Nonlinear vibration phenomenon of an aircraft rub-impact rotor system due to hovering flight

This paper focuses on the nonlinear vibration phenomenon caused by aircraft hovering flight in a rub-impact rotor system supported by two general supports with cubic stiffness. The effect of aircraft hovering flight on the rotor system is considered as a maneuver load to formulate the equations of motion, which might result in periodic response instability to the rotor system even the eccentricity is small. The dynamic responses of the system under maneuver load are presented by bifurcation diagrams and the corresponding Lyapunov exponent spectrums. Numerical analyses are carried out to detect the periodic, sub-harmonic and quasi-periodic motions of the system, which are presented by orbit diagrams, phase trajectories, Poincare maps and amplitude power spectrums. The results obtained in this paper will contribute an understanding of the nonlinear dynamic behaviors of aircraft rotor systems in maneuvering flight.     

A new nonlinear dynamic analysis method of rotor system supported by oil-film journal bearings

The strong nonlinear behavior usually exists in rotor systems supported by oil-film journal bearings. In this paper, the partial derivative method is extended to the second-order approximate extent to predict the nonlinear dynamic stiffness and damping coefficients of finite-long journal bearings. And the nonlinear oil-film forces approximately represented by dynamic coefficients are used to analyze nonlinear dynamic performance of a symmetrical flexible rotor-bearing system via the journal orbit, phase portrait and Poincaré map. The effects of mass eccentricity on dynamic behaviors of rotor system are mainly investigated. Moreover, the computational method of nonlinear dynamic coefficients of infinite-short bearing is presented. The nonlinear oil-film forces model of finite-long bearing is validated by comparing the numerical results with those obtained by an infinite-short bearing-rotor system model. The results show that the representation method of nonlinear oil-film forces by dynamic coefficients has universal applicability and allows one easily to conduct the nonlinear dynamic analysis of rotor systems.     

非线性系统动力分析的模态综合技术

各种模态综合方法已广泛应用于线性结构的动力分析,但是,一般都不适用于非线性系统. 本文基于[20][21]提出的方法,将一种模态综合技术推广到非线性系统的动力分析.该法应用于具有连接件耦合的复杂结构系统,以往把连接件简化为线性弹簧和阻尼器.事实上,这些连接件通常具有非线性弹性和非线性阻尼特性.例如,分段线性弹簧、软特性或硬特性弹簧、库伦阻尼、弹塑性滞后阻尼等.但就各部件而言,仍属线性系统.可以通过计算或试验或兼由两者得到一组各部件的独立的自由界面主模态信息,且只保留低阶主模态.通过连接件的非线性耦合力,集合各部件运动方程而建立成总体的非线性振动方程.这样问题就成为缩减了自由度的非线性求解方程,可以达到节省计算机的存贮和运行时间的目的.对于阶次很高的非线性系统,若能缩减足够的自由度,那么问题就可在普通的计算机上得以解决. 由于一般多自由度非线性振动系统的复杂性,一般而言,这种非线性方程很难找到精确解.因此,对于任意激励下系统的瞬态响应,可以采用数值计算方法求解缩减的非线性方程.     

Using a Volterra system model to analyze nonlinear response in video-packet transmission over IP networks

This paper presents a Volterra system-based nonlinear analysis of video-packet transmission over IP networks. With the Volterra system, which is applicable to the modeling of nonlinear dynamic systems from sets of input and output data, we applied a time-series analysis of measured data for network response evaluation. In a test-bed connected to the Internet, we measured two parameters: the time intervals between consecutive packets from a video server at the originating side, and the transmission time of packets between originating and terminating sides. We used these as input and output data for the Volterra system and confirmed that the relative error of this model changed with conditions of network systems, which suggested that the packet transmission process affected the degree of nonlinearity of the system. The proposed method can reproduce the time-series responses observed in video-packet transmission over the Internet, reflecting nonlinear dynamic behaviors such that the obtained results provided us with an effective depiction of network conditions at different times.     

