Dynamics of piecewise linear interval maps with hysteresis

We study a special case of multistate maps, piecewise linear interval maps with hysteresis. The main object of study is the global attractor. We find the conditions for it to be lower semicontinuous. We also prove that it coincides with the non-wandering set and prove several facts about omega-limit sets of the discontinuity points of the map.     

Estimation of parameters of a multifractal process

Multifractal functions are widely used to model irregular signals such as turbulence, data stream or road traffic. Here, we consider multifractal functions defined as lacunar wavelet series observed in a white noise model. These random functions are statistically characterized by two parameters. The first parameter governs the intensity of the wavelet coefficients while the second one governs its sparsity. We construct estimators of these two parameters and discuss statistical properties of this important model: the rate of the Fisher information and a testing procedure to check the multifractal feature of an observed noisy signal.      

基于多重分形去趋势波动的机械故障诊断新方法

针对旋转机械系统故障信号的非平稳性、非线性等复杂特征,给出一种基于多重分形去趋势波动分析的机械故障诊断方法。该方法首先对原始信号进行去趋势处理,再结合多重分形理论提取多重分形谱面积和多重分形熵两个分形参数,并将其作为故障诊断的新判据,最后通过实验结果证明了方法的有效性,从而为机械故障诊断提供了一种新的方法。     

The irregular set for maps with the specification property has full topological pressure

Let (X, d) be a compact metric space, f : X ? X be a continuous map with the specification property and ? : X ? ? a continuous function. We consider the set of points for which the Birkhoff average of ? does not exist (which we call the irregular set for ?) and show that this set is either empty or carries full topological pressure (in the sense of Pesin and Pitskel see, for example [Y.B. Pesin, Dimension Theory in Dimensional Systems: Contemporary Views and Applications, University of Chicago Press, Chicago, 1997]). We formulate various equivalent natural conditions on ? that completely describe when the latter situation holds and give examples of interesting systems to which our results apply but were not previously known. As an application, we show that for a suspension flow over a continuous map with specification, the irregular set carries full topological entropy.     

基于多重分形去趋势波动分析的齿轮箱故障特征提取方法

齿轮箱故障信号通常是具有多标度行为的非平稳信号,去趋势波动分析(Detrended Fluctuation Analysis, DFA)不能准确揭示隐藏在这类信号中的动力学行为。多重分形去趋势波动分析(Multifractal Detrended Fluctuation Analysis, MF-DFA)是DFA方法的拓展,能够有效地揭示隐藏在多标度非平稳信号中的动力学行为。利用MF-DFA计算齿轮箱故障信号的多重分形奇异谱,而多重分形奇异谱的宽度、最大奇异指数、最小奇异指数和极值点对应的奇异指数都具有明确的物理意义,能够表征齿轮箱故障信号的内在动力学机制,适合作为齿轮箱振动信号的故障特征。提出一种基于MF-DFA的齿轮箱故障特征提取方法,将该方法用于包含正常、轻度磨损、中度磨损和断齿故障齿轮箱的故障诊断,并与DFA方法的结果进行了对比。结果表明,提出的方法对齿轮箱故障状态的变化非常敏感,能够完全分离相近的故障模式,有效地克服了传统DFA方法存在的缺陷,为齿轮箱的故障特征提取提供了一种新方法。     

A numerical study of infinitely renormalizable area-preserving maps

It has been shown in Gaidashev and Johnson [D. Gaidashev and T. Johnson, Dynamics of the universal area-preserving map associated with period doubling: stable sets, J. Mod. Dyn. 3(4) (2009), pp. 555–587.] and Gaidashev et al. [D. Gaidashev, T. Johnson, and M. Martens, Rigidity for infinitely renormalizable area-preserving maps, in preparation.] that infinitely renormalizable area-preserving maps admit invariant Cantor sets with a maximal Lyapunov exponent equal to zero. Furthermore, the dynamics on these Cantor sets for any two infinitely renormalizable maps is conjugated by a transformation that extends to a differentiable function whose derivative is Hölder continuous of exponent α?>?0. In this article we investigate numerically the specific value of α. We also present numerical evidence that the normalized derivative cocycle with the base dynamics in the Cantor set is ergodic. Finally, we compute renormalization eigenvalues to a high accuracy to support a conjecture that the renormalization spectrum is real.     

