‐mode resonance

A wideband low profile H‐shaped microstrip patch antenna (MPA) with reallocated quadruple‐mode resonance is presented for indoor wireless communication application. In this paper, the TM₂₀ (mode 1), TM₀₂ (mode 2), TM₂₂ (mode 3), and additionally notch mode 4 of the proposed MPA are simultaneously employed. First, the rectangular radiating patch is reshaped as an H‐shaped radiator so as to separate a pair of degenerate modes (mode 1 and mode 2). Then, a pair of linear notches is cut on the diagonal of the patch to excite an additional notch resonance (mode 4). Finally, in order to improve the frequency of mode 1, four shorting pins are placed at the four corners of the H‐shaped patch. Therefore, the bandwidth of the antenna is dramatically increased up by utilizing four resonant modes (modes 1, 2, 3, and 4). A prototype of H‐shaped patch antenna with notches and shorting pins is manufactured and measured. The results show that the antenna achieves a broad bandwidth of about 31.7% (2.31‐3.18 GHz), and its profile is only 0.036 wavelength of center frequency. It is particularly noticed that a relative high gain of around 9.8 dBi is successfully acquired, while keeping relative stable dual‐beam radiation patterns.     

受面内激励和横向外激励联合作用下蜂窝夹层板的双Hopf分叉

随着航空航天事业的发展,对各种材料性能的要求也越来越高.而蜂窝夹层板在结构和性能上具有许多优点,已在航空航天等领域应用广泛,并在一些重要结构中充当承力部件,但由于其特殊的蜂窝结构,相对于一般的板,在受力时会发生比较大的变形,所以用非线性理论研究蜂窝夹层板结构,并考察不同参数对非线性振动特性的影响,具有重要的理论和实际意义.如今,蜂窝夹层板的几何非线性问题已引起更多学者的关注.在一般均质理论的假设下,一些学者已经研究了各向同性蜂窝夹层板板的非线性动力学特性.本文研究了一类受面内激励和横向外激励联合作用下的四边简支蜂窝夹层板在主参数共振-1:2内共振时的双Hopf分叉问题.首先利用多尺度法得到系统的平均方程,然后结合分叉理论得到了系统的分叉响应方程,根据对分叉响应方程的分析,得到了六种不同的分叉响应曲线并给出了系统产生双Hopf分叉的条件.利用数值方法得到系统在参数平面的分叉集,通过对不同分叉区域的分析发现,随着参数的变化系统平衡点会分叉为两类周期解,随后周期解会通过广义静态分叉为准周期解,或者通过广义Hopf分叉为3D环面.     

A MOSFET-based model of a class 2 nerve membrane

We have constructed a nerve membrane using MOSFET circuitry, which can be a basic element of an FET-based neural system. Its mechanism of action potentials generation is designed to reproduce that of the Hodgkin-Huxley equations. The responses to singlet, doublet, repetitive pulse, and sustained stimuli are analyzed to show that it exhibits similar properties to the Hodgkin-Huxley equations; namely, 1) excitable dynamics with generation of action potentials, 2) the existence of a chaotic response to periodic stimuli, and 3) Class 2 excitability. It is known that Class 2 excitability is generated by an inverted Hopf bifurcation. We have applied Hopf bifurcation theory to our nerve membrane's system equations and have shown a routine for ascertaining whether a certain parameter set generates an inverted Hopf bifurcation.     

Conjugate gradient type methods for semilinear elliptic problems with symmetry

We study block conjugate gradient methods in the context of continuation methods for bifurcation problems. By exploiting symmetry in certain semilinear elliptic differential equations, we can decompose the problems into small ones and reduce computational cost. On the other hand, the associated centered difference discretization matrices on the subdomains are nonsymmetric. We symmetrize them by using simple similarity transformations and discuss some basic properties concerning the discretization matrices. These properties allow the discrete pure mode solution paths branching from a multiple bifurcation point [0, λ^m,n] of the centered difference analogue of the original problem to be represented by the solution path branching from the first simple bifurcation point (0, μ^1,1) of the counterpart of the reduced problem. Thus, the structure of a multiple bifurcation is preserved in discretization, while its treatment is reduced to those for simple bifurcation of problems on subdomains. In particular, we can adapt the continuation-Lanczos algorithm proposed in [1] to trace simple solution paths. Sample numerical results are reported.     

