Mean-square stability of linear stochastic dynamical systems under parametric wide-band excitations

A new simplified condition is developed for determining the exponential meansquare stability margins of linear stochastic dynamical systems. It is well-known that under parametric wide-band noise disturbances, the governing equations of motion of such a system can be approximated by linear Itô stochastic differential equations (SDE). A necessary and sufficient condition for exponential mean square stability of the resulting ltô SDE is that the real parts of all the eigenvalues of the matrix describing the system of second-order moments are negative. Equivalently, the Routh-Hurwitz procedure provides conditions for stability in the form of several inequalities. In this study, it is shown that a necessary condition for the system configuration to correspond to a point on the stability boundary is that the determinant of the matrix describing the system of secondorder moments be zero. This condition is a single algebraic expression allowing for the straightforward calculation of all candidate stability boundaries. In addition, the topological properties of the stability domain are presented and shown to be useful in identifying stability boundaries and stability domains from the developed single stability boundary condition. This simplified condition provides significant advantages in the analytical and numerical estimation of the stability border and stability region of dynamical systems. The usefulness and superiority of the new condition is demonstrated by applications to example dynamical systems, including a long-span bridge model subjected to turbulent wind.     

Effects of distributed delays on the stability of structures under seismic excitation and multiplicative noise

The effect of seismic excitation and multiplicative noise (arising from environmental fluctuations) on the stability of a single degree of freedom system with distributed delays are investigated. The system is modelled in the form of a stochastic integro-differential equation interpreted in Stratonovich sense. Both deterministic stability and stochastic moment stability are examined for the system in the absence of seismic excitation. The model is also extended to incorporate effects of symmetric nonlinearity. The simulation of stochastic linear and nonlinear systems is carried out by resorting to numerical techniques for the solution of stochastic differential equations. Part of the work was done during the visit of the second author to the Jawaharal Nehru University, New Delhi in December 1993, for which the authors would like to thank the University.     

Linear systems driven by martingale noise

A method is developed for calculating second moment properties and moments of order three and higher of the state X of a linear filter driven by martingale noise. The martingale noise is interpreted as the formal derivative of a square integrable martingale with continuous samples. The Gaussian white noise is an example of a martingale noise. It is shown that the differential equations of the mean and correlation functions of the state X developed in the paper resemble the corresponding equations of the classical linear random vibration and coincide with these equations if the input is a Gaussian white noise. The moment equations are derived by (1) the Itô formula for semimartingales and (2) the classical Itô formula applied to a diffusion process whose coordinates include X. An advantage of the second method is use of more familiar concepts. However, this method requires to calculate unnecessary moments and can be applied only for a class of martingale noise processes. Examples are presented to illustrate and evaluate the two methods for calculating moments of X and demonstrate the use of these methods in linear random vibration.     

有界窄带激励柴油机轴系扭振系统主参数共振

研究柴油机轴系扭振强非线性系统在有界窄带激励下的主参数共振响应和稳定性问题。应用改进多尺度法得到在有界窄带随机激励下柴油机轴系扭振系统的幅频响应方程,导出系统的Ito随机微分方程。通过矩法得到系统随机均方响应的近似表达式,分析各个参数对柴油机轴系扭振系统主参数共振均方值的影响。结果表明,主参数共振稳态解稳定的充分必要条件与系统二阶矩稳定的充分必要条件是一样的;随着阻尼值的增大,系统主参数共振振幅的均方值减小;随着曲轴扭转刚度的减小,系统主参数共振的均方响应曲线的斜率增大;随着随机扰动强度的增大,系统时间响应曲线和相图变化微小。     

一类具离散时变时滞和分布时滞神经网络的指数稳定性(英文)

本文讨论一类具离散时变时滞和分布时滞神经网络的指数稳定性。利用非线性测度,本文得到一个与时滞无关的充分条件,它保证了平衡点的存在性、唯一性和指数稳定性。既然新稳定准则不要求激活函数的有界性、单调性及可微性和随时间改变的传递延迟函数的可微性,那么它是某些已有结果的推广。此外,本文的方法的另一个优点是给出了解的指数收敛速度。最后,给出的例子说明我们的方法是有效的。     

Linear systems with polynomials of filtered Poisson processes

Moment equations are calculated exactly for the response of linear systems subjected polynomials of filtered Poisson processes. The Itô formula for stochastic differential equations driven by Poisson white noise is applied to derive moment equations. It is shown that the set of moment equations is closed. The proposed method is used to calculate moments up to the fourth order for the response of two linear systems subjected to quadratic forms of filtered Poisson processes. Results by Monte Carlo simulations are also presented for comparison.     

A quasi‐linear reproducing kernel particle method

Reproducing kernel particle method (RKPM) has been applied to many large deformation problems. RKPM relies on polynomial reproducing conditions to yield desired accuracy and convergence properties but requires appropriate kernel support coverage of neighboring nodes to ensure kernel stability. This kernel stability condition is difficult to achieve for problems with large particle motion such as the fragment‐impact processes that commonly exist in extreme events. A new reproducing kernel formulation with ‘quasi‐linear’ reproducing conditions is introduced. In this approach, the first‐order polynomial reproducing conditions are approximately enforced to yield a nonsingular moment matrix. With proper error control of the first‐order completeness, nearly second‐order convergence rate in L2 norm can be achieved while maintaining kernel stability. The effectiveness of this quasi‐linear RKPM formulation is demonstrated by modeling several extremely large deformation and fragment‐impact problems. Copyright © 2016 John Wiley & Sons, Ltd.     

单摆系统的多时滞控制与特征根分析

对单摆系统在位置反馈和时滞位置反馈下的镇定问题进行了研究。考虑到控制器本身存在滞后现象这一实际因素,构建了含有两个时滞的二阶时滞微分方程模型;建立了系统的特征方程,采用特征根分析方法以及解析函数的零点重数之和关于参数的连续依赖性,得到了系统参数及各个时滞值与系统稳定性之间的关系,给出了与时滞相关的稳定性条件及与时滞无关的稳定性条件;结合代数方程求根技巧,分析了特征根的重数,得到了特征根的重数至多为 4 的参数条件,并证明了三重特征根至多有两个;在系统参数满足稳定性条件的前提下,证明了当特征根的模趋于无穷大时,必有特征根的实部趋于负无穷大;利用渐近分析技巧,计算出了当特征根的实部趋于负无穷大时特征根的渐近表达式。最后,通过 Matlab 数值仿真验证了结论的有效性,这一方法可进一步推广到含有多个时滞的 $n$ 阶微分方程系统。     

Stochastic stability of SDOF linear viscoelastic system under wideband noise excitation

The stochastic moment stability and almost-sure stability of a single-degree-of-freedom (SDOF) viscoelastic system subject to parametric fluctuation is investigated by using the method of higher-order stochastic averaging. The stochastic parametric excitation is modeled as a wideband noise, which is taken as Gaussian white noise and real noise. The viscoelastic material is assumed to follow ordinary Maxwell linear constitutive relation. For small damping and weak stochastic fluctuation, analytical expressions are derived for the moment Lyapunov exponent and the Lyapunov exponent, which indicate moment stability and almost-sure stability respectively. The effects of various system and loading parameters on the stochastic stability are discussed. Both analytical and simulation results show that higher-order stochastic averaging improves the accuracy compared with the first-order stochastic averaging. However, results of the third-order averaging are almost overridden by those of second-order averaging and the third-order averaging involves far more calculation. It is advisable to consider a balance between accuracy achievement and calculation endeavor when using higher-order stochastic averaging.     

Stochastic modal interaction in linear and nonlinear aeroelastic structures

The linear and autoparametric modal interactions in a three defree-of-freedom structure under wide band random excitation are examined. For a structure with constant parameters the linear response is obtained in a closed form. When the structure stiffness matrix involves random fluctuations, the governing equations of motion, in terms of the normal coordinates, are found to be coupled through parametric terms. The structural response is mainly governed by the condition of mean square stability. The boundary of stable-unstable responses is obtained as a function of the internal detuning parameter. The results of the linear system with constant parameters are used as a reference to measure the deviation of the system response when the nonlinear inertia coupling is included. In the neighbourhood of combination internal resonance the system random response is determined by using the Fokker Planck equation approach together with the Gaussian closure scheme. This approach results in 27 coupled first order differential equations in the first and second response moments. These equations are solved numerically. The response is found to deviate significantly from the linear solution when the system internal detuning is close to the exact internal resonance. The autoparametric interaction is found to depend significantly on the system damping ratios and a nonlinear coupling parameter. In the vicinity of combination internal resonance, the second normal mode mean square exhibits an increase associated with a corresponding decrease in the first and third normal modes.     

A Comparison of Stability and Bifurcation Criteria for a Compressible Elastic Cube

A version of Rivlin’s cube problem is considered for compressible materials. The cube is stretched along one axis by a fixed amount and then subjected to equal tensile loads along the other two axes. A number of general results are found. Because of the homogeneous trivial and non-trivial deformations exact bifurcation results can be found and an exact stability analysis through the second variation of the energy can be performed. This problem is then used to compare results obtained using more general methods. Firstly, results are obtained for a more general bifurcation analysis. Secondly, the exact stability results are compared with stability results obtained via a new method that is applicable to inhomogeneous problems. This new stability method allows a full nonlinear stability analysis of inhomogeneous deformations of arbitrary, compressible or incompressible, hyperelastic materials. The second variation condition expressed as an integral involving two arbitrary perturbations is replaced with an equivalent nonlinear third order system of ordinary differential equations. The positive definiteness condition is thereby reduced to the simple numerical evaluation of zeros of a well behaved function.     

Asymptotic Lyapunov stability with probability one of Duffing oscillator subject to time-delayed feedback control and bounded noise excitation

The asymptotic Lyapunov stability with probability one of a Duffing system with time-delayed feedback control under bounded noise parametric excitation is studied. First, the time-delayed feedback control force is expressed approximately in terms of the system state variables without time delay. Then, the averaged Itô stochastic differential equations for the system are derived by using the stochastic averaging method and the expression for the Lyapunov exponent of the linearized averaged Itô equations is derived. It is inferred that the Lyapunov exponent so obtained is the first approximation of the largest Lyapunov exponent of the original system, and the asymptotic Lyapunov stability with probability one of the original system can be determined approximately by using the Lyapunov exponent. Finally, the effects of time delay in feedback control on the Lyapunov exponent and the stability of the system are analyzed. The theoretical results are well verified through digital simulation.     

Some aspects of chaotic and stochastic dynamics for structural systems

In this paper, the bifurcation behaviour of an externally excited four-dimensional nonlinear system is examined. Throughout this paper, a two-degree-of-freedom shallow arch structure under either a periodic or a stochastic excitation will be considered. For the case when the excitation is periodic, the local and global behaviour is examined in the presence of principalsubharmonic resonance and1:2 internal resonance. The method of averaging is used to obtain the first order approximation of the response of the system under resonant conditions. A standard Melnikov type perturbation method is used to show analytically that the system may exhibit chaotic dynamics in the sense of Smale horseshoe for the 1:2 internal resonance case in the absence of dissipation. In the case of stochastic excitation, the stability of the stationary solution is examined by determining themaximal Lyapunov exponent andmoment Lyapunov exponent in terms of system parameters. An asymptotic method is used to obtain explicit expressions for various exponents in the presence of weak dissipation and noise intensity. These quantities provide almost-sure stability boundaries in parameter space. When the system parameters lie outside these boundaries, it is essential to understand the nonlinear behaviour. The method of stochastic averaging is applied to obtain a set of approximate Itô equations which are then examined to describe the local bifurcation behaviour.     

Investigation of the lipschitz stability via limiting equations

This work is dedicated to an effective method for investigation of the stability of solutions of ordinary differential equations. The notions of strong Lipschitz stability and strong uniform Lipschitz stability are introduced. Two theorems are proved. The first contains sufficient conditions under which the strong Lipschitz stability of the zero solution of each of the respective limiting equations implies stability of the zero solution of the initial equation. In the second theorem sufficient conditions are given under which from the uniform Lipschitz stability of the zero solution of the initial equation there follows uniform Lipschitz stability of the zero solution of each one of the respective limiting equations     

A finite element model for plane elasticity problems using the complementary energy theorem

A finite element stress analysis capability for plane elasticity problems, employing the principle of stationary complementary energy, is developed. Two models are investigated. The first is a 24 d.o.f. rectangular finite element. The second model consists of an 18 d.o.f. triangular element. In order to allow for self-equilibrating stresses which are continuous within the element, the well-known Airy stress function ø is used. The function ø is represented by means of quintic Hermitian polynomials within the finite element. The values of the ø function and its derivatives up to order two are used as nodal parameters. For matching the stress function with the prescribed boundary tractions, additional equations are developed considering the force and moment equilibrium equations on the boundary consistent with the assumed stress function. These additional boundary equations are incorporated into the system equations using the Lagrangian multiplier technique. Excellent results are obtained for linear elastic problems even with coarse finite element discretization. Some examples of plane elasticity problems are solved and results compared.     

