Double-mode modeling of chaotic and bifurcation dynamics for a simply supported rectangular plate in large deflection

This paper presents a new approach to characterize the conditions that can possibly lead to chaotic motion for a simply supported large deflection rectangular plate by utilizing the criteria of the fractal dimension and the maximum Lyapunov exponent. The governing partial differential equation of the simply supported rectangular plate is first derived and simplified to a set of two ordinary differential equations by the Galerkin method. Several different features including Fourier spectra, state-space plot, Poinca?e map and bifurcation diagram are then numerically computed by using a double-mode approach. These features are used to characterize the dynamic behavior of the plate subjected to various excitation conditions. Numerical examples are presented to verify the validity of the conditions that lead to chaotic motion and the effectiveness of the proposed modeling approach. The numerical results indicate that large deflection motion of a rectangular plate possesses many bifurcation points, two different chaotic motions and some jump phenomena under various lateral loading. The results of numerical simulation indicate that the computed bifurcation points can lead to either a transcritical bifurcation or a pitchfork bifurcation for the motion of a large deflection rectangular plate. Meanwhile, the points of pitchfork bifurcation can gradually lead to chaotic motion in some specific loading conditions. The modeling result thus obtained by using the method proposed in this paper can be employed to predict the instability induced by the dynamics of a large deflection plate.     

Overview of dynamic modelling and analysis of rolling element bearings with localized and distributed faults

Rolling element bearings (REBs) play a most critical role in various industrial machinery. A clearly and in-depth understanding of vibration characteristics of REBs is very helpful for condition monitoring and diagnosis applications for the machinery. This work presented a comprehensive review of dynamic modelling and analysis methods for predicting the vibration characteristics of REBs with and without localized and distributed faults. Main capabilities and limitations of those methods for both the localized and distributed faults have been described. Explanations for the generations of the bearing vibrations were also reviewed. A summary of the literature was given followed by the recommendations of current and future research works. Moreover, recent challenges, directions and implications in research works on the dynamic modelling and analysis methods of the localized and distributed faults in REBs have also been conducted.     

Experimental and numerical investigation of chaotic regions in the triple physical pendulum

The experimental and numerical analysis of triple physical pendulum is performed. The experimental setup of the triple pendulum with the first body externally excited by the square function and the widely used LabView measure-programming system, which is designed especially for measure data processing and acquisition, are described. The mathematical model of the system is then introduced. The parameters of the model are estimated by minimization of the sum of squares of deviations between the signal from the simulation and the signal from the experiment. A good agreement between results from experiment and from simulation is shown in few examples, including periodic as well as chaotic solutions.     

Pseudo-rigid ball impact on an oscillating rigid foundation

The problem of dissipative normal impact of a linearly elastic pseudo-rigid ball on a sinusoidally oscillating rigid foundation is considered. This problem yields a four-dimensional smooth impact map that produces sticking, regular and chaotic responses. Parameter values that furnish such responses are identified with the aid of a perturbation analysis.     

Dynamics of pseudo-rigid ball impact on rigid foundation

This paper concerns the dynamics induced by the ideally elastic normal impact of a linearly elastic pseudo-rigid sphere on a rigid, stationary foundation. An impact map is derived and studied by numerical and analytical means. Periodic, quasi-periodic, and chaotic response is observed consistently with the symplectic nature of the map.     

Deterministic Behaviour of Short Time Series

We present a new method for detecting a lowdimensional deterministic character of very short discrete time series. The algorithm depends on two parameters, that can be selected according to a simple criterion. Experiments show that the method is sensitive to noise levels as low as 2%. In addition, our technique allows us to estimate the value of the largest Lyapunov exponent.Sommario: In questo articolo si presenta un nuovo metodo per stabilire il carattere deterministico di serie tempodiscrete molto corte. L'algoritmo dipende da due parametri che possono essere selezionati secondo un semplice criterio. Gli esperimenti mostrano che il metodo è sensibile a livelli di rumore del 2%. Inoltre, tale tecnica consente di stimare il valore del massimo esponente di Lyapunov.     

The non-linear phenomena in vibration cutting system: The establishment of dynamic model

It is well-known that vibration cutting is an effective method to cut materials with accuracy. However, it often suffers from unevenness of surface roughness caused by the non-linear vibration phenomena in tool-work vibration system. To solve this problem, we propose a vibration cutting model described by two-dimensional differential equations to analyze the non-linear phenomena. This non-linear model reproduces the basic features of the real vibration cutting systems, that is, the reduction of the cutting force and the displacement of work by pulsing of the cutting force. In this report, our experiment shows that various non-linear phenomena exist in the vibration cutting system. Our non-linear model for the vibration cutting produce behaviors which qualitatively agree with these various non-linear phenomena. These behaviors can be classified by the Lyapunov exponents.     

Lyapunov exponents in discrete modelling of a cantilever beam impacting on a moving base

The dynamic behaviour of a cantilever beam of an unnegligible large mass and with a concentrated mass fixed at its end, which impacts on a movable base according to Hertz's damp law, is studied. A new finite element reference model of the system and its lower-dimensional substitutive models with one degree or two degrees of freedom are developed. The qualitative-type as well as quantitative-type applicability limits of these substitutive models are discussed - the latter ones are described in terms of the corresponding spectra of Lyapunov exponents.     

