A Frequency-Domain Approach to Optimal Fractional-Order Damping

In this paper, we will consider the single term optimal fractional-order damper for an otherwise undamped oscillator. First, we will find the single term damper that minimizes the time domain integral of the squared step error (2-norm) and the integral of the time-weighted squared error (Hilbert–Schmidt–Hankel norm). Next we will consider a more intuitive frequency domain approach that insures the maximally flat magnitude response. Time and frequency domain plots are given for comparison with the integer-order solutions. Further improvements in performance are shown to be possible using multiple active fractional-order dampers.     

A modified fractional-order thermo-viscoelastic model and its application to a polymer micro-rod heated by a moving heat source

Classical thermo-viscoelastic models may be challenged to predict the precise thermo-mechanical behavior of viscoelastic materials without considering the memorydependent effect. Meanwhile, with the miniaturization of devices, the size-dependent effect on elastic deformation is becoming more and more important. To capture the memory-dependent effect and the size-dependent effect, the present study aims at developing a modified fractional-order thermo-viscoelastic coupling model at the microscale...     

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.     

基于LMI的框架结构分数阶控制方法研究

以框架结构为研究对象,采用分数阶状态方程进行控制器设计,提出了基于LMI的分数阶控制器设计方法。首先,把原系统振动方程改为阶次为1～2的分数阶状态方程,同时,构造同阶次分数阶控制器状态方程,进而组装为一个整体状态方程;其次,根据分数阶系统稳定性条件确定满足系统渐进稳定性的不等式。为了获得可行解,在不等式中附加了两个参数,同时,令控制器参数矩阵为对称;再次,从仿真结果来看,控制效果过好,但控制力偏大,为此,控制力计算时附加了调节因子,以期在满足控制效果的基础上,降低所需控制力;最后给出了一个算例说明本文方法的可行性。     

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.     

分数阶Duffng振子的超谐共振

研究了含分数阶微分项的Duffing振子的超谐共振,通过平均法得到了系统的一阶近似解. 提出了超谐共振时等效线性 阻尼和等效线性刚度的概念,分析了分数阶微分项的系数和阶次对等效线性阻尼和等效线性刚度的影响. 建立了超谐共振解的幅频曲线的解析表达式和稳定性判断准则,对分数阶Duffing振子与传统整数阶Duffing振子的超谐共振解进行了比较. 最后通过数值仿真研究了分数阶微分项的参数对超谐共振幅频曲线的影响.     

A Suggestion of Fractional-Order Controller for Flexible Spacecraft Attitude Control

A controller design method for flexible spacecraft attitude control isproposed. The system is first described by partial differential equationwith internal damping. Then the frequency response is analyzed, and thethree basic characteristics of the flexible system, namely, averagefunction, lower bound and upper bound are defined. On this basis, afractional-order controller is proposed, which functions as phasestabilization control for lower frequency and smoothly enters toamplitude stabilization at higher frequency by proper amplitudeattenuation. It is shown that the equivalent damping ratio increases inproportion to the square of frequency.     

分数阶达芬振子的超谐与亚谐联合共振

研究了含分数阶微分项的达芬(Duffing)振子的超谐与亚谐联合共振.采用平均法得到了系统的一阶近似解析解,提出了超、亚谐联合共振时等效线性阻尼和等效线性刚度的概念.建立了联合共振定常解幅频曲线的解析表达式,并对联合共振幅频响应的近似解析解和数值解进行了比较,二者吻合良好,证明了求解过程及近似解析解的正确性.然后,将等效线性阻尼和等效线性刚度的概念与传统整数阶系统进行比较,证明分数阶微分项不仅起着阻尼的作用同时还起着刚度的作用.最后,通过数值仿真研究了不同的分数阶微分项系数和阶次对联合共振幅频曲线多值性和跳跃现象的影响,并与单一频率下超谐共振或亚谐共振进行了对比.研究发现,分数阶微分项系数与阶次不仅影响着系统的响应幅值、共振频率,同时还对系统的周期解个数、发生区域面积、发生先后等有重要影响.并且,在不同的基本参数下该系统分别表现出单独超谐共振、单独亚谐共振以及超谐共振和亚谐共振同时存在的现象.这些结果对系统动力学特性的研究具有重要意义.     

