Detecting nonlinearity from a continuous dynamic system based on the delay vector variance method and its application to gear fault identification

Many natural time series and signals collected from engineered systems are continuous dynamical signals. In practice, it is necessary to study nonlinearities in such continuous dynamic systems under different sampling conditions. In this paper, nonlinearity tests based on the delay vector variance (DVV) and iterative amplitude adjusted Fourier transform (IAAFT) based surrogates are first applied to Lorenz time series under various sampling rates. Traditional nonlinearity measures, such as third-order autocovariance and asymmetry due to time reversal, are shown to be sensitive to different sampling conditions whereas the relative insensitivity of the DVV proves promising. This insensitivity makes DVV a desirable nonlinearity measure for describing continuous dynamic signals and herein it is shown to be a suitable nonlinear fault diagnostic method for use in the gearbox condition monitoring. Moreover, by applying the DVV rank test, faults in gears can be identified effectively.     

Identifying ill-behaved nonlinear processes without metrics: use of symbolic dynamics

Given ill-behaved psychological data that are unlikely to satisfy metric axioms, the use of encoding in symbolic dynamics, and hence leading into Markov analyses, is explored. Various measures of entropy are calculated. The tractability of entropic measures for categorizing the trajectories of nonlinear dynamics that may be present and chaotic is considered, with a focus on the case where there are two attractors and at least one heteroclinic orbit between them. Fast/slow dynamics are treated as a special case. The problem of identification is in other contexts the problem of diagnosis in time-varying pathologies. Some real data, selected for their psychological relevance in clinical, forensic and psychophysical processes, that are apparently edge-of chaos and nonstationary, are for comparison analysed both as metric and discrete and in symbolic encoding.     

Describing Function Analysis of Systems with Impacts and Backlash

This paper analyzes the dynamical properties of systems with backlashand impact phenomena based on the describing function method. It isshown that this type of nonlinearity can be analyzed in the perspectiveof the fractional calculus theory. The fractional-order dynamics isillustrated using the Nyquist plot and the results are compared withthose of standard models.     

Nonlinear delay difference equations for housing dynamics assuming heterogeneous backward-looking expectations

China's first interest rate hike during the last decade, aiming to cool down the seemingly overheated real estate market, had aroused more caution on housing market. This paper aims to analyze the housing price dynamics after an unanticipated economic shock, which was believed to have similar properties with the back ward- looking expectation models. The analysis of the housing price dynamics is based on the cobweb model with a simple user cost affected demand and a stock-flow supply assumption. Several nth-order delay rational difference equations are set up to illustrate the properties of housing dynamics phenomena, such as the equilibrium or oscillations, overshoot or undershoot and convergent or divergent, for a kind of heterogeneous backward-looking expectation models. The results show that demand elasticity is less than supply elasticity is not a necessary condition for the occurrence of oscillation. The housing price dynamics will vary substantially with the heterogeneous backward-looking expectation assumption and some other endogenous factors.     

A heterogeneous boundedly rational expectation model for housing market

This research aims to test the housing price dynamics when considering heterogeneous boundedly rational expectations such as naive expectation, adaptive expectation and biased belief. The housing market is investigated as an evolutionary system with heterogeneous and competing expectations. The results show that the dynamics of the expected housing price varies substantially when heterogeneous expectations are considered together with some other endogenous factors. Simulation results explain some stylized phenomena such as equilibrium or oscillation, convergence or divergence, and over-shooting or under-shooting. Furthermore, the results suggest that variation of the proportion of groups of agents is basically dependent on the selected strategies. It also indicates that control policies should be chosen carefully in consistence with a unique real estate market during a unique period since certain parameter portfolio may increase or suppress oscillation.     

A novel construction of chaotic dynamics base gliders in diffusion rule B2/S7

The dynamical behaviors of gliders (mobile localizations) in diffusion rule B2/S7 are quantitatively analyzed from the theory of symbolic dynamics in two-dimensional symbolic sequence space. Their intrinsic complexity is demonstrated by exploiting the relationship between one-dimensional and two-dimensional subshifts. Based on this rigorous approach and technique, the underlying chaos of the extant gliders and their combinations is characterized in a subtle way. It is demonstrated that they can be identified to distinct subsystems with very rich and complicated dynamics; that is, diffusion rule is topologically mixing and possesses positive topological entropy on each subsystem. This analytical assertion provides the fact that diffusion rule is covered with complex subsystems “almost everywhere”. Finally, it is worth mentioning that the procedure proposed in this paper is also applicable to all other gliders arising from the two-dimensional cellular automata therein. It is an extended discovery in both cellular automata and chaos theory.     

