Steady-State Analysis for a Class of Sliding Mode Controlled Systems Using Describing Function Method

In this paper, the analysis of the steady-state response of the slidingmode control system is presented. The nonlinearity of the switching termin the control law is approximately characterized by using itsequivalent describing function. The parasitic dynamics is modeled as afirst-order lag transfer function, and a possible transport delay isconsidered. Subsequently, a frequency domain method is used for theprediction of limit cycles. The stability-equation method together withthe parameter plane method is proposed to predict graphically limitcycles in the system coefficient plane. Four common types of switchingfunctions are investigated. This analysis further provides an approachof switching control gain selection for suppressing the limit cycle inthe sliding mode.     

机床非线性颤振的描述函数分析

本文采用描述函数方法分析机床的非线性颤振.以描述函数刻划切削过程复杂的非线性特性,并将非线性振动分析所依赖的复平面推广到三维空间进行稳定性分析。从而求出颤振振幅和颤振频率。该法不仅可揭示切削过程等效刚度系数和等效阻尼系数的变化,而且还可考察各切削参数对颤振频率和振幅的影响.结果表明,描述函数方法是分析机床颤振这个极其复杂的非线性系统的一个有效的手段.     

Analytical Stability Bound for a Class of Delayed Fractional-Order Dynamic Systems

Delayed Linear Time-Invariant (LTI) fractional-order dynamic systems areconsidered. The analytical stability bound is obtained by using Lambertfunction. Two examples are presented to illustrate the obtainedanalytical results.     

Dynamics and Control of Initialized Fractional-Order Systems

Due to the importance of historical effects in fractional-order systems,this paper presents a general fractional-order system and control theorythat includes the time-varying initialization response. Previous studieshave not properly accounted for these historical effects. Theinitialization response, along with the forced response, forfractional-order systems is determined. The scalar fractional-orderimpulse response is determined, and is a generalization of theexponential function. Stability properties of fractional-order systemsare presented in the complex w-plane, which is a transformation of thes-plane. Time responses are discussed with respect to pole positions inthe complex w-plane and frequency response behavior is included. Afractional-order vector space representation, which is a generalizationof the state space concept, is presented including the initializationresponse. Control methods for vector representations of initializedfractional-order systems are shown. Finally, the fractional-orderdifferintegral is generalized to continuous order-distributions whichhave the possibility of including all fractional orders in a transferfunction.     

Identification of Weakly Nonlinear Systems Using Describing Function Inversion

In this paper describing functions inversion is used and the restoring force of a nonlinear element in a MDOF system is characterized. The describing functions can be obtained using linearized frequency response functions (FRFs). The response of the system to harmonic excitation forces at distinct frequencies close to the resonant frequency results in linearized FRFs. The nonlinear system can be approximated at each excitation frequency by an equivalent linear system. This approximation leads to calculation of the first-order describing functions. By having the experimental describing functions calculated and the system’s responses corresponding to the nonlinear element (measured or interpolated), nonlinear parameter identification can be performed. Two numerical and experimental case studies are provided to show the applicability of this method.     

Limit Cycle Oscillation and Orbital Stability in Aeroelastic Systems with Torsional Nonlinearity

The paper treats the question of the existence of limit cycleoscillations of prototypical aeroelastic wing sections with structuralnonlinearity using the describing function method. The chosen dynamicmodel describes the nonlinear plunge and pitch motion of a wing. Themodel includes an asymmetric structural nonlinearity in the pitchdegree-of-freedom. The dual-input describing functions of thenonlinearity are derived for the limit cycle analysis. Analyticalexpressions for the average value, and the amplitude and frequency ofoscillation of pitch and plunge responses are obtained. Based on ananalytical approach as well as the Nyquist criterion, stability of thelimit cycles is examined. Numerical results are presented for a set ofvalues of the flow velocities and the locations of the elastic axiswhich show that the predicted limit cycle oscillation amplitude andfrequency as well as the mean value are quite close to the actualvalues. Furthermore, for the chosen model with linear aerodynamics, itis seen that the amplitude of the pitch limit cycle oscillation does notalways increase with the flow velocity for certain elastic axislocations.     

D/A Converters and Iterated Function Systems

This paper discusses A/D and D/A converters as applications of iterated function systems. We describe operation of basic converters in terms of iterate function systems and introduce typical examples for binary and Gray encodings/decodings. We then comment that converters in the field of analog signal processing have different character from iterated function systems in the field of nonlinear dynamical systems. Simple implementation circuits for the binary and Gray D/A converters are also shown.     

An Eigenvalue Method for Calculation of Stability and Limit Cycles in Nonlinear Systems

In the calculation of periodic oscillations of nonlinear systems –so-called limit cycles – approximative and systematic engineeringmethods of linear system analysis are known. The techniques, working inthe frequency domain, perform a quasi-linearization of the nonlinear system,replacing nonlinearities by amplitude-dependent describing functions.Frequently, the resulting equations for the amplitude and frequency ofpresumed limit cycles are solved directly by a graphical procedure in aNyquist plane or by solving the nonlinear equations or a parameteroptimization problem. In this paper, an indirect numerical approach isdescribed which shows that, for a system of nonlinear differentialequations, the eigenvalues of the quasi-linear system simply indicateall limit cycles and, additionally, yield stability regions for thelinearized case. The method is applicable to systems with multiplenonlinearities which may be static or dynamic. It is demonstrated foran example of aircraft nose gear shimmy dynamics in the presence ofdifferent nonlinearities and the results are compared with those fromsimulation.     