Identification of nonlinear dynamics using a general spatio-temporal network

The so-called spatio-temporal neural network is considered. This is a neural network where the conventional weight multiplication operation is replaced by a linear filtering operation. General learning algorithms are derived for such a network, both in the discrete-time and in the continuous-time domains. The problem of deterministic nonlinear system identification is considered as an application of spatio-temporal neural networks. Nonlinear system identification is one of the challenging problems in the field of dynamic systems, with limited successful results using conventional methods. Neural network approaches have so far been encouraging, but further exploration is needed. The capabilities of the derived algorithms and of the considered architectures to effectively identify deterministic nonlinear systems is demonstrated through examples.     

非线性系统运动稳定性的一种判据

对于自治的非线性系统来说,只要其线性部分系数矩阵的特征值不属于临界情形,其无扰运动在其足够小的邻域内的稳定性完全可以由其线性部分的特征值确定.关于线性系统的稳定性,已有不少简单易行的判别方法,而关于非线性系统的稳定性,很多数学家和力学家作了大量的研究工作;但大都是针对特殊类型的非线性系统解决了一些问题,直到现在为止,还没有普遍适用于任何的非线性系统的简单易行的判别方法.本文所给的是判别非线性系统稳定性的充要条件,常用的克拉索夫斯基方法只是这一方法的一个特例[1],[2].     

Stability properties of autonomous homogeneous polynomial differential systems

A geometrical approach is used to derive a generalized characteristic value problem for dynamic systems described by homogeneous polynomials. It is shown that a nonlinear homogeneous polynomial system possesses eigenvectors and eigenvalues, quantities normally associated with a linear system. These quantities are then employed in studying stability properties. The necessary and sufficient conditions for all forms of stabilities characteristic of a two-dimensional system are provided. This result, together with the classical theorem of Frommer, completes a stability analysis for a two-dimensional homogeneous polynomial system.     

An improved efficiency of fuzzy logic control of PMBLDC for PV pumping system

This paper presents an analysis by which the dynamic performances of a permanent magnet brushless DC (PMBLDC) motor is controlled through a hysteresis current loop and an outer speed loop with different controllers. The dynamics of the photovoltaic pumping drive system with (PI) and a fuzzy logic (FL) speed controllers are presented. In order to optimize the overall system efficiency, a maximum power point tracker is also used. Simulation is carried out by formatting the mathematical model for photovoltaic source, MPPT, motor and pump load. The results for such complicated and nonlinear system, with FL speed controller show improvement in transient response of PMBLDC drive over conventional PI. The effectiveness of the FL controller is also demonstrated.     

Design of a nonsingular adaptive fuzzy backstepping controller for electrostatically actuated microplates

This paper adopts some alternative strategies to design a nonlinear controller for double electrostatically actuated microplates. The novel design is carried out to solve the singularity problem reported in many articles due to the use of the Taylor expansion to simplify the electrostatic force. The nonlinear governing partial differential equation is converted to the modal equation using the Galerkin method. Then, based on the Lyapunov stability criterion, a fuzzy backstepping controller facilitated by prescribed performance functions is applied to the non-affine system to extend the travel range beyond the pull-in region and capture the structural and nonstructural uncertainties that exist in the practical systems. The present work also aims to bring satisfactory transient and steady-state performance indices to the system. Moreover, unknown time-varying delays as the indispensable part of practical systems are considered in the proposed control scheme to suppress the delays occurring in the measurement of the states by constructing Lyapunov–Krasovskii function. The accuracy of the modal equation in both the static and dynamic analysis is verified through a meshless method as a direct solution of the partial differential equation. The proposed controller guarantees that all the closed-loop signals are semi-globally, uniformly ultimately bounded, and the error evolves within the decaying prescribed bounds. Finally, the proposed controller demonstrates its feasibility to extend the travel range within and beyond the pull-in range despite the unknown uncertainties and time-varying delays which exist in the system.     