Describing function of two masses with backlash

This paper analyzes the dynamical properties of systems with backlash and impact phenomena based on the describing function method. It is shown that this type of nonlinearity can be analyzed in the perspective of the fractional calculus theory. The fractional dynamics is compared with that of standard models.     

Propagation properties of optical soliton in an erbium-doped tapered parabolic index nonlinear fiber: soliton control

In this paper, with the aid of symbolic computation, we investigate the generalized nonlinear Schrödinger Maxwell–Bloch equation, which describes the propagation of the optical soliton through an inhomogeneous two-level dielectric tapered fiber medium. By virtue of the Darboux transformation method, two-soliton solutions are generated based on the constructed Lax pair and figures are plotted to illustrate the properties of the obtained solutions. Moreover, through manipulating the dispersion and nonlinearity profiles, various soliton control systems are investigated which is promising for potential applications in the design of soliton compressor, soliton amplification and high-speed optical devices in ultralarge capacity transmission systems. This means that we are able to control the soliton types with suitably selected values of the parameters. Additionally more soliton control techniques are proposed and investigated. We expect that the above analysis could be observed in future experiments.     

Dynamic response of slacked single-walled carbon nanotube resonators

This paper presents an investigation of the dynamics of electrically actuated single-walled carbon nanotube (CNT) resonators including the effect of their initial curvature due to fabrication (slack). A nonlinear shallow arch model is utilized. A perturbation method, the method of multiple scales, is used to obtain analytically the forced vibration response due to DC and small AC loads for various slacked CNTs of higher and lower aspect ratio. Results of the perturbation method are verified with those obtained by numerically integrating the equations of a multi-mode reduced-order model based on the Galerkin procedure. The effective nonlinearity of the CNT is calculated as a function of the slack level and the DC load. To handle computational problems associated with CNTs of small radiuses, results based on a nonlinear cable model are also demonstrated. The results have indicated that the quadratic nonlinearity due to slack has dominant effect on the dynamic behavior of the CNT.     

A symbolic algorithm for the automatic computation of multitone-input harmonic balance equations for nonlinear systems

A symbolic algorithm is developed for the automatic generation of harmonic balance equations for multitone input for a class of nonlinear differential systems with polynomial nonlinearities. Generalized expressions are derived for the construction of balance equations for a defined multitone signal form. Procedures are described for determining combinations for a given output frequency from the desired set obtained from box truncated spectra and their permutations to automate symbolic algorithm. An application of method is demonstrated using the well-known Duffing–Van der Pol equation. Then the obtained analytical results are compared with numerical simulations to show the accuracy of the approach. The computation times for both the numerical solutions of equations versus the number of frequency components and the symbolic generation of the equations versus the power of nonlinearity are also investigated.     

Critical experiments used to determine flow relations for rate-dependent metal deformation

This article addresses issues in the experimental evaluation of flow relations for rate-dependent metal deformation using 1145 commercially pure aluminum as the test material. Both testing procedures and test interpretation are critically examined, and many current approaches to both these issues are shown to be inadequate. Experimentally, the main difficulty with current work is a lack of appreciation of internal test machine dynamics and how they influence observed material response. This can produce spurious transient phenomena and completely invalid data in some cases. This problem is solved in this work, and very accurate measurements of intrinsic stress and strain rate histories are presented. The issue of test interpretation centers on the evaluation of flow equations and material rate sensitivity from so-called constant structure tests. Relaxation tests, rate increments, and rate decrements are studied in this work. Rate decrements are shown to best capture all thermally activated flow phenomena in 1145, but may well contain some contamination from anelastic effects. Relaxation tests are problematic on account of contamination from machine dynamics, and rate increment tests show some dependence on stress rate, which excludes their use in flow equation determination. A new and extremely general test interpretation scheme is proposed that assumes certain functional dependencies for the flow relation, and not specific functional forms. Power law forms are shown to be inadequate, and using rate decrements, a methodology is suggested for measuring the true material rate-dependence.     