含分数阶Bingham模型的阻尼减振系统时滞半主动控制

针对基于磁流变液阻尼器的半主动控制系统中存在的时滞问题, 采用了一种将可控的时滞变量引入半主动控制切换条件的控制策略, 研究了考虑时滞的天棚阻尼控制切换条件对半主动阻尼减振系统的影响, 分析了含有分数阶Bingham模型的线性刚度系统在基础激励下的振动特性. 利用平均法得到了系统在含时滞半主动控制策略下主共振响应的近似解析解, 根据Lyapunov理论分析了系统的稳定性. 通过数值解验证了近似解析解的准确性, 二者具有较好的一致性. 利用近似解析解分析了固定激励频率下时滞对系统幅频响应特性的影响, 以及主共振峰值响应和共振频率随时滞变化的特性规律. 结果表明, 含时滞的半主动控制系统存在一个小时滞区间, 使得系统的振幅在主共振峰对应的频率附近低于不考虑时滞时系统的振幅, 且存在最优时滞使得系统的振幅大幅度降低; 而大时滞的引入会加剧系统的振动, 导致系统的颤振. 确定了基于分数阶Bingham模型的线性刚度系统在天棚阻尼半主动控制下的时滞选取原则, 为振动系统半主动阻尼控制中的时滞选取提供了参考.      

Fractional-order Maxwell model of seabed mud and its effect on surface-wave damping

A fractional-order Maxwell model is used to describe the viscoelastic seabed mud.The experimental data of the real mud well fit the results of the fractional-order Maxwell model that has fewer parameters than the traditional model.The model is then used to investigate the effect of the mud on the surface-wave damping.The damping rate of a linear monochromatic wave is obtained.The elastic resonance of the mud layer is observed,which leads to the peaks in the damping rate.The damping rate is a sum of the m...     

Analogue Realizations of Fractional-Order Controllers

An approach to the design of analogue circuits, implementingfractional-order controllers, is presented. The suggestedapproach is based on the use of continued fraction expansions;in the case of negative coefficients in a continued fractionexpansion, the use of negative impedance converters is proposed.Several possible methods for obtaining suitable rational appromixationsand continued fraction expansions are discussed. An exampleof realization of a fractional-order I controlleris presented and illustrated by obtained measurements.The suggested approach can be used for the control of veryfast processes, where the use of digital controllers isdifficult or impossible.     

基于速度反馈分数阶PID控制的达芬振子的主共振.

与传统整数阶比例-积分-微分(PID)控制器相比,分数阶比例-积分-微分控制器由于增加了两个控制参数,因此能够更灵活地控制受控对象.研究了基于速度反馈分数阶比例-积分-微分控制的达芬振子的主共振,利用平均法获得了系统的近似解析解.研究发现分数阶比例-积分-微分控制器的比例环节以等效线性阻尼的形式影响系统的共振振幅,积分环节以等效线性阻尼和等效线性刚度的形式影响系统的动力学特性,微分环节以等效线性阻尼和等效质量的形式影响系统的动力学特性.建立了主共振幅频响应方程的解析表达式和稳定性判断准则,并对主共振幅频响应的近似解析解和数值解进行了比较,二者吻合良好,验证了求解过程和近似解析解的正确性.分析了分数阶比例-积分-微分控制器的比例环节系数、积分环节系数、微分环节系数以及分数阶阶次变化时,对系统主共振幅频响应的影响.对分数阶比例-积分-微分控制器与传统整数阶比例-积分-微分控制器的控制效果进行了对比,发现当控制器各个环节的系数相同时,基于速度反馈的分数阶比例-积分-微分控制对达芬振子主共振的控制效果要优于传统整数阶比例-积分-微分控制.      

ENERGY PRINCIPLE OF THREE DIMENSIONAL J INTEGRAL

Various exprressions of three-dimensional J integral are proposed. We derive here three-dimensional J integral by means of general potential energy principle and Green's theomem. Its physical meaning and application are shown in the paper, the results are true both for infinitesimal and finite deformation.     

网纹红土分数阶应力松弛模型

网纹红土松弛特性具有非线性。基于分数阶微积分理论,探讨了能更准确描述网纹红土松弛非线性特性和全过程的FVMS(Fractional Voigt and Maxwell model in series)松弛模型和FVMP(Fractional Voigt and Maxwell model in parallel)松弛模型及其理论解,进而应用提出的模型对三轴松弛试验实测数据进行反演,讨论了分数阶阶数的敏感性,并与西原模型和Burgers模型进行对比分析。研究结果表明,建立的四元件分数阶松弛本构模型应用于网纹红土应力松弛特性分析是有效可行的,模型灵活且精度更高,参数确定简便,发现分数阶阶数对应力松弛量的影响较大,但其对FVMS模型和FVMP模型松弛速率的影响不同,为实际工程长期稳定性分析提供了参考。     

High-order numerical methods of fractional-order Stokes' first problem for heated generalized second grade fluid

The high-order implicit finite difference schemes for solving the fractionalorder Stokes' first problem for a heated generalized second grade fluid with the Dirichlet boundary condition and the initial...     

High-order numerical methods of fractional-order Stokes’ first problem for heated generalized second grade fluid

The high-order implicit finite difference schemes for solving the fractionalorder Stokes’ first problem for a heated generalized second grade fluid with the Dirichlet boundary condition and the initial condition are given. The stability, solvability, and convergence of the numerical scheme are discussed via the Fourier analysis and the matrix analysis methods. An improved implicit scheme is also obtained. Finally, two numerical examples are given to demonstrate the effectiveness of the mentioned schemes.     

A method for conformal mapping of a two-connected region onto an annulus

This paper presents a method for conformal of a two-connectedregion onto an annulus.The philosophy of the method is toconvert the problem into a Dirichlet problem and to prove thereal part of the analytic function transformation should be aharmonic function satisfying certain boundary conditions.Ac-cording to the theory of harmonic function we can determine theinner radius of the annulus from the condition that the harmonic function defined in two-connected region should be single-valued.It is then easy to see that the imaginary part can directly beobtained with the aid of Cauchy-Riemann conditions.The unknownconstants of integration only influence the argument of imagepoints and can easily be derived by using the one-to-one mappingof region onto an annulus.Without loss of generality,themethod can be used to conformally map other two-connected regionsonto an annulus if they can be subdivided into several rectang-ulars.The method has been programmed for a digital computer.It is demonstrated that the meth     

Free vibration of thermo-elastic microplate based on spatiotemporal fractional-order derivatives with nonlocal characteristic length and time

A fractional-order thermo-elastic model taking into account the small-scale effects of the thermo-elastic coupled behavior is developed to study the free vibration of a higher-order shear microplate. The nonlocal strain gradient theory is modified with the introduction of the fractional-order derivatives and the nonlocal characteristic length. The Fourier heat conduction is replaced by the non-Fourier heat conduction with the introduction of the fractional order and the memory characteristic tim...     

Fractional Order Control of a Hexapod Robot

This paper studies the performance of integer and fractional order controllers in a hexapod robot with joints at the legs having viscous friction and flexibility. For that objective the robot prescribed motion is characterized in terms of several locomotion variables. The walking performance is analysed through the Nyquist stability criterion and several indices that reflect the system dynamical properties. A set of model-based experiments reveals the influence of the different controller implementations upon the proposed metrics.