非线性分数阶双稳系统逻辑随机共振的研究

在整数阶逻辑随机共振的郎之万方程基础上构建了分数阶情况下的郎之万方程。对该方程描述的非线性分数阶双稳系统进行了仿真验证,分析分数阶阶次和系统参数的改变对逻辑随机共振现象的影响。结果表明当分数阶阶次小于临界值时,即使没有外加高斯白噪声或微弱周期信号也能观察到逻辑随机共振现象;当分数阶阶次大于临界值时,需要外加高斯白噪声或微弱周期信号才能实现逻辑随机共振,选择合适的噪声强度、微弱周期信号振幅、频率等可以提高逻辑输出的成功率。     

分数阶具有不确定项和外扰的超混沌金融系统的滑模同步

本文利用滑模控制与比例积分滑模控制技巧研究了分数阶具有不确定项和外扰的一类超混沌金融系统的同步问题,运用分数阶微积分设计出滑模函数,通过设计适应规则构造出适应控制器,得到了分数阶不确定超混沌金融系统取得滑模同步和积分滑模同步的两个充分性条件．Matlab 数值仿真验证了理论结果．文中滑模函数的设计、控制器的构造及适应规则的选取对研究整数阶超混沌金融系统的滑模同步具有可移植性,所使用的方法为研究分数阶混沌系统提供了思路,同时分数阶系统的相关结果可以移植到整数阶系统的同步问题．     

基于量子粒子群算法的主动悬架分数阶控制策略

结合分数阶微积分理论,针对1/4车辆主动悬架模型,设计一种以车身垂向加速度为反馈变量的分数阶控制器。综合车身垂向加速、悬架动挠度和车轮相对动载荷3个指标,建立优化指标的无量纲评价函数。将分数阶控制器的比例系数、积分系数、积分阶次、微分系数及微分阶次当作五维空间粒子,采用量子粒子群算法(QPSO)确定最优粒子。利用MATLAB软件建立悬架系统仿真模型,分别对被动悬架、含整数阶控制器的主动悬架及含分数阶控制器的主动悬架进行时域和频域仿真研究,对比结果表明,相对于整数阶主动悬架与被动悬架,含分数阶控制器的主动悬架明显改善了汽车平顺性。基于量子粒子群算法的主动悬架分数阶控制策略能更有效地抑制车身共振、改善汽车乘坐舒适性。     

Atangana-Baleanu分数阶双稳系统的随机共振现象

针对基于常用分数阶微积分对随机共振现象的研究存在奇异性的问题,提出了基于Atangana-Baleanu分数阶微积分的双稳系统随机共振现象的研究方法。首先,根据Atangana-Baleanu分数阶微积分的定义构造了用于描述随机共振系统的Langevin方程;其次,通过改进的Oustaloup算法对其近似化求解;最后,编写仿真程序,利用控制单一变量法研究参数变化对随机共振的影响。仿真结果表明:噪声强度一定时改变分数阶求导阶次,分数阶求导阶次与输出信号的功率谱值呈非线性关系且存在一个最佳分数阶求导阶次使系统产生随机共振;分数阶求导阶次一定时改变噪声强度,噪声强度与输出信号的功率谱值呈非线性关系且存在一个最佳噪声强度使系统产生随机共振。     

基于分数阶微积分的油气悬架建模与试验分析

根据油气悬架的多相介质力学特点,引入了分数阶微积分理论,在油气悬架运动微分方程基础上建立其分数阶Bagley-Torvik方程。设计Oustaloup算法低通滤波器进行运算,求得非线性分数阶微分方程数值解。将微分方程中最优阶次的选取转化为单变量最优问题,确立位移差方均值为评价目标进行求解。搭建等比例试验台和建立仿真模型,进行理论、试验、整数阶仿真数据的对比。改变激励频率和振幅观察最优阶次变化及误差变化趋势。结果表明试验油气悬架在激励频率1 Hz,振幅5 mm下分数阶次取0.912时能更好地反映油气悬架运动特性。最优分数阶次随着激励幅值和频率的增加而减小并最终趋于稳定,在高频振动下分数阶次趋于0.9,在高幅振动下分数阶次为0.86。在多个频率及振幅激励的试验条件下,分数阶结果误差都要小于整数阶结果,论证了分数阶微积分在油气悬架建模上的有效性。     

不同阶数不同维数的分数阶混沌系统的时滞混合投影同步

本文主要讨论了不同阶数不同维数的分数阶混沌系统的时滞混合投影同步.基于分数阶微积分的基本性质,将两个不同阶数的分数阶混沌系统转换为两个同阶的分数阶混沌系统.然后,利用分数阶线性系统的稳定性理论,并构造一个非线性控制器,得到了同步的一般方法.数值模拟证实了设计方法的有效性.     

不同阶数不同维数的分数阶混沌系统的时滞混合投影同步（英文）

本文主要讨论了不同阶数不同维数的分数阶混沌系统的时滞混合投影同步.基于分数阶微积分的基本性质,将两个不同阶数的分数阶混沌系统转换为两个同阶的分数阶混沌系统.然后,利用分数阶线性系统的稳定性理论,并构造一个非线性控制器,得到了同步的一般方法.数值模拟证实了设计方法的有效性.     

基于粒子群算法的被动分数阶汽车悬架参数优化设计

利用粒子群算法研究了被动分数阶汽车悬架参数的优化设计。分数阶汽车悬架系统是指运动微分方程中含有分数阶微分项的汽车悬架系统。建立了被动分数阶悬架系统的仿真模型,利用Oustaloup滤波器算法实现了该模型中分数阶微积分的近似计算。利用粒子群算法寻找一组最优的悬架参数来协调汽车操纵稳定性和乘坐舒适性的关系以到达最优的悬架性能。对比原悬架系统和优化后悬架系统在A、B、C、D共四级路面输入下的响应及其频率特性。研究结果表明,利用该方法对被动分数阶悬架参数进行优化设计,在保证汽车操纵稳定性的前提下乘坐舒适性得到明显改善。     

含分数阶惯容器隔振系统的临界阻尼设计

为了研究用含分数阶导数描述的液力惯容器的非线性隔振系统特性,建立无量纲动力学模型。在无阻尼控制时,通过对比力传递率特性指标讨论参数对隔振系统性能的影响。结果表明分数阶惯容器能够反映液力式惯容器的多相特性,既有惯性作用又有一定阻尼作用,相比整数阶惯容器在隔振效果上有一定优势,但同样不能有效抑制非线性效应。在忽略非线性项的前提下,考虑系统含分数导数的特殊情况,详细介绍分数阶临界阻尼设计过程。仿真试验结果表明,所设计临界阻尼可以保证系统在自由振动时单调递减,而且考虑非线性项后,相比整数阶惯容器能够更好抑制非线性效应。     

分数阶惯容悬架动态特性研究

采用分数阶微分理论描述液力式惯容器的力学特性，建立分数阶惯容悬架系统动力学模型。采用拉普拉斯变换方法研究分数阶惯容悬架系统在简谐激励下的动态响应，分析系统结构参数对悬架动态特性的影响规律，并结合数值解验证解析解的精确性。研究分数阶惯容悬架系统在随机路面激励下的动态特性，并与整数阶惯容悬架系统进行对比。结果表明相较整数阶惯容悬架系统，运用分数阶惯容悬架系统能进一步减小车身振动，提高系统动态特性。研究为将分数阶微分运用于惯容悬架系统提供理论指导。     

On a quasiclassical Langevin equation

It is shown that the motion of a quantum mechanical particle coupled to a dissipative environment can be described by a Langevin equation where the stochastic force is generalized such that its power spectrum is in accordance with the fluctuation-dissipation theorem. This generalized Langevin equation has an interesting range of applicability. It includes the quasiclassical regime provided that the damping, that is, the coupling of the particle to its environment, is sufficiently strong.     

First passage failure of MDOF quasi-integrable Hamiltonian systems with fractional derivative damping

The first passage failure of multi-degree-of-freedom (MDOF) quasi-integrable Hamiltonian systems with damping described by a fractional derivative is studied. The stochastic averaging procedure is applied to derive the averaged equations for first integrals. The conditional reliability function and the conditional mean of first passage failure time are obtained by solving the associated backward Kolmogorv equation and Pontryagin equation together with suitable boundary conditions and initial condition, respectively. One example of two coupled nonlinear oscillators with fractional derivative damping is given to illustrate the proposed procedure. The accuracy of the method is substantiated by comparing the analytical results with those from Monte Carlo simulation. Effects of some parameters of fractional order, damping coefficients and nonlinear strength on the system??s reliability are examined.     

Stochastic response of fractionally damped beams

This paper aims at introducing the governing equation of motion of a continuous fractionally damped system under generic input loads, no matter the order of the fractional derivative. Moreover, particularizing the excitation as a random noise, the evaluation of the power spectral density performed in frequency domain highlights relevant features of such a system.Numerical results have been carried out considering a cantilever beam under stochastic loads. The influence of the fractional derivative order on the power spectral density response has been investigated, underscoring the damping effect in reducing the power spectral density amplitude for higher values of the fractional derivative order. Finally, the fractional derivative term introduces in the system dynamics both effective damping and effective stiffness frequency dependent terms.     

Stochastic stability of Duffing oscillator with fractional derivative damping under combined harmonic and white noise parametric excitations

The stochastic stability of a Duffing oscillator with fractional derivative damping of order α (0 < α < 1) under parametric excitation of both harmonic and white noise is studied. First, the averaged Itô equations are derived by using the stochastic averaging method for an SDOF strongly nonlinear stochastic system with fractional derivative damping under combined harmonic and white noise excitations. Then, the expression for the largest Lyapunov exponent of the linearized averaged Itô equations is obtained and the asymptotic Lyapunov stability with probability one of the original system is determined approximately by using the largest Lyapunov exponent. Finally, the analytical results are confirmed by using those from a Monte Carlo simulation of the original system.     

A bounded optimal control for maximizing the reliability of randomly excited nonlinear oscillators with fractional derivative damping

In this paper, a bounded optimal control for maximizing the reliability of randomly excited nonlinear oscillators with fractional derivative damping is proposed. First, the partially averaged It? equations for the energy processes of individual degree of freedom are derived by using the stochastic averaging method. Second, the dynamical programming equations for the control problems of maximizing the reliability function and maximizing the mean first passage time are established from the partially averaged It? equations by using the dynamical programming principle. The optimal control law is derived from the dynamical programming equation and control constraints. Third, the conditional reliability function and mean first passage time of the optimally controlled system are obtained by solving the backward Kolmogorov equation and Pontryagin equation associated with the fully averaged It? equation, respectively. The application of the proposed procedure and effectiveness of the control strategy are illustrated by using two examples. Besides, the effect of fractional derivative order on the reliability of the optimally controlled system is examined.     

Investigation of mode coupling in low and high NA step index plastic optical fibres using the Langevin equation

By employing the Langevin equation, we have examined a mode coupling in low and high NA step index plastic optical fibres. The numerical integration of the Langevin equation is based on the computer-simulated Langevin force. The solution matches the experimental data reported previously. We have shown that by solving the Langevin equation (stochastic differential equation) one can treat a mode coupling in multimode low and high NA step index plastic optical fibres, which is the result of fibre's intrinsic random perturbations.     

Solution of mode coupling in step-index optical fibers by the Fokker-Planck equation and the Langevin equation

The power-flow equation is approximated by the Fokker-Planck equation that is further transformed into a stochastic differential (Langevin) equation, resulting in an efficient method for the estimation of the state of mode coupling along step-index optical fibers caused by their intrinsic perturbation effects. The inherently stochastic nature of these effects is thus fully recognized mathematically. The numerical integration is based on the computer-simulated Langevin force. The solution matches the solution of the power-flow equation reported previously. Conceptually important steps of this work include (i) the expression of the power-flow equation in a form of the diffusion equation that is known to represent the solution of the stochastic differential equation describing processes with random perturbations and (ii) the recognition that mode coupling in multimode optical fibers is caused by random perturbations.     

Approximate solution for nonlinear random vibration problems by partial stochastic linearization

The accuracy of the stochastic linearization methods is improved by the proposed method of partial stochastic linearization, in which only the nonlinear damping force in the original system is replaced by a linear viscous damping, while the nonlinear restoring force remains unchanged. The replacement is based on the criterion of equal mean work, performed by the nonlinear damping force in the original system and its linear counterpart. The resulting nonlinear stochastic differential equation is then solved exactly, keeping the equivalent damping coefficient as a parameter, which can be determined for a specific system by solving a nonlinear algebraic equation.