基于分数阶模型的非完整系统的Mei对称性及其守恒量

为了进一步揭示非完整系统的对称性和守恒量之间的内在关系，提出并研究基于分数阶模型的非完整系统的Mei对称性及其守恒量．首先，根据分数阶d’Alembert-Lagrange原理建立基于分数阶模型的非完整系统的动力学方程．其次，根据动力学方程中的动力学函数经无限小变换后仍满足原方程的不变性，建立分数阶模型下非完整系统的Mei对称性定理，给出Mei守恒量．再次，讨论了几个特例：分数阶Hamilton系统、经典非完整系统和受非完整约束的分数阶Lagrange系统的Mei对称性定理．文末举例说明结果的应用．     

Caputo导数下分数阶Birkhoff系统的准对称性与分数阶Noether定理

应用分数阶模型可以更准确地描述和研究复杂系统的动力学行为和物理过程,同时Birkhoff力学是Hamilton力学的推广,因此研究分数阶Birkhoff系统动力学具有重要意义.分数阶Noether定理揭示了Noether对称变换与分数阶守恒量之间的内在联系,但是当变换拓展为Noether准对称变换时,该定理的推广遇到了很大的困难.本文基于时间重新参数化方法提出并研究Caputo导数下分数阶Birkhoff系统的Noether准对称性与守恒量.首先,将时间重新参数化方法应用于经典Birkhoff系统的Noether准对称性与守恒量研究,建立了相应的Noether定理;其次,基于分数阶Pfaff作用量分别在时间不变的和一般单参数无限小变换群下的不变性给出分数阶Birkhoff系统的Noether准对称变换的定义和判据,基于Frederico和Torres提出的分数阶守恒量定义,利用时间重新参数化方法建立了分数阶Birkhoff系统的Noether定理,从而揭示了分数阶Birkhoff系统的Noether准对称性与分数阶守恒量之间的内在联系.分数阶Birkhoff系统的Noether对称性定理和经典Birkhoff系统的Noether定理是其特例.最后以分数阶Hojman-Urrutia问题为例说明结果的应用.     

分数阶微分方程的Haar小波算法研究

在BPFs的Caputo分数阶微分算子矩阵的基础上,建立了Haar小波的分数阶微分算子矩阵,提出了一种有效的求解分数阶微分方程的Haar小波数值方法,并将该方法应用于线性和非线性分数阶常微分方程求解中.数值算例表明,该算法简单,数值精确度高,是一种高效的数值求解方法.     

分数阶Birkhoff系统Noether对称性的摄动与绝热不变量

20世纪90年代以来,分数阶微积分理论与方法已被广泛地应用到自然科学和社会科学的各个领域,动力学与控制是其中的一个重要应用领域.为了进一步研究分数阶力学系统,本文基于Riemann-Liouville分数阶导数,讨论了分数阶Birkhoff系统Noether对称性的摄动与绝热不变量问题.首先,给出分数阶Birkhoff系统的运动微分方程及精确不变量;其次,给出绝热不变量的定义,并研究分数阶Birkhoff系统的绝热不变量;文末举例说明结果的应用.     

非线性非完整系统的运动方程及其广义能量积分

本文研究了一阶非线性非完整系统的运动规律,由 Routh 方程出发,用了矩阵的右乘零因子的有关理论,给出了该系统的不含待定乘子的一般方程,本文还讨论了一阶非线性非完整系统的运动方程存在广义能量积分的条件,并给出了相应的广义能量积分.     

非线性粘弹性Timoshenko梁动力学行为的分布

本文讨论了有限变形粘弹性Timoshenko梁的动力学行为。首先由Timoshenko梁的理论和分数导数型本构关系给出了梁的控制方程。其次为了便于求解,采用Galerkin方法对系统进行了简化,并比较了1阶和2阶截断系统的动力学性质,它们具有相同的定性性质,说明Galerkin方法的合理性。给出了求解包含分数积分的积分－微分方程的一种新方法,以便求解系统的长时间的解。综合利用非线性动力系统中的经典方法,揭示了梁在有限变形情况下丰富的动力学行为,并分别考察了载荷参数的材料参数对结构的动力学行为的影响。     

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.      