Finite element analysis of bifurcation in nonlocal strain softening solids

Progressive damage in brittle heterogeneous materials produces at the macroscopic level strain softening from which theoretical difficulties arise (e.g. ill-posedness and multiple bifurcation points). This characteristics behavior favours spurious strain localization in numerical analyses and calls for the implementation of localization limiters, for instance nonlocal damage constitutive relations. The issue of possible (stable or unstable) equilibrium paths, multiple localization zones, and of the detection of bifurcation points has, however, never been addressed in the context of nonlocal constitutive laws. We extend here the eigenmode analysis and perturbation method proposed by De Borst to the study of the bifurcation and post-bifurcation response of discrete nonlocal strain softening solids. Numerical applications on beams show that bifurcation and instability may occur in the post-peak regime. As opposed to the case of local constitutive relations, the number of possible solutions at a bifurcation point is restricted due to the constraint introduced by the localization limiter and these solutions are shown to be mesh independent.     

Finding nondegenerate Hopf bifurcation points for four-dimensional two-parametric systems

We establish a numerically feasible algorithm to find a simplicial approximation A to a certain part of the boundary of the set of stable (or Hurwitz) polynomials of degree 4. Moreover, we have that . Using this, we build an algorithm to find a piecewise-linear approximation to the intersection curve of a given surface contained in ⁴ with . We have also devised an efficient computer program to perform all these operations. The main motivation is to find the curve of nondegenerate bifurcation points in parameter space for a given 2-parametric Hopf bifurcation problem of dimension 4.     

Anomalous Statistics for Type-III Intermittency

The statistics for the distribution of laminar phases in type-IIIintermittency is examined for the map . Due to a strongly nonuniform reinjectionprocess, characteristic deviations from the normal statistics are observed.There is an enhancement of relatively long laminar phases followed by anabrupt cut-off of laminar phases beyond a certain length. The paper alsoexamines the bifurcation structure of two symmetrically coupled maps, eachdisplaying a subcritical period-doubling bifurcation. The conditions forsuch a pair of coupled maps to exhibit type-II intermittency are discussed.     

转子-密封系统中气流激振力的非线性动力学特性分析

在高参数汽轮机组和航空发动机等旋转机械中，转子-密封中的气流激振力对转子非线性动力学特性的影响不容忽视．本研究中建立了转子-密封系统三维流场模型，应用计算流体动力学（CFD）软件对可压缩气流流场进行模拟计算，获得了密封流场特性．由流场计算结果进一步获得了Muszynska气流激振力模型中的相关经验系数，使得此模型更加适用于气流激振力的计算．在对转子一密封系统进行非线性动力学分析过程中应用幂级数展开形式建立了系统幂级数模型．利用平均法得到气流激振力的1：2亚谐共振分岔方程，进一步应用奇异性理论和Hopf分岔理论研究了系统1：2亚谐共振的转迁集和系统超临界Hopf分岔与亚临界Hopf分岔的存在条件．通过参数控制方法抑制了转子-密封系统出现亚临界分岔的出现，使得系统稳定性提高．本文的分析结果对工程设计和操作具有一定的指导作用和意义．     

四阶血细胞生成系统的双参数分岔分析

动态血液病表现为血液细胞的数量出现大范围的波动,其主要病因是造血干细胞数量的异常变化.本文针对四阶造血系统,针对于临床上表现为动态血液病的血细胞数量变化的振荡模式,通过造血系统的双参数分岔分析表明:当一个参数发生变化而导致系统出现振荡解时,可以通过调整另外一个参数将振荡解转化为稳定的平衡解,从而维持病人的血细胞数量的稳定.