On the asymptotic properties of piecewise contracting maps

We are interested in the phenomenology of the asymptotic dynamics of piecewise contracting maps. We consider a wide class of such maps and we give sufficient conditions to ensure some general basic properties, such as the periodicity, the total disconnectedness or the zero Lebesgue measure of the attractor. These conditions show in particular that a non-periodic attractor necessarily contains discontinuities of the map. Under this hypothesis, we obtain numerous examples of attractors, ranging from finite to connected and chaotic, contrasting with the (quasi-)periodic asymptotic behaviours observed so far.     

New methods for estimating the torsional rigidity of composite bars

We obtain general bounds on the torsional rigidity and give examples on how these bounds can be used to obtain estimates and simple formulae which yield more accurate information than the classical treatment. Moreover, we use the homogenization theory to estimate the limit of the torsional rigidity when the characteristic length of the composite microstructures decreases.     

A comparison of piecewise linear programming formulations for stochastic disassembly line balancing

Recently, several mathematical programming formulations and solution approaches have been developed for the stochastic disassembly line balancing problem (DLBP). This paper aims at finding optimal solutions for the stochastic DLBP. Two second-order cone programming (SOCP1 and SOCP2) models and five piecewise linear mixed integer programming (PwLP) models are presented. The PwLP formulations involve two specially ordered sets of type 2 (S1 and S2) models and three convex combination (CC1, CC2 and CC3) models. In each modelling category, the latter models strengthen the initial S1 and CC1 models. Our computational analysis of a total 240 instances of ten problems demonstrates that all the seven models can be used to solve practical-sized DLBP problems to optimality using GUROBI. The SOCP2 model and the strengthened S2 and CC2 models lead to lower computation times, compared to SOCP1, S1, CC1 and CC3, respectively. Using the strengthened S2 and CC2 formulations, the CPU times of the CC3 model available in the literature can be reduced by 50 and 40%, respectively. Besides analysing the optimal solutions and the differences of the computation times, we present insights gained from our results.     

Multifractality in steady-state accommodation is robust to dynamic correction of aberrations using adaptive optics

When fixating on a stationary object, the human eye exhibits microfluctuations in accommodation. Changes in the magnitude of these fluctuations reflect changes in the accommodation control system. We used adaptive optics to determine the effect of monochromatic aberration dynamics on the control of steady-state accommodation of four subjects. The subjects viewed a stationary stimulus at 2 D while selective Zernike aberrations were corrected. The fluctuations in accommodation were characterised using a wavelet-based multifractal formalism approach. We found that for all subjects, and all experimental conditions, the accommodative fluctuations were multifractal. For one subject, we found that the width of the multifractal spectrum was statistically significantly larger when even-order aberrations were corrected as compared to no aberrations corrected. Hence, in general, for the subjects tested, the multifractal nature of steady-state accommodation control is unaffected by the manipulation of monochromatic aberration dynamics. Averaging across all subjects and experimental conditions, the mean spectrum was right-skewed with a most frequently occurring Hölder exponent of 0.31?±?0.08. Future applications of multifractal analysis to accommodation control are discussed.     

The influence of surface morphology and rigidity of the substrata on cell motility

The purpose of this study is to measure the cell motility under different stiffness and pattern of the substrata. Human melanoma cells were used for this study. Three surface patterns including flat, 6 µm-cone, and 6 µm-groove on polydimethylsiloxane (PDMS) were created by the micro-electro-mechanical system (MEMS). Glass dish was used as control substrate. After cells were seeded for 30 min, the cell images were taken every minute for 2 h. Each group had 4-5 dishes and 27-38 cells were calculated. The cell motility was 1.13 ± 1.30, 2.12 ± 1.21, 2.39 ± 1.11 and 3.08 ± 1.49 µm/min on glass, flat, cone, and groove PDMS, respectively. Cells in PDMS groups moved significantly faster than the control group (glass) due to smaller stiffness of the former substrates. More than 80% of cells on grooved PDMS moved along the grooves, indicating the grooved surface morphology could control the direction of cell movement. Our results display that substrate modulus and pattern can influence cell motility.     