非线性转子系统稳定性量化分析方法

转子轴承系统是一类多自由度非线性动力系统,广泛应用于工程实际．设计观念和维修体制的变革提出了稳定性量化分析的要求．本文利用轨线保稳降维方法提出了转子系统稳定性的量化分析方法．首先,对高维非线性非自治转子系统进行数值积分,将n维空间的轨线映射为一系列一维的映象轨线,并将各自由度的运动方程中除该自由度外的所有状态变量用积分结果代换,得到n个互相解耦,含有多个时变参数的单自由度方程．然后,在一维观察空间的外力位移扩展相平面上定义了动态中心点,研究转子系统中常见的几种运动的动态中心点动能差序列的特点,给出了上述典型运动形式的轨线稳定裕度的定量评估指标,应用灵敏度分析技术快速有效地预测周期运动的倍周期分岔点和Hopf分岔点．以一个具有非线性支承的滑动轴承柔性转子模型为例,证明了该方法的有效性．     

A Class of Learning/Estimation Algorithms Using Nominal Values: Asymptotic Analysis and Applications

A class of estimation/learning algorithms using stochastic approximation in conjunction with two kernel functions is developed. This algorithm is recursive in form and uses known nominal values and other observed quantities. Its convergence analysis is carried out; the rate of convergence is also evaluated. Applications to a nonlinear chemical engineering system are examined through simulation study. The estimates obtained will be useful in process operation and control, and in on-line monitoring and fault detection.     

Structural design and analysis for a sensor-integrated ball bearing

The structural design issues associated with the development of a sensor-integrated ball bearing were investigated. A finite element model was established to analyze the structural integrity of a ball bearing whose outer ring was modified to accommodate a miniaturized load sensor module. The sensor module was designed to monitor dynamic load variations within the bearing, which is an important indicator of a bearings safe operation. This structural integration of “intelligence” is significant in that it bring's automated, on-line, self-diagnosis capability to a dynamic system for which reliable, safe operation is of critical importance. A design tool was developed to evaluate the effect of outer ring modification on the dynamic load-carrying capability of the bearing. The tool also provided quantitative guidelines for the design of load and other types of sensor modules that will be structurally embedded into the outer ring of a conventional bearing for industrial applications.     

A nonlinear iteration method for solving a two-dimensional nonlinear coupled system of parabolic and hyperbolic equations

A nonlinear iteration method for solving a class of two-dimensional nonlinear coupled systems of parabolic and hyperbolic equations is studied. A simple iterative finite difference scheme is designed; the calculation complexity is reduced by decoupling the nonlinear system, and the precision is assured by timely evaluation updating. A strict theoretical analysis is carried out as regards the convergence and approximation properties of the iterative scheme, and the related stability and approximation properties of the nonlinear fully implicit finite difference (FIFD) scheme. The iterative algorithm has a linear constringent ratio; its solution gives a second-order spatial approximation and first-order temporal approximation to the real solution. The corresponding nonlinear FIFD scheme is stable and gives the same order of approximation. Numerical tests verify the results of the theoretical analysis. The discrete functional analysis and inductive hypothesis reasoning techniques used in this paper are helpful for overcoming difficulties arising from the nonlinearity and coupling and lead to a related theoretical analysis for nonlinear FI schemes.     

Linear and nonlinear dynamics of a circular cylindrical shell connected to a rigid disk

The dynamics of a circular cylindrical shell carrying a rigid disk on the top and clamped at the base is investigated. The Sanders–Koiter theory is considered to develop a nonlinear analytical model for moderately large shell vibration. A reduced order dynamical system is obtained using Lagrange equations: radial and in-plane displacement fields are expanded by using trial functions that respect the geometric boundary conditions.The theoretical model is compared with experiments and with a finite element model developed with commercial software: comparisons are carried out on linear dynamics.The dynamic stability of the system is studied, when a periodic vertical motion of the base is imposed. Both a perturbation approach and a direct numerical technique are used. The perturbation method allows to obtain instability boundaries by means of elementary formulae; the numerical approach allows to perform a complete analysis of the linear and nonlinear response.     

Numerical Solution of A Class of Systems of Volterra Polynomial Equations of the First Kind

In this paper, a class of second-order systems of Volterra nonlinear integral equations is considered. This class is related to a problem of automatic control of a dynamic object with vector inputs and outputs. A numerical solution technique based on the Newton–Kantorovich method is considered. To verify the efficiency of the algorithms developed, a series of test calculations are carried out.     