Effects of tension on vortex-induced vibration (VIV) responses of a long tensioned cylinder in uniform flows

The effects of tension on vortex-induced vibra-tion (VIV) responses for a tension-dominated long cylinder with an aspect ratio of 550 in uniform flows are experimen-tally investigated in this paper. The results show that elevated tension suppresses fluctuations of maximum displacement with respect to flow velocity and makes chaotic VIV more likely to appear. With respect to periodic VIV, if elevated tension is applied, the dominant vibration frequency in the in-line (IL) direction will switch from a fundamental vibra-tion frequency to twice the value of the fundamental vibration frequency, which results in a ratio of the dominant vibration frequency in the IL direction to that in the cross-flow direc-tion of 2.0. The suppression of the elevated tension in the fluctuation of the maximum displacement causes the axial tension to become an active control parameter for the VIV maximum displacement of a tension-dominated long riser or tether of an engineering structure in deep oceans. However, the axial tension must be optimized before being used since the high dominant vibration frequency due to the elevated tension may unfavorably affect the fatigue life of the riser or tether.     

基于离散元-有限元模型的冰激锥体海洋平台结构振动分析

冰荷载是影响海洋平台结构及其上部设备作业安全的重要因素。针对海洋平台与海冰相互作用的动力过程,采用具有粘接-破碎性能的离散元方法(DEM)对海冰的破碎特性进行分析,采用有限元方法(FEM)对海洋平台结构的动力响应进行计算;同时考虑海冰与海洋平台结构的耦合作用,将海冰的离散元方法与海洋平台结构的有限元方法相结合,建立了冰激海洋平台结构振动的DEM-FEM模型,并由此计算海冰作用下海洋平台结构的振动响应以及冰荷载特性。     

基于遗传RBF网络的惯性测量组合模拟电路故障诊断

为解决惯性测量组合模拟电路的诊断不易定位到元件级故障的问题,提出了一种基于遗传RBF网络的智能诊断方法。该方法首先利用RBF神经网络快速准确识别故障的能力,以RBF的训练均方误差为遗传算法的适应度函数,依靠遗传算法强大的全局寻优能力实现故障特征选择。在特征选择的过程中,同时记录使训练均方误差达到最小的最优RBF网络,然后直接利用特征选择过程中训练好的最优RBF网络诊断故障,而无需利用特征选择后的训练数据对RBF网络进行再训练,简化了诊断步骤,同时增强了网络的抗干扰能力。仿真结果表明,该方法能有效去除冗余特征,准确诊断惯性测量组合模拟电路的故障,并有较好的抗噪能力,证明了该方法的有效性和可行性。     

The fractal research and predicating on the times series of sunspot relative number

In this paper, with the theory of nonlinear dynamic systems, It is analyzed that the dynamic behavior and the predictabilityfor the monthly mean variationsof the sunspot relative number recorded from January 1891 to December 1996. Inthe progress, the fractal dimension (D=3. 3±0.2) for the variation process wascomputed. This helped us to determine the embedded dimension [2×D+1]=7.By computing the Lyapunov index (λ₁=0.863), it was indicated that the variationprocess is a chaotic system. The Kolmogorov entropy (K=0.0260) was also computed, which provides, theoretically, the predicable time scale. And at the end, according to the result of the analysis above, an experimental predication is made,whose data was a part cut from the sample data.     

基于小波有限元的车辆-轨道-桥梁系统竖向振动分析

基于小波有限元建立了车辆-轨道-桥梁系统竖向运动方程。将车辆、轨道和桥梁作为一个整体系统,钢轨和桥梁采用区间B样条小波单元离散,钢轨与桥梁之间的钢轨基础采用均布的弹簧和阻尼模拟,采用虚功原理建立了基于小波有限元的四轴车辆-轨道-桥梁竖向振动分析模型。结果表明,采用区间B样条小波单元可较大程度上减小系统的自由度数,缩减计算量,节省计算时间。     

Effect of gravity on the vibration of vertical cantilevers

The free vibration of a vertically-oriented, thin, prismatic cantilever is influenced by weight. That is, the natural frequencies (and to a lesser extent, mode shapes) are affected by the application of a linearly varying axial load. A beam with an “upward” orientation, i.e., with the free end above the clamped end, will experience a de-stiffening effect, up to the point of self-weight buckling (at zero effective stiffness). A beam in a “downward” orientation will be stiffened by the weight of the beam. This technical note describes some simple experiments on very slender strips and their (vertical) orientation and shows a close correlation with theory.     

Explanation of scaling phenomenon based on fractal fragmentation

A significant ‘size effect' is observed in tensile strength of solid particles, such as ice, rock, ceramics and concrete: the tensile strength is not independent of the fragment size, but decreases with increasing size. The Weibull statistical theory was universally used to calculate the size effect observed in solid particles. Recent developments in fractal theory suggest that fractals may provide a more realistic representation of solid particles. In this paper, the scaling phenomenon of ice mechanics is explained using the fractal model for ice particle fragmentation. The Weibull statistics is modified using the fractal crushing of ice, and is compared with the conventional one. Goodness-of-fit statistics show that the modified Weibull statistics fits the experimental data of ice much better than the conventional one. The modified Weibull statistics has only one parameter, the fractal dimension of the fragment size distribution, which has a general value of 2.50 for the ice fragmentations.     