爬壁机器人系统的Noether对称性和守恒量

爬壁机器人的运动是一种模仿壁虎爬行的运动, 爬壁机器人的运动可分解为四肢带动身体的运动, 先前的研究都是基于牛顿力学的方法. 本文采用Lagrange 力学的方法建立爬壁机器人系统的运动方程, 并运用Lie群分析方法建立该系统的Noether对称性理论, 得出爬壁机器人的运动规律. 首先, 给出非完整爬壁机器人系统的动能、势能和Lagrange函数以及所受的非完整约束, 从而建立了非完整爬壁机器人系统的Lagrange方程; 其次, 引入关于时间和广义坐标的无限小变换, 提出了非完整爬壁机器人系统的Hamilton作用量和Hamilton作用量的基本变分公式; 第三, 给出爬壁机器人系统 Noether对称性变换和广义准对称变换的定义, 判据和存在的Noether守恒量, 并提出了非保守完整系统和非保守非完整爬壁机器人系统的Noether定理; 最后, 以圆锥面上爬壁机器人为例, 对给出的守恒量直接进行积分给出圆锥面上爬壁机器人整体运动的精确解和四肢运动的数值解, 发现了该爬壁机器人的运动规律, 很好地验证了非完整爬壁机器人系统的Noether对称性理论. 本文的研究为Lie群分析方法应用于其他复杂的机器人系统以及柔性机器人系统的对称性求解提出了一种新的对称性求解方法.      

基于分数因子法的分数阶约束Hamilton系统的Lie对称性和守恒量研究

采用基于分数因子的分数阶导数计算方法,结合Hamilton变分原理推导出保守分数阶奇异系统的正则方程;进一步探讨无穷小变换下系统微分和代数方程的不变性,给出Lie对称性的判据方程;构造Lie对称性的结构方程,得到系统相空间守恒量的形式.最后举例说明文中方法的应用.     

分数阶Chen混沌系统的完全同步与反相同步

以分数阶 Chen 系统为例,分析了该系统的混沌特性,并对其分别进行了完全同步与反相同步控制.首先,基于分数阶系统稳定性理论和非线性动力学理论,构造出相应的非线性控制器,由此实现了分数阶 Chen 混沌系统的完全同步与反相同步控制;其次,利用分数阶稳定性理论,对上述同步给出了严格的数学证明;最后,借助于 Adams-Bashforth-Moultom 算法,利用数值模拟验证了所提方法的有效性.     

Fractional order spring/spring-pot/actuator element in a multibody system: Application of an expansion formula

Fractional order models of a spring/spring-pot and spring/spring-pot/actuator element connected into a multibody system are proposed in order to represent smart materials and components in adaptronic systems by introducing new tuning parameter. The models are introduced into dynamic equations via generalized forces and using the Lagrange's equations of the second kind in covariant form. Generalized forces are derived by taking into account fractional order derivatives in force–displacement relations and by using the principle of virtual work. The numerical scheme for solving fractional order differential equations proposed in Atanacković and Stanković (2008) is used in order to approximate fractional order derivative of a composite function appearing in the presented fractional order model. Numerical example for the multibody system with three degrees of freedom is presented. The results obtained for generalized forces are compared for different values of parameters in the fractional order derivative model.     

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.     

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.     

Correcting the initialization of models with fractional derivatives via history-dependent conditions

Fractional differential equations are more and more used in modeling memory(history-dependent,nonlocal,or hereditary) phenomena.Conventional initial values of fractional differential equations are define at a point,while recent works defin initial conditions over histories.We prove that the conventional initialization of fractional differential equations with a Riemann–Liouville derivative is wrong with a simple counter-example.The initial values were assumed to be arbitrarily given for a typical fractional differential equation,but we fin one of these values can only be zero.We show that fractional differential equations are of infinit dimensions,and the initial conditions,initial histories,are define as functions over intervals.We obtain the equivalent integral equation for Caputo case.With a simple fractional model of materials,we illustrate that the recovery behavior is correct with the initial creep history,but is wrong with initial values at the starting point of the recovery.We demonstrate the application of initial history by solving a forced fractional Lorenz system numerically.     

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.     

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.     

Galerkin Projections and Finite Elements for Fractional Order Derivatives

Ordinary differential equations (ODEs) with fractional order derivatives are infinite dimensional systems and nonlocal in time: the history of the state variable is needed to calculate the instantaneous rate of change. This nonlocal nature leads to expensive long-time computations (O(t ²) computations for solution up to time t). A finite dimensional approximation of the fractional order derivative can alleviate this problem. We present one such approximation using a Galerkin projection. The original infinite dimensional system is replaced with an equivalent infinite dimensional system involving a partial differential equation (PDE). The Galerkin projection reduces the PDE to a finite system of ODEs. These ODEs can be solved cheaply (O(t) computations). The shape functions used for the Galerkin projection are important, and given attention. The approximation obtained is specific to the fractional order of the derivative; but can be used in any system with a derivative of that order. Calculations with both global shape functions as well as finite elements are presented. The discretization strategy is improved in a few steps until, finally, very good performance is obtained over a user-specifiable frequency range (not including zero). In particular, numerical examples are presented showing good performance for frequencies varying over more than 7 orders of magnitude. For any discretization held fixed, however, errors will be significant at sufficiently low or high frequencies. We discuss why such asymptotics may not significantly impact the engineering utility of the method.     

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．