Simple Testing System for Pure Bending Tests with Large Deflections

The potential uses of new materials in structural elements are growing constantly. The focus is on the use of composites with high strengths and strains [1], among other properties. This means that their mechanical properties and failure modes need to be described. Bending tests are widely used for these purposes due to their advantages compared to other types of tests. In particular, pure bending tests are very useful when calculating mechanical parameters and failure modes. These tests do not superimpose different types of loading on the specimens and do not introduce shear stresses which often make the test difficult to carry out properly (for example, in sandwich beams that exhibit great changes in curvature and therefore have a small core thickness). Nonetheless, if long specimens with large allowable strains or curved specimens need to be tested, then traditional bending test methods may fail to apply; making the displacement of support points and load application necessary. This work proposes a pure bending test method that overcomes these disadvantages. It is based on a pulley system which, in addition to introducing the necessary rotations, allows the reduction of the distance between test specimen supports and guarantees pure bending. After the corresponding kinematic study that shows the foregoing, the tests carried out on a test specimen yield uniform characteristics known to confirm the validity of the design.     

Topological chaos of universal elementary cellular automata?rule

The dynamical behaviors of elementary cellular automata rule 110 are analyzed from the viewpoint of symbolic dynamics in the space of bi-infinite symbolic sequences. This paper conducts a rigorous analysis of the relationship between rules 110, 170 and 240 by applying blocking transformation and releasing transformation. Based on this result, the topological chaos of T ₁₁₀ induced by rule 110 is evaluated; that is, $T_{110}^{9}$ and $T_{110}^{16}$ are topologically mixing and possess the positive topological entropies on their respective subsystems. Therefore, it is natural to argue that the intrinsic complexity of rule 110 is high according to the usual measure of complexity organized around the symbolic dynamics of stationary symbol sequences. Finally, it is worth mentioning that the method presented in this paper is also applicable to other blocking transformation equivalences therein.     

Chaotic behaviour of the general symbolic dynamics

This paper extends symbolic dynamics to general cases. Some chaotic properties and applications of the general symbolic dynamics (∑(X),σ) and its special cases are discussed, where X is a separable metric space.     

Nonlinear flutter behavior of a plate with motion constraints in subsonic flow

This paper is aimed at presenting the nonlinear flutter peculiarities of a cantilevered plate with motion-limiting constraints in subsonic flow. A non-smooth free-play structural nonlinearity is considered to model the motion constraints. The governing nonlinear partial differential equation is discretized in space and time domains by using the Galerkin method. The equilibrium points and their stabilities are presented based on qualitative analysis and numerical studies. The system loses its stability by flutter and undergoes the limit cycle oscillations (LCOs) due to the nonlinearity. A heuristic analysis scheme based on the equivalent linearization method is applied to theoretical analysis of the LCOs. The Hopf and two-multiple semi-stable limit cycle bifurcation bifurcations are supercritical or subcritical, which is dependent on the location of the motion constraints. For some special cases the bifurcations are, interestingly, both supercritical and subcritical. The influence of varying parameters on the dynamics is discussed in detail. The results predicted by the analysis scheme are in good agreement with the numerical ones.     

Solitary-wave solutions of the Klein-Gordon equation with quintic nonlinearity

In this paper, the (G′/G)-expansion method is used to obtain exact solitary-wave and periodic-wave solutions for nonlinear evolution equations arising in mathematical physics with the aid of symbolic computations, namely, the Klein-Gordon equation with quintic nonlinearity. Our work is motivated by the fact that the (G′/G)-expansion method provides not only more general forms of solutions, but also periodic and solitary waves. As a result, hyperbolic function solutions and trigonometric function solutions with parameters are obtained. The method is straightforward and concise, and its application is promising for other nonlinear evolution equations in mathematical physics.     