Effects of land use patterns on soil aggregate stability in Sichuan Basin, China

Soil aggregate stability as a key indicator of soil structure, is a product of interactions between soil environment, management practices, and land use patterns. The objective of this study was to analyze the impact of various land use patterns on soil aggregate stability in Sichuan Basin of southwestern China. The dry- and water-stable aggregate size distributions were determined by manual dry sieving procedure and Yoder＇s wet sieving procedure, respectively, while microaggregates and its mechanical and chemical stabilities by Kachisky＇s method, oscillator method, and citrate-dithionate （C-D） reagent method, separately. The results indicated that fractal dimension and surface fractal dimension were useful indicators to reflect soil aggregate distribution. Land use patterns have an obvious influence on soil aggregate stability. In the study area, water stability, mechanical stability, and chemical stability followed the sequence, Barren land 〉 forestland 〉 orchard 〉 cropland, and the original stability and collapse velocity were sensitive to soil properties and soil structure. The difference of aggregate stability under different land use patterns is mainly due to the intensity of human disturbance and cultivation. Improper land use patterns will lead to breakdown of unstable aggregates, producing finer and more-easily transportable particles and microaggregates. In the future, inappropriate cultivation and land use patterns should be changed to protect soil structure, to improve soil aggregate stability and soil fertility in Sichuan Basin.     

A comparative study on algorithms of boundary element solution for free torsional vibration of bodies of revolution

In this paper the functionu=rsin in cylindrical coordinates (r,,z) is introduced into the equation for free torsional vibration of bodies of revolution (where=v / r represents the angle of twist). With the static fundamental solution (–1 /R) a mixed BEM / FEM equation is derived. The domain integral term in the equation is discretized by Serendipity elements instead of commonly used constant value finite elements in the literature. The equation is an algebraic eigenvalue one. The dynamic fundamental solution (e ^1R /R) is also used for deriving the other mixed BEM / FEM equation. An appropriate iterative solution procedure is described. An algebraic eigenvalue equation can be obtained and its solution accuracy is almost interior meshing independent. A number of examples are studied. The results show the good economy and high accuracy of the algorithms proposed.The Project is Supported by National Natural Science Foundation of China.     

Effect of non-homogeneity on natural frequencies of vibration of elliptic plates

This paper presents the effect of non-homogeneity of the material of plate structures on the vibration frequencies. The non-homogeneity of the plate is characterized by taking a variety of combinations of linear as well as quadratic variations in the Young’s modulus and density of the material. Boundary characteristic orthogonal polynomials using Gram-Schmidt procedure have been employed in the Rayleigh Ritz method. The results have been included for the first few frequencies of the plate element for all the boundary conditions viz. clamped, simply supported and free. Related Tables and graphs are incorporated to show the effect of the non-homogeneity parameter on the frequencies of the vibration of elliptic and circular plates.     

Effect of the geometry on the non-linear vibration of circular cylindrical shells

The non-linear vibration of simply supported, circular cylindrical shells is analysed. Geometric non-linearities due to finite-amplitude shell motion are considered by using Donnell's non-linear shallow-shell theory; the effect of viscous structural damping is taken into account. A discretization method based on a series expansion of an unlimited number of linear modes, including axisymmetric and asymmetric modes, following the Galerkin procedure, is developed. Both driven and companion modes are included, allowing for travelling-wave response of the shell. Axisymmetric modes are included because they are essential in simulating the inward mean deflection of the oscillation with respect to the equilibrium position. The fundamental role of the axisymmetric modes is confirmed and the role of higher order asymmetric modes is clarified in order to obtain the correct character of the circular cylindrical shell non-linearity. The effect of the geometric shell characteristics, i.e., radius, length and thickness, on the non-linear behaviour is analysed: very short or thick shells display a hardening non-linearity; conversely, a softening type non-linearity is found in a wide range of shell geometries.     

Free vibration characteristics of sectioned unidirectional/bidirectional functionally graded material cantilever beams based on finite element analysis

Advancements in manufacturing technology, including the rapid development of additive manufacturing (AM), allow the fabrication of complex functionally graded material (FGM) sectioned beams. Portions of these beams may be made from different materials with possibly different gradients of material properties. The present work proposes models to investigate the free vibration of FGM sectioned beams based on onedimensional (1D) finite element analysis. For this purpose, a sample beam is divided into discrete elements, and the total energy stored in each element during vibration is computed by considering either the Timoshenko or Euler-Bernoulli beam theory. Then, Hamilton's principle is used to derive the equations of motion for the beam. The effects of material properties and dimensions of FGM sections on the beam's natural frequencies and their corresponding mode shapes are then investigated based on a dynamic Timoshenko model (TM). The presented model is validated by comparison with three-dimensional (3D) finite element simulations of the first three mode shapes of the beam.