考虑抗弯刚度影响的斜拉索面内非线性自由振动分析

研究了考虑抗弯刚度影响的斜拉索面内非线性自由振动，根据小垂度斜拉索的受力特性，求解出了反映抗弯刚度影响的斜拉索静态曲线方程，同时推导了它在平面内发生非线性自由振动的运动微分方程，针对斜拉索发生单模态振动的情况，将上述方程转化为一个带平方项的Duffing方程，并运用KBM法给出了反映该平方项影响的近似解析解；最后，结合具体工程实例分析了斜拉索的有关动力特性。     

Torsional rigidity of a circular bar with multiple circular inclusions using the null-field integral approach

In this article, a systematic approach is proposed to calculate the torsional rigidity and stress of a circular bar containing multiple circular inclusions. To fully capture the circular geometries, the kernel function is expanded to the degenerate form and the boundary density is expressed into Fourier series. The approach is seen as a semi-analytical manner since error purely attributes to the truncation of Fourier series. By collocating the null-field point exactly on the real boundary and matching the boundary condition, a linear algebraic system is obtained. Convergence study shows that only a few number of Fourier series terms can yield acceptable results. Finally, torsion problems are revisited to check the validity of our method. Not only the torsional rigidities but also the stresses of multiple inclusions are also obtained by using the present approach.     

Studies on identification of an efficient and accurate integration scheme for piecewise linear dynamic problems

Engineering structures often have nonlinear characteristics. Many of these nonlinear systems could be modelled as piecewise linear. The dynamic analysis of such systems can be carried out by the direct integration of the mathematical model of the system. In this article, solution schemes are identified and qualified studying a single degree-of-freedom system and evaluated studying real-life engineering structures. As an alternative, a modified Newmark method with iteration for nonlinear forces and half-step error monitoring is presented. The modified scheme has been found to be efficient and accurate.     

Multifractal detection of overlaps based on a stereometric analysis of fracture surfaces: Application to fractures of sintered carbides

A methodology is presented of a multifractal image analysis which was applied to real stereometric files from profilographometric examinations of the fractures of WC-Co sintered carbides. The analysis was conducted based on the research scheme and theoretical basis discussed in [Stach S, Cybo J. Multifractal detection of overlaps based on a stereometric analysis of fracture surface: assumptions, Materials Characterization (in this issue).[1]]. The solution presented enables the detection of the locations of overlaps on a fracture surface and their graphic presentation.     

A multifractal analysis of fatigue crack growth and its application to concrete

As is well-known, strength of materials is influenced by the specimen or structure size. In particular, several experimental campaigns have shown a decrease of the material strength under static or fatigue loading with increasing structure size, and some theoretical arguments have been proposed to interpret such a phenomenon. As far as fatigue crack growth is concerned, limited information on size effect is available in the literature, particularly for so-called quasi-brittle materials like concrete. In the present paper, by exploiting concepts of fractal geometry, some definitions of fracture energy and stress intensity factor based on physical dimensions different from the classical ones are discussed. A multifractal size-dependent fatigue crack growth law (expressing crack growth rate against stress intensity factor range) is proposed and used to interpret relevant experimental data related to concrete.     

瓦楞纸板基于抗弯刚度的一种简化模型

基于等效抗弯刚度原理,对瓦楞纸板进行了简化,得到了瓦楞纸板的简化模型——等效板,并利用An-sys Workbench有限元软件对等效板有限元模型进行了屈曲分析,最后通过试验进行了验证。结果表明,通过抗弯刚度得到瓦楞纸板简化模型的方法简单可行。     

On rainflow cycles and the distribution of the number of interval crossings by a Markov chain

This paper treats the number of crossings of an interval by a Markov chain observed for finite time. The exact marginal distribution is derived, in the form of its probability generating function. A numerical example is presented. Interval crossings have an important application in fatigue of materials. For analysing complex fatigue loads, the so-called rainflow cycle counting method is widely used, which is equivalent to counting crossings of intervals.     

确定建筑物刚心的一种动测方法

介绍一种利用实测振动记录确定已有建筑物的刚度中心的方法。该方法以建筑物刚度中心处的平移运动与扭转运动的相干函数最小为依据，分别对建筑物不同位置的运动作对于扭转的相干函数并积分求面积，面积最小所对应的结构坐标即为建筑特的刚度中心。