In-plane and out-of-plane dynamics of a curved pipe conveying pulsating fluid

This paper investigates the in-plane and out-of-plane dynamics of a curved pipe conveying fluid. Considering the extensibility, von Karman nonlinearity, and pulsating flow, the governing equations are derived by the Newtonian method. First, according to the modified inextensible theory, only the out-of-plane vibration is investigated based on a Galerkin method for discretizing the partial differential equations. The instability regions of combination parametric resonance and principal parametric resonance are determined by using the method of multiple scales (MMS). Parametric studies are also performed. Then the differential quadrature method (DQM) is adopted to discretize the complete pipe model and the nonlinear dynamic equations are carried out numerically with a fourth-order Runge–Kutta technique. The nonlinear dynamic responses are presented to validate the out-of-plane instability analysis and to demonstrate the influence of von Karman geometric nonlinearity. Further, some numerical results obtained in this work are compared with previous experimental results, showing the validity of the theoretical model developed in this paper.     

Nonlinear dynamics of imperfectly-supported pipes conveying fluid

In this paper, the nonlinear dynamics of a pipe imperfectly supported at the upstream end and free at the other and conveying fluid is investigated. The imperfect support is modelled via cubic translational and rotational springs. The equation of motion is obtained via Hamilton’s principle for an open system, and the Galerkin method is used for discretizing the resulting partial differential equation. The dynamics of a system with either strong rotational or strong translational stiffness is examined in details. Numerical results show that similarly to a cantilevered pipe, the system undergoes a supercritical Hopf bifurcation leading to period-1 limit cycle oscillations. The Hopf bifurcation may, however, occur at a much lower flow velocity compared to the perfect system. At higher flow velocities, quasi-periodic and chaotic-like motions may be observed. The amplitude of transverse displacement is generally much higher than that for a cantilevered pipe, mainly due to large-amplitude rigid-body motion. In addition, effects of the mass ratio, internal dissipation, hardening- or softening-type nonlinearity, as well as concentrated- or distributed-type nonlinearity on the dynamics of the system are examined.     

基于ED-LSTM的智能汽车神经网络横向动力学建模与控制

车辆动力学建模过程中通常会进行简化和假设, 导致模型在某些工况下无法准确反映车辆的实际动态特性, 影响控制精度甚至安全性. 鉴于此, 该文提出了一种基于数据驱动的非线性建模与控制方法, 建立了新型神经网络车辆横向动力学多步预测模型, 实现了智能汽车对参考轨迹的跟踪控制. 首先, 在分析车辆单轨模型并考虑轮胎非线性和纵向负载转移的基础上, 基于编码器?解码器结构设计神经网络横向动力学模型. 其中, 使用串行排列来扩展微分方程描述不完全的动力学信息, 隐藏层神经元学习车辆的高度非线性和强耦合特性, 进而提高模型全局计算精度. 利用所构建的数据集进行模型训练和测试, 结果表明, 相比于物理模型, 所提出的模型在不同路面附着系数条件下均具有更高的建模精度, 具有隐式预测路面摩擦条件能力. 其次, 利用提出的模型设计轨迹跟踪控制算法, 根据车辆稳态转向假设, 计算所需的前轮转向角和稳态质心侧偏角, 将稳态质心侧偏角纳入基于路径误差的转向反馈中, 实现参考轨迹跟踪控制. 最后, 使用CarSim/Simulink联合仿真及HIL实验测试进行不同工况试验的对比分析, 对所提出的基于神经网络模型的控制算法进行评价, 结果表明, 该模型能够实现智能汽车在高速下精确的跟踪控制效果, 并具有良好的横向稳定性.      

Chaos in a coupled oscillators system with widely spaced frequencies and energy-preserving non-linearity

This paper is a sequel to Tuwankotta [Widely separated frequencies in coupled oscillators with energy-preserving nonlinearity, Physica D 182 (2003) 125-149.], where a system of coupled oscillators with widely separated frequencies and energy-preserving quadratic non-linearity is studied. We analyze the system for a different set of parameter values compared with those in Tuwankotta [Widely separated frequencies in coupled oscillators with energy-preserving nonlinearity, Physica D 182 (2003) 125-149.]. In this set of parameters, the manifold of equilibria are non-compact. This turns out to have an interesting consequence to the dynamics. Numerically, we found interesting bifurcations and dynamics such as torus (Neimark-Sacker) bifurcation, chaos and heteroclinic-like behavior. The heteroclinic-like behavior is of particular interest since it is related to the regime behavior of the atmospheric flow which motivates the analysis in Tuwankotta [Widely separated frequencies in coupled oscillators with energy-preserving nonlinearity, Physica D 182 (2003) 125-149.] and this paper.