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.     

An automatic formulation of inverse free second moment method for algebraic systems

In systems with probabilistic uncertainties, an estimation of reliability requires at least the first two moments. In this paper, we focus on probabilistic analysis of linear systems. The important tasks in this analysis are the formulation and the automation of the moment equations. The main objective of the formulation is to provide at least means and variances of the output variables with at least a second-order accuracy. The objective of the automation is to reduce the storage and computational complexities required for implementing (automating) those formulations. This paper extends the recent work done to calculate the first two moments of a set of random algebraic linear equations by developing a stamping procedure to facilitate its automation. The new method has an additional advantage of being able to solve problems when the mean matrix of a system is singular. Lastly, from storage and computational complexities and accuracy point of view, a comparison between the new method and another recently developed first order second moment method is made with numerical examples.     

Finite element formulation of exact non‐reflecting boundary conditions for the time‐dependent wave equation

A modified version of an exact Non‐reflecting Boundary Condition (NRBC) first derived by Grote and Keller is implemented in a finite element formulation for the scalar wave equation. The NRBC annihilate the first N wave harmonics on a spherical truncation boundary, and may be viewed as an extension of the second‐order local boundary condition derived by Bayliss and Turkel. Two alternative finite element formulations are given. In the first, the boundary operator is implemented directly as a ‘natural’ boundary condition in the weak form of the initial–boundary value problem. In the second, the operator is implemented indirectly by introducing auxiliary variables on the truncation boundary. Several versions of implicit and explicit time‐integration schemes are presented for solution of the finite element semidiscrete equations concurrently with the first‐order differential equations associated with the NRBC and an auxiliary variable. Numerical studies are performed to assess the accuracy and convergence properties of the NRBC when implemented in the finite element method. The results demonstrate that the finite element formulation of the (modified) NRBC is remarkably robust, and highly accurate. Copyright © 1999 John Wiley & Sons, Ltd.     

Comparison of Dynamic Differential Evolution and Self-Adaptive Dynamic Differential Evolution for Buried Metallic Cylinder

The application of two techniques for the reconstruction of shape reconstruction of a metallic cylinder from scattered field measurements is studied in this paper. These techniques are applied to two-dimensional configurations, for which the method of moment (MoM) is applied to solve the integral equations. Considering that the microwave imaging is recast as a nonlinear optimization problem, an objective function is defined by the norm of the difference between the measured scattered electric fields and those calculated for each estimated metallic cylinder. Thus, the shape of a metallic cylinder can be obtained by minimizing the objective function. In order to solve this inverse scattering problem, two techniques are employed. The first one is based on dynamic differential evolution (DDE) algorithm, while the second one is an improved version of the DDE algorithm with self-adaptive control parameters, called SADDE. Both techniques are tested for the simulated data contaminated by additive white Gaussian noise. Numerical results indicate that SADDE algorithm outperforms DDE algorithm in terms of reconstruction accuracy and convergence speed.     

Properties of an oscillator slaved to a periodically interrogated atomic resonator

In advanced atomic resonators, such as those using a fountain of cold cesium atoms or an ensemble of stored ions, the atomic medium is interrogated periodically, and the control signal of the slaved oscillator is updated at equally spaced time intervals. We analyze the properties of the output frequency of these frequency standards. We establish the equations that describe the time behavior of this frequency. We give the stability condition and the transient response of the frequency feedback loop, the response to systematic frequency changes of the free running oscillator, the frequency stability for given free-running oscillator noise and given optical detection noise, and the limitation of the frequency stability by down-conversion of the intrinsic oscillator frequency noise (Dick effect). We point out that a second integration in the feedback loop may not improve significantly the rejection of slow perturbations, unless a condition relative to the timing of the atom-field interaction is verified.