Synchronization of fractional order chaotic systems using active control method

In this article, the active control method is used for synchronization of two different pairs of fractional order systems with Lotka–Volterra chaotic system as the master system and the other two fractional order chaotic systems, viz., Newton–Leipnik and Lorenz systems as slave systems separately. The fractional derivative is described in Caputo sense. Numerical simulation results which are carried out using Adams–Bashforth–Moulton method show that the method is easy to implement and reliable for synchronizing the two nonlinear fractional order chaotic systems while it also allows both the systems to remain in chaotic states. A salient feature of this analysis is the revelation that the time for synchronization increases when the system-pair approaches the integer order from fractional order for Lotka–Volterra and Newton–Leipnik systems while it reduces for the other concerned pair.     

An analysis of heart rhythm dynamics using a three-coupled oscillator model

Rhythmic phenomena represent one of the most striking manifestations of the dynamic behavior in biological systems. Understanding the mechanisms responsible for biological rhythms is crucial for the comprehension of the dynamics of life. Natural rhythms could be either regular or irregular over time and space. Each kind of dynamical behavior may be related to both normal and pathological physiological functioning. The cardiac conducting system can be treated as a network of self-excitatory elements and, since these elements exhibit oscillatory behavior, they can be modeled as nonlinear oscillators. This paper proposes a mathematical model to describe heart rhythms considering three modified Van der Pol oscillators connected with time delay couplings. Therefore, the heart dynamics is represented by a system of differential difference equations. Numerical simulations are carried out presenting qualitative agreement with the general heart rhythm behavior. Normal and pathological rhythms represented by the ECG signals are reproduced. Pathological rhythms are generated by either the coupling alterations that represents communications aspects in the heart electric system or forcing excitation representing external pacemaker excitation.     

Visualization of four limit cycles of two-dimensional quadratic systems in the parameter space

This paper deals with the visualization of a domain that contains four limit cycles for quadratic dynamical systems of first-order differential equations with real coefficients. The visualization of the domain is carried out in the three-dimensional space of coefficients corresponding to the nonlinear part of the quadratic system. Theoretical and practical aspects of the numerical solution of the Cauchy problem for unstable systems are discussed.     

Suppressing grazing chaos in impacting system by structural nonlinearity

In this note we investigate the influence of structural nonlinearity of a simple cantilever beam impacting system on its dynamic responses close to grazing incidence by a means of numerical simulation. To obtain a clear picture of this effect we considered two systems exhibiting impacting motion, where the primary stiffness is either linear (piecewise linear system) or nonlinear (piecewise nonlinear system). Two systems were studied by constructing bifurcation diagrams, basins of attractions, Lyapunov exponents and parameter plots. In our analysis we focused on the grazing transitions from no impact to impact motion. We observed that the dynamic responses of these two similar systems are qualitatively different around the grazing transitions. For the piecewise linear system, we identified on the parameter space a considerable region with chaotic behaviour, while for the piecewise nonlinear system we found just periodic attractors. We postulate that the structural nonlinearity of the cantilever impacting beam suppresses chaos near grazing.     

Fuzzy guaranteed cost controller for trajectory tracking in nonlinear systems

In this paper, we propose a fuzzy logic based guaranteed cost controller for trajectory tracking in nonlinear systems. Takagi–Sugeno (T–S) fuzzy model is used to represent the dynamics of a nonlinear system and the controller design is carried out using this fuzzy model. State feedback law is used for building the fuzzy controller whose performance is evaluated using a quadratic cost function. For designing the fuzzy logic based controller which satisfies guaranteed performance, linear matrix inequality (LMI) approach is used. Sufficient conditions are derived in terms of matrix inequalities for minimizing the performance function of the controller. The performance function minimization problem with polynomial matrix inequalities is then transformed into a problem of minimizing a convex performance function involving standard LMIs. This minimization problem can be solved easily and efficiently using the LMI optimization techniques. Our controller design method also ensures that the closed-loop system is asymptotically stable. Simulation study is carried out on a two-link robotic manipulator tracking a reference trajectory. From the results of the simulation study, it is observed that our proposed controller tracks the reference trajectory closely while maintaining a guaranteed minimum cost.