First passage failure of SDOF nonlinear oscillator with lightly fractional derivative damping under real noise excitations

The first passage failure of single-degree-of-freedom (SDOF) nonlinear oscillator with lightly fractional derivative damping under real noise excitations is investigated in this paper. First, the system state is approximately represented by one-dimensional time-homogeneous diffusive Markov process of amplitude through stochastic averaging. Then, the backward Kolmogorov equation governing the conditional reliability function and the Pontryagin equation governing the conditional mean of first passage time are established from the averaged Itô equation for Hamiltonian. The conditional reliability function, the conditional probability density and mean of the first passage time are obtained by solving these equations together with suitable initial condition and boundary conditions. Finally, two examples are worked out in detail and the analytical solutions are checked by those from the Monte Carlo simulation of original systems.     

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.     

First passage failure of quasi integrable-Hamiltonian systems under combined harmonic and white noise excitations

The first passage failure of multi-degree-of-freedom (MDOF) quasi integrable-Hamiltonian systems under combined harmonic and white noise excitations in the case of external resonance is studied. First, a stochastic averaging method for quasi integrable-Hamiltonian systems under combined harmonic and white noise excitations using generalized harmonic functions is reviewed briefly. Then, a backward Kolmogorov equation governing the conditional reliability function and a Pontryagin equation governing the conditional mean of the first passage time are established from the averaged Itô equations, respectively. The conditional reliability function, and the conditional probability density and conditional mean of the first passage time are obtained from solving these equations together with suitable initial condition and boundary conditions. The comparison between the analytical results and those from Monte Carlo simulation for an example shows that the proposed method works very well. It is also shown by using Monte Carlo simulation that the reliability of the system in the case of external resonance is much lower than that in the non-resonant case.     

First-passage failure of quasi linear systems subject to multi-time-delayed feedback control and wide-band random excitation

A procedure for studying the first-passage failure of quasi-linear systems subject to multi-time-delayed feedback control and wide-band random excitation is proposed. The stochastic averaging method for quasi-integrable Hamiltonian systems is first introduced. The backward Kolmogorov equation governing the conditional reliability function and a set of generalized Pontryagin equations governing the conditional moments of first-passage time are then established. The conditional reliability function, the conditional probability density and moments of first-passage time are obtained by solving the backward Kolmogorov equation and generalized Pontryagin equations with suitable initial and boundary conditions. An example is given to illustrate the proposed procedure and the results from digital simulation are obtained to verify the effectiveness of the proposed procedure. The effects of time delay in feedback control forces on the conditional reliability function, conditional probability density and moments of first-passage time are analyzed.     

粘弹性阻尼隔振体的非线性振动分析

 研究了由基础振动激励、粘弹性材料隔离的被动隔振体的非线性动力响应.用变形的三次多项式函数表征隔振材料的非线性刚度,用分数阶算子表征阻尼,建立被动隔振体的分数阶非线性动力学方程.用谐波平衡法研究其非线性动力响应特性,得出频率响应方程和幅频曲线,分析了非线性因素对系统的影响.最后,用Floquet理论讨论了周期解的稳态性和稳定区间.分析结果表明,含分数阶算子的动力学模型能够准确地描述粘弹性材料隔振器的动态特性.忽略隔振材料的阻尼非线性、刚度非线性将导致隔振设计和隔振效果分析的明显误差.分析方法和结论为精确进行粘弹性材料的被动隔振设计和隔振效果评价提供理论参考.     

含分数阶导数项的随机Duffing振子的稳态响应分析

本文对一个含有分数阶导数项阻尼的、Gaussian白噪声激励下的Duffing振子进行了稳态响应分析。首先,基于能量平衡理论,运用等效线性化方法,计算等效系统的线性阻尼及自然频率,建立统计意义下的等效线性化系统。然后,利用平均法建立随机Ito方程,得到随机响应的Markovian近似;给出描述振子振幅概率密度函数演化的Fokker-Planck方程,并得到它的稳态解。进一步,对于含有响应振幅的等效线性系统,借助由Laplace变换得到的转换函数,得到原系统的条件功率谱密度,结合振幅的稳态概率密度作为权重函数,给出原系统功率谱密度的估计,以及响应的统计量的估计。数值模拟的结果说明所提出的功率谱密度的近似解析表达式是可靠的,它甚至适用于Duffing振子具有强非线性回复力的情形,因为它可以较好的表现出功率谱密度共振频谱加宽及多峰现象的出现。     

Nonlinear differential equations with fractional damping with applications to the 1dof and 2dof pendulum

Summary. Existence, uniqueness and dissipativity is established for a class of nonlinear dynamical systems including systems with fractional damping. The problem is reduced to a system of fractional-order differential equations for numerical integration. The method is applied to a non-linear pendulum with fractional damping as well as to a nonlinear pendulum suspended on an extensible string. An example of such a fractional damping is a pendulum with the bob swinging in a viscous fluid and subject to the Stokes force (proportional to the velocity of the bob) and the Basset-Boussinesq force (proportional to the Caputo derivative of order 1/2 of the angular velocity). An existence and uniqueness theorem is proved and dissipativity is studied for a class of discrete mechanical systems subject to fractional-type damping. Some particularities of fractional damping are exhibited, including non-monotonic decay of elastic energy. The 2:1 resonance is compared with nonresonant behavior.     

Smart damping of geometrically nonlinear vibrations of composite shells using fractional order derivative viscoelastic constitutive relations

In this article, a three-dimensional fractional order derivative model has been developed for the constrained viscoelastic layer of the active constrained layer damping (ACLD) treatment of laminated composite shells undergoing geometrically nonlinear vibrations. The constraining layer of the ACLD treatment is made of vertically/obliquely reinforced 1–3 piezoelectric composites and acts as the distributed actuator. A three-dimensional smart nonlinear finite element model has been developed. Several numerical results are presented to check the accuracy of the present three-dimensional fractional derivative model of the constrained viscoelastic layer for smart damping of geometrically nonlinear vibrations of laminated composite shells.     

Homogenised finite element for transient dynamic analysis of unconstrained layer damping beams involving fractional derivative models

This paper presents a homogenised finite element formulation for the transient dynamic analysis of asymmetric and symmetric unconstrained layer damping beams in which the viscoelastic material is characterised by a five-parameter fractional derivative model. This formulation is based on the weighted residual method (Galerkin’s approach) providing a fractional matrix equation of motion. The application of Grünwald-Letnikov’s definition of the fractional derivatives allows to solve numerically the fractional equation by means of two different implicit formulations. Numerical examples for a cantilever beam with viscoelastic treatment are presented comparing the response provided by the proposed homogenised formulation with that of Padovan, based on the principle of virtual work. Different damping levels and load cases are analysed, as well as the influence of the truncation and time-step. From the numerical applications it can be concluded that the presented formulation allows to reduce significantly the degrees of freedom and consequently the computational time and storage needs for the transient dynamic analysis of structural systems in which damping treatments have been applied by means of viscoelastic materials characterised by fractional derivative models.     

Subset simulation for structural reliability sensitivity analysis

Based on two procedures for efficiently generating conditional samples, i.e. Markov chain Monte Carlo (MCMC) simulation and importance sampling (IS), two reliability sensitivity (RS) algorithms are presented. On the basis of reliability analysis of Subset simulation (Subsim), the RS of the failure probability with respect to the distribution parameter of the basic variable is transformed as a set of RS of conditional failure probabilities with respect to the distribution parameter of the basic variable. By use of the conditional samples generated by MCMC simulation and IS, procedures are established to estimate the RS of the conditional failure probabilities. The formulae of the RS estimator, its variance and its coefficient of variation are derived in detail. The results of the illustrations show high efficiency and high precision of the presented algorithms, and it is suitable for highly nonlinear limit state equation and structural system with single and multiple failure modes.     

Nonlinear dynamics of offshore structures under sea wave and earthquake forces

A study of dynamic response of offshore structures in random seas to inputs of earthquake ground motions is presented. Emphasis is placed on the evaluation of nonlinear hydrodynamic damping effects due to sea waves for the earthquake response. The structure is discretized using the finite element method. Sea waves are represented by Bretschneider’s power spectrum and the Morison equation defines the wave forcing function. Tajimi-Kanai’s power spectrum is used for the horizontal ground acceleration due to earthquakes. The governing equations of motion are obtained by the substructure method. Response analysis is carried out using the frequency-domain random-vibration approach. It is found that the hydrodynamic damping forces are higher in random seas than in still water and sea waves generally reduce the seismic response of offshore structures. Studies on the first passage probabilities of response indicate that small sea waves enhance the reliability of offshore structures against earthquakes forces.     

一类含有分数阶导数的参数激励振动问题

研究了一类具有分数阶导数阻尼的参数激励振动问题。对含有由Riemann-Liouville定义的分数阶导数的Mathieu振动方程构造渐近解。利用多重尺度法,在激励参数取不同值的情况下,求得渐近解,得到分数阶指数对解的影响。     

分数阶Ver Del Pol-Duffing系统的非线性动力学行为分析

利用分数导数本构模型模拟系统的阻尼特性,构造了分数阶Ver Del Pol-Duffing系统,探讨了系统的动力特性随特征参数的变化规律。分析发现：该非线性振子具有与经典Ver Del Pol系统相似的自激振动特性,但其非线性强弱受分数导数阶值以及阻尼系数和非线性大位移系数的影响;在简谐荷载作用下,随着外荷载幅值的增大或阻尼系数的减小,系统由拟周期振动变为周期三振动最后发展为单周期振动;在地震荷载作用下,分数导数阶值的变化能改变系统的输出能量。     

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.     

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.     

Survival probability determination of nonlinear oscillators with fractional derivative elements under evolutionary stochastic excitation

An approximate analytical technique based on a combination of statistical linearization and stochastic averaging is developed for determining the survival probability of stochastically excited nonlinear/hysteretic oscillators with fractional derivative elements. Specifically, approximate closed form expressions are derived for the oscillator non-stationary marginal, transition, and joint response amplitude probability density functions (PDF) and, ultimately, for the time-dependent oscillator survival probability. Notably, the technique can treat a wide range of nonlinear/hysteretic response behaviors and can account even for evolutionary excitation power spectra with time-dependent frequency content. Further, the corresponding computational cost is kept at a minimum level since it relates, in essence, only to the numerical integration of a deterministic nonlinear differential equation governing approximately the evolution in time of the oscillator response variance. Overall, the developed technique can be construed as an extension of earlier efforts in the literature to account for fractional derivative terms in the equation of motion. The numerical examples include a hardening Duffing and a bilinear hysteretic nonlinear oscillators with fractional derivative terms. The accuracy degree of the technique is assessed by comparisons with pertinent Monte Carlo simulation data.     

非广延等离子体中非线性离子声波的数值研究：分数阶模型（英）

本文的主要目的是研究非磁化等离子体中离子声波的非局部非线性行为,该非磁化等离子体是由正离子、满足非广延分布的电子、以及带有负电荷的静止的尘埃颗粒构成的．在流体力学基本方程组中,本文引入修正的Riemann-Liouville分数阶导数并建立了分数阶模型,结合约化摄动法推导出描述离子声波运动的Korteweg　de Vries (Kdv)方程．本文采用Chebyshev-Legendre-Galerkin (CLG)拟谱方法数值求解该方程,并分析等离子体参数对离子孤立声波结构的影响．本文的研究结果表明：提高分数阶导数的阶数能够提升孤立波的振幅．该结果将有助于更好地理解天体物理和实验室等离子体中的非线性波动现象．     

具有分数导数本构关系的粘弹性浅拱的非线性动力学行为

利用分数导数本构模型描述材料的粘弹性特性,建立了粘弹性浅拱在横向荷载作用下的动力学方程。利用Galerkin截断法并结合边界条件分别得到了一阶和二阶Galerkin系统的控制微分方程。通过数值计算,分析了简谐激励下一阶Galerkin系统的非线动力学行为。研究表明:随着外激励幅值的变化,粘弹性浅拱系统可以通过倍周期分岔或阵发性两条路径进入混沌;固定外激励幅值、频率以及阻尼系数等状态参数,不同初始条件下,系统可以出现多周期解共存、周期解与混沌解共存的现象。     

A Wiener path integral technique for determining the stochastic response of nonlinear oscillators with fractional derivative elements: A constrained variational formulation with free boundaries

A technique based on the concept of Wiener path integral (WPI) is developed for determining approximately the joint response probability density function (PDF) of nonlinear oscillators endowed with fractional derivative elements. Specifically, first, the dependence of the state of the system on its history due to the fractional derivative terms is accounted for, alternatively, by augmenting the response vector and by considering additional auxiliary state variables and equations. In this regard, the original single-degree-of-freedom (SDOF) nonlinear system with fractional derivative terms is cast, equivalently, into a multi-degree-of-freedom (MDOF) nonlinear system involving integer-order derivatives only. From a mathematics perspective, the equations of motion referring to the latter can be interpreted as constrained. Second, to circumvent the challenge of increased dimensionality of the problem due to the augmentation of the response vector, a WPI formulation with mixed fixed/free boundary conditions is developed for determining directly any lower-dimensional joint PDF corresponding to a subset only of the response vector components. This can be construed as an approximation-free dimension reduction approach that renders the associated computational cost independent of the total number of stochastic dimensions of the problem. Thus, the original SDOF oscillator joint PDF corresponding to the response displacement and velocity is determined efficiently, while circumventing the computationally challenging task of treating directly equations of motion involving fractional derivatives. Two illustrative numerical examples are considered for demonstrating the reliability of the developed technique. These pertain to a nonlinear Duffing and a nonlinear vibro-impact oscillators with fractional derivative elements subjected to combined stochastic and deterministic periodic loading. Note that alternative standard approximate techniques, such as statistical linearization, need to be significantly modified and extended to account for such cases of combined loading. Remarkably, it is shown herein that the WPI technique exhibits the additional advantage of treating such types of excitation in a straightforward manner without the need for any ad hoc modifications. Comparisons with pertinent Monte Carlo simulation data are included as well.     

Stability,control and reliability of a ship crane payload motion

This paper investigates the stochastic dynamics, stability and control of a ship-based crane payload motion, as well as the first time passage type of failure. The simplified nonlinear model of the payload motion is considered, where the excitation of a suspension point is imposed due to the heaving motion of waves. The latter enters the system parametrically, leading to a Mathieu type nonlinear equation. The stability boundaries are numerically calculated, using the Lyapunov exponent approach. The control strategy, based on the feedback bang–bang control policy, is implemented to minimize the load's swinging motion. Finally, the first time passage problem is addressed employing Monte-Carlo sampling of the failure process.