An Improved Closure Method for Analysis of Nonlinear Stochastic Systems

The probability density function (PDF) and the mean up-crossing rate (MCR) of the stationary responses of nonlinear stochastic systems excited by white noise is analyzed based on the assumption that the PDF of the responses is approximated with an exponential function of a polynomial in the state variables. Based upon the approximate PDF, a new technique is developed for the approximate PDF solution of Fokker–Planck–Kolmogorov equation, and consequently, the MCR of the system responses is analyzed. Numerical results showed that the approximate PDFs and MCRs approach to the exact ones as the degree of the polynomial increases.     

刚柔耦合约束多体系统的动力分析

本文提出了描述柔性多体系统的牵连坐标系统。该系统由惯性参考系,牵连坐标系,物体坐标系及单元坐标系组成,实现了对刚体平动,刚体转动及弹性运动的连续分解,最大限度地消除了由于刚体大角度转动导致的非线性特性。以有限元法为基础,应用拉格朗日方程建立了在该坐标下的刚柔耦合约束多体系统的动力学控制方程。该方程具有耦合程度小、易于推导、编程及求解等优点,为大规模约束多体系统的动力分析提供了新的途径。本文还讨论了平面铰链约束的约束形式及约束方程,最后给出了一个典型多体系统的数值算例。     

On the Implementation of the Eigenvalue Method for Limit Cycle Determination in Nonlinear Systems

In many practical systems, limit cycles can be predicted with suitable precision by frequency domain methods using describing functions. Within such an approach, limit cycles can be predicted using the “eigenvalue method” [Somieski, G., Nonlinear Dynamics 26(1), 2001, 3–22]. This contribution presents a novel and advantageous implementation of this method, using singular value instead of eigenvalue calculations, and enhancing computational efficiency by avoiding a so called “frequency iteration”. An erratum to this article is available at .     

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.     

Affine Transformation in Random Iterated Function Systems

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Modeling and Dynamic Analysis of Telescopic Systems of Structural Members with Clearance

The dynamic behavior of multi-section constructions with clearance during extending and retracting motion of the sections is analyzed. First, an appropriate physical modeling is introduced before next, the governing boundary value problem is derived by applying Hamilton's principle. Then, a classical discretization procedure is used to generate a coupled system of nonlinear ordinary differential equations as the corresponding truncated mathematical model. Performing appropriate simulations to be verified by results of an alternative software package and partly validated by some preliminary experiments, the vibrational behavior of the suggested type of multi-section telescoping systems can be studied in detail.     

Standing Pulse Solutions in Reaction-Diffusion Systems with Skew-Gradient Structure

The purpose of this paper is to introduce a reaction-diffusion system with skew-gradient structure and discuss the stability of standing pulse solutions. In short, the skew-gradient system is a reaction-diffusion system which resembles a gradient system but has nonlinearities with different sign. We assume the existence of a standing pulse solution and define its orientation in some geometrical manner. Then we show that the stationary solution becomes unstable if time constants satisfy some inequality. The Evans function plays a crucial role for the stability analysis.     

A General Data-Based Approach for Developing Reduced-Order Models of Nonlinear MDOF Systems

A general procedure is presented for analyzing dynamic response measurements from complex multi-degree-of-freedom nonlinear systems incorporating arbitrary types of nonlinear elements. The analysis procedure develops a reduced-order, nonlinear model whose format is convenient for numerical simulation studies. No information about the systems mass properties is needed, and only the applied excitations and corresponding response are needed to develop the model whose dimension is compatible with the number of available sensors. The utility of the approach is demonstrated by means of a three-degree-of-freedom system incorporating polynomial-type nonlinear features with hardening as well as softening characteristics.     

On the systematic analytic-numeric bifurcation analysis

Many technical stituations are adequately described only by means of nonlinear mathematical models. Long-term or steady-state behavior of such systems can have a periodic, quasi-periodic, or chaotic character. Changes of the qualitative behavior are characterized by local and global bifurcations. The aim of this paper is to present a systematic analysis approach for the study of different bifurcation phenomena in nonlinear engineering systems. The applicability of the approach is demonstrated with examples.     

Identification of Fractional Systems Using an Output-Error Technique

An original method for modeling, simulation and identification of fractional systems in the time domain is presented in this article. The basic idea is to model the fractional system by a state-space representation, where conventional integration is replaced by a fractional one with the help of a non-integer integrator. This operator is itself approximated by a N-dimensional system composed of an integrator and of a phase-lead filter. An output-error technique is used in order to estimate the parameters of the model, including the fractional order N. Simulations exhibit the properties of the identification algorithm. Finally, this methodology is applied to the modeling of the dynamics of a real heat transfer system.     

控制存在时滞的线性系统主动控制的滑移模态方法

本文研究控制存在时滞的线性系统的滑模控制方法。在控制时滞量为采样周期的整数倍和非整数倍的两种情况下，通过采用零阶保持器，将包含时滞的连续系统转化为形式上不包含时滞的标准离散线性系统，然后分别进行切换函数和控制律的设计。所得出的切换面和控制律表达式中，除了含有当前的状态反馈外，还包含有前若干步控制项的线形组合。最后对某三自由度结构模型进行仿真计算，验证了所给控制方法的有效性。     

Higher Order Approximation of the Period-energy Function for Single Degree of Freedom Hamiltonian Systems

In 1985 Franz Rothe [J. Reine Angew Math. 355 (1985) 129–138] found, by means of the thermodynamical equilibrium theory, an asymptotic estimate of the period of solutions of ordinary differential equations originated by predator–prey Volterra–Lotka model. We extend some of the Rothe's ideas to more general systems: