Fractional Trigonometry and the Spiral Functions

This paper develops two related fractional trigonometries based on the multi-valued fractional generalization of the exponential function, the R-function. The trigonometries contain the traditional trigonometric functions as proper subsets. Also developed are relationships between the R-function and the new fractional trigonometric functions. Laplace transforms are derived for the new functions and are used to generate solution sets for various classes of fractional differential equations. Because of the fractional character of the R-function, several new trigonometric functions are required to augment the traditional sine, cosine, etc. functions. Fractional generalizations of the Euler equation are derived. As a result of the fractional trigonometry a new set of phase plane functions, the Spiral functions, that contain the circular functions as a subset, is identified. These Spiral functions display many new symmetries.     

On Fractional Adaptive Control

Introducing fractional operators in the adaptive control loop, and especially in Model Reference Adaptive Control (MRAC), has proven to be a good mean for improving the plant dynamics with respect to response time and disturbance rejection. The idea of introducing fractional operators in adaptation algorithms is very recent and needs to be more established, that is why many research teams are working on the subject. Previously, some authors have introduced a fractional model reference in the adaptation scheme, and then fractional integration has been used to deal directly with the control rule. Our original contribution in this paper is the use of a fractional derivative feedback of the plant output, showing that this scheme is equivalent to the fractional integration, one with a certain benefit action on the system dynamical behaviour and a good robustness effect. Numerical simulations are presented to show the effectiveness of the proposed fractional adaptive schemes.     

Variable Order and Distributed Order Fractional Operators

Many physical processes appear to exhibit fractional order behavior thatmay vary with time or space. The continuum of order in the fractionalcalculus allows the order of the fractional operator to be considered asa variable. This paper develops the concept of variable and distributedorder fractional operators. Definitions based on the Riemann–Liouvilledefinition are introduced and the behavior of the new operators isstudied. Several time domain definitions that assign different argumentsto the order q in the Riemann–Liouville definition are introduced. Foreach of these definitions various characteristics are determined. Theseinclude: time invariance of the operator, operator initialization,physical realization, linearity, operational transforms, and memorycharacteristics of the defining kernels.A measure (m ₂) for memory retentiveness of the order history isintroduced. A generalized linear argument for the order q allows theconcept of `tailored' variable order fractional operators whose m ₂ memory may be chosen for a particular application. Memory retentiveness (m ₂) andorder dynamic behavior are investigated and applications are shown.The concept of distributed order operators where the order of thetime based operator depends on an additional independent (spatial)variable is also forwarded. Several definitions and their Laplacetransforms are developed, analysis methods with these operators aredemonstrated, and examples shown. Finally operators of multivariable anddistributed order are defined and their various applications areoutlined.     

分数因子与分数阶完整力学系统的运动方程和循环积分

引入分数因子和分数增量,给出了分数阶微积分的定义和性质;基于分数阶导数的定义,证明了含有分数因子的等时变分与分数阶算子的交换关系;提出了分数阶完整保守和非保守系统的Hamilton原理;建立了分数阶完整保守系统和非保守系统的运动微分方程;得到了分数阶完整保守系统的循环积分;并利用分数阶循环积分导出分数阶罗兹方程．最后给出了两个例子．研究表明利用分数因子给出的分数阶微分方程是一个含有分数因子的通常的微分方程,那么分数阶系统运动微分方程的求解都可以采用通常微分方程的求解方法．     

Solution for a Fractional Diffusion-Wave Equation Defined in a Bounded Domain

A general solution is given for a fractional diffusion-wave equation defined in a bounded space domain. The fractional time derivative is described in the Caputo sense. The finite sine transform technique is used to convert a fractional differential equation from a space domain to a wavenumber domain. Laplace transform is used to reduce the resulting equation to an ordinary algebraic equation. Inverse Laplace and inverse finite sine transforms are used to obtain the desired solutions. The response expressions are written in terms of the Mittag–Leffler functions. For the first and the second derivative terms, these expressions reduce to the ordinary diffusion and wave solutions. Two examples are presented to show the application of the present technique. Results show that for fractional time derivatives of order 1/2 and 3/2, the system exhibits, respectively, slow diffusion and mixed diffusion-wave behaviors.     

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.     

The Time-Scaled Trapezoidal Integration Rule for Discrete Fractional Order Controllers

In this paper, the time-scaled trapezoidal integration rule for discretizing fractional order controllers is discussed. This interesting proposal is used to interpret discrete fractional order control (FOC) systems as control with scaled sampling time. Based on this time-scaled version of trapezoidal integration rule, discrete FOC can be regarded as some kind of control strategy, in which strong control action is applied to the latest sampled inputs by using scaled sampling time. Namely, there are two time scalers for FOC systems: a normal time scale for ordinary feedback and a scaled one for fractional order controllers. A new realization method is also proposed for discrete fractional order controllers, which is based on the time-scaled trapezoidal integration rule. Finally, a one mass position 1/s^k control system, realized by the proposed method, is introduced to verify discrete FOC systems and their robustness against saturation non-linearity.     

Finite Element Formulation of Viscoelastic Constitutive Equations Using Fractional Time Derivatives

Fractional time derivatives are used to deduce a generalization ofviscoelastic constitutive equations of differential operator type. Theseso-called fractional constitutive equations result in improvedcurve-fitting properties, especially when experimental data from longtime intervals or spanning several frequency decades need to be fitted.Compared to integer-order time derivative concepts less parameters arerequired. In addition, fractional constitutive equations lead to causalbehavior and the concept of fractional derivatives can be physicallyjustified providing a foundation of fractional constitutive equations.First, three-dimensional fractional constitutive equations based onthe Grünwaldian formulation are derived and their implementationinto an elastic FE code is demonstrated. Then, parameter identificationsfor the fractional 3-parameter model in the time domain as well as inthe frequency domain are carried out and compared to integer-orderderivative constitutive equations. As a result the improved performanceof fractional constitutive equations becomes obvious. Finally, theidentified material model is used to perform an FE time steppinganalysis of a viscoelastic structure.     

Fractional Derivative Viscoelasticity at Large Deformations

A time domain viscoelastic model for large three-dimensional responses underisothermal conditions is presented. Internal variables with fractional orderevolution equations are used to model the time dependent part of the response. By using fractional order rate laws, the characteristics of the timedependency of many polymeric materials can be described using relatively fewparameters. Moreover, here we take into account that polymeric materials are often used in applications where the small deformations approximation does nothold (e.g., suspensions, vibration isolators and rubber bushings). A numerical algorithm for the constitutive response is developed and implemented into a finite element code forstructural dynamics. The algorithm calculates the fractional derivatives by means of the Grünwald–Lubich approach.Analytical and numerical calculations of the constitutive response in the nonlinearregime are presented and compared. The dynamicstructural response of a viscoelastic bar as well as the quasi-static response of athick walled tube are computed, including both geometrically and materiallynonlinear effects. Moreover, it isshown that by applying relatively small load magnitudes, the responses ofthe linear viscoelastic model are recovered.     

Fractional complex modulus manifested in asphalts

In recent years, a considerable effort has been made to develop a new generation of asphaltic materials based on a combination of polymers and asphalts. Regular and polymer-modified asphalts are studied via fractional relaxation processes, represented here by a fractional rational form of the complex modulus, G ^*. Basic properties of this complex modulus and the forms of generated constitutive equations are studied. Relaxation times of the model are related via a pseudospectrum to the phase angle lag.     

Fractional Discrete-Time Signal Processing: Scale Conversion and Linear Prediction

A generalization of the linear prediction for fractional steps isreviewed, widening well-known concepts and results. This prediction isused to derive a causal interpolation algorithm. A reconstructionalgorithm for the situation where averages are observed is alsopresented. Scale conversion of discrete-time signals is studied takingas base the fractional discrete-time system theory. Some simulationresults to illustrate the behaviour of the algorithms will be presented.A new algorithm for performing the zoom transform is also described.     

Discrete and Continuous Random Walk Models for Space-Time Fractional Diffusion

A mathematical approach to anomalous diffusion may be based on generalized diffusion equations (containing derivatives of fractional order in space or/and time) and related random walk models. A more general approach is however provided by the integral equation for the so-called continuous time random walk (CTRW), which can be understood as a random walk subordinated to a renewal process. We show how this integral equation reduces to our fractional diffusion equations by a properly scaled passage to the limit of compressed waiting times and jumps. The essential assumption is that the probabilities for waiting times and jumps behave asymptotically like powers with negative exponents related to the orders of the fractional derivatives. Illustrating examples are given, numerical results and plots of simulations are displayed.     

Application of Fractional Derivatives in Thermal Analysis of Disk Brakes

This paper presents a Fractional Derivative Approach for thermal analysis of disk brakes. In this research, the problem is idealized as one-dimensional. The formulation developed contains fractional semi integral and derivative expressions, which provide an easy approach to compute friction surface temperature and heat flux as functions of time. Given the heat flux, the formulation provides a means to compute the surface temperature, and given the surface temperature, it provides a means to compute surface heat flux. A least square method is presented to smooth out the temperature curve and eliminate/reduce the effect of statistical variations in temperature due to measurement errors. It is shown that the integer power series approach to consider simple polynomials for least square purposes can lead to significant error. In contrast, the polynomials considered here contain fractional power terms. The formulation is extended to account for convective heat loss from the side surfaces. Using a simulated experiment, it is also shown that the present formulation predicts accurate values for the surface heat flux. Results of this study compare well with analytical and experimental results.     

Nonlinear Fractional Order Viscoelasticity at Large Strains

In this paper, we formulate a fractional order viscoelastic model for large deformations and develop an algorithm for the integration of the constitutive response. The model is based on the multiplicative split of the deformation gradient into elastic and viscous parts. Further, the stress response is considered to be composed of a nonequilibrium part and an equilibrium part. The viscous part of the deformation gradient (here regarded as an internal variable) is governed by a nonlinear rate equation of fractional order. To solve the rate equation the finite element method in time is used in combination with Newton iterations. The method can handle nonuniform time meshes and uses sparse quadrature for the calculations of the fractional order integral. Moreover, the proposed model is compared to another large deformation viscoelastic model with a linear rate equation of fractional order. This is done by computing constitutive responses as well as structural dynamic responses of fictitious rubber materials.     

Impulse Responses of Fractional Damped Systems

Considered are systems of single-mass oscillators with different fractional damping behaviour. Similar to the classical model, where the damping terms are represented by first derivatives, the eigensystem can be used to decompose the fractional system in frequency domain, if mass, stiffness and damping matrices are linearly dependent. The solution appears as a linear combination of single-mass oscillators. This is true even in the general case such that stability and causality are insured by the same argumentation as used in the linear dependent case.     

A Fractional Linear System View of the Fractional Brownian Motion

A definition of the fractional Brownian motion based on the fractional differintegrator characteristics is proposed and studied. It is shown that the model enjoys the usually required properties. A discrete-time version based in the backward difference and in the bilinear transformation is considered. Some results are presented.     

Fractional Derivatives Applied to Phase-Space Reconstructions

The concept and application of phase-space reconstructions are reviewed. Fractional derivatives are then proposed for the purpose of reconstructing dynamics from a single observed time history. A procedure is presented in which the fractional derivatives of time series data are obtained in the frequency domain. The method is applied to the Lorenz system. The ability of the method to unfold the data is assessed by the method of global false nearest neighbors. The reconstructed data is used to compute recurrences and correlation dimensions. The reconstruction is compared to the commonly used method of delays in order to assess the choice of reconstruction parameters, and also the quality of results.     

Fractional Order Disturbance Observer for Robust Vibration Suppression

For the first time, the fractional order disturbance observer (FO-DOB) is proposed for vibration suppression applications such as hard disk drive servo control. It has been discovered in a recently published US patent application (US20010036026) that there is a tradeoff between phase margin loss and strength of the low frequency vibration suppression. Given the required cutoff frequency of the low pass filter, also known as the Q-filter, it turns out that the relative degree of the Q-filter is the major tuning knob for this tradeoff. The solution in US20010036026 was based on an integer order Q-filter with a variable relative degree. This actually motivated the use of a fractional order Q-filter. The fractional order disturbance observer is based on the fractional order Q-filter. The implementation issue is also discussed. The nice point of this paper is that the traditional DOB is extended to the fractional order DOB with the advantage that the FO-DOB design is now no longer conservative nor aggressive, i.e., given the cutoff frequency and the desired phase margin, we can uniquely determine the fractional order of the low pass filter.     

Fractional relaxation and the time-temperature superposition principle

Relaxation processes in complex systems like polymers or other viscoelastic materials can be described by equations containing fractional differential or integral operators. In order to give a physical motivation for fractional order equations, the fractional relaxation is discussed in the framework of statistical mechanics. We show that fractional relaxation represents a special type of a non-Markovian process. Assuming a separation condition and the validity of the thermo-rheological principle, stating that a change of the temperature only influences the time scale but not the rheological functional form, it is shown that a fractional operator equation for the underlying relaxation process results.     

分数导数粘弹性模型描述的土中管桩竖向振动的频域解

在分数导数粘弹性本构模型的基础上综合考虑桩周土和桩芯土的平衡方程和几何方程建立了桩周土和桩芯土的竖向运动的控制方程．在频率域内利用分离变量法和分数导数的性质求解了桩周土和桩芯土竖向振动控制方程．考虑管桩与桩周土、管桩与桩芯土的边界连续性条件以及三角函数的正交性得到了分数导数粘弹性模型描述的土中管桩的竖向振动,通过数值分析研究了管桩和土体模型参数和几何参数对管桩的桩顶复刚度的影响规律．结果显示：桩芯土本构模型的分数导数的阶数对管桩竖向振动的影响较桩周土本构模型的阶数要小,且与频率有一定关系;桩芯土与桩周土的模型参数比τ1 和τ2 对等效阻尼的影响较对刚度因子的影响要大;管桩桩周和桩芯的直径比d 越小,管桩复刚度的实部和虚部就越大;土体力学性能对管桩竖向振动的影响要比管桩桩身力学性能的影响小．