Wavelet basis expansion-based spatio-temporal Volterra kernels identification for nonlinear distributed parameter systems

In practical industries, there are many systems belong to nonlinear distributed parameter systems (DPS); unfortunately, modeling of nonlinear DPS is a challenging task because of the infinite-dimensional and nonlinear properties. To model the nonlinear DPS, a spatio-temporal Volterra model is presented with a series of spatio-temporal kernels. It can be considered as a spatial extension of the traditional Volterra model. One question involved in modeling a spatio-temporal functional relationship between the input and output of nonlinear distributed parameter systems using spatio-temporal Volterra series is to identify the spatio-temporal Volterra kernel functions. In addition, in order to derive a low-order model, the Karhunen–Loève (KL) decomposition is used for the time/space separation. The basic routine of the approach is that, first, from the system outputs, KL decomposition is used for the time/space separation, where the spatio-temporal output is decomposed into a few dominant spatial basis functions with temporal coefficients. Second, according to temporal coefficients of outputs under multilevel excitations, the Volterra series outputs of different orders are estimated with the wavelet balance method. Third, the Volterra kernel functions of different orders are separately estimated through their corresponding Volterra series outputs by expanding them with four-order B-spline wavelet on the interval (BSWI). Finally, the spatio-temporal Volterra model can be reconstructed using the time/space synthesis. The simulation studies verify the effectiveness of the presented identification method.     

Wavelet basis expansion-based Volterra kernel function identification through multilevel excitations

Volterra series is a powerful mathematical tool for nonlinear system analysis, which extends the convolution integral for linear system to nonlinear system. There is a wide range of nonlinear engineering systems and structures which can be modeled as Volterra series. One question involved in modeling a functional relationship between the input and output of a system using Volterra series is to identify the Volterra kernel functions. In this article, a wavelet balance method-based approach is proposed to identify the Volterra kernel functions from observations of the in- and outgoing signals. The basic routine of the approach is that, from the system outputs under multilevel excitations, the Volterra series outputs of different orders are first estimated with the wavelet balance method, and then the Volterra kernel functions of different orders are separately estimated through their corresponding Volterra series outputs by expanding them with four-order B-spline wavelet on the interval. The simulation studies verify the effectiveness of the proposed Volterra kernel identification method.     

A nonlinear analysis framework for bluff-body aerodynamics: A Volterra representation of the solution of Navier-Stokes equations

A nonlinear analysis framework for bluff-body aerodynamics based on Volterra theory is introduced to capture the linear and nonlinear aerodynamic effects. The Volterra kernels based on the impulse function concept are identified by way of the simulation of Navier-Stokes equations using computational fluid dynamics (CFD). The computational schemes used here are validated through theoretical consideration, i.e., Blasius solution for the steady-state and Theodorsen solution for the system dynamic-state simulation. The source of nonlinearities in the aerodynamics of bluff bodies is systematically investigated. The simulation of bluff-body aerodynamics based on the Volterra reduced-order modeling scheme is obtained by the convolution of the identified kernels with the external inputs, e.g., turbulent inflow or body motion for aerodynamic or aeroelastic response, respectively. It is demonstrated that the Volterra theory-based nonlinear analysis framework for bluff-body aerodynamics combined with the identification of kernels using CFD promises to capture the salient features of bluff-body aerodynamics and offers an accurate reduced-order approximation of the Navier-Stokes equations with reduced level of computational effort.     

Volterra Series Approximation of a Class of Nonlinear Dynamical Systems Using the Adomian Decomposition Method

The relationship between the Adomian decomposition and the Volterra series is investigated and it is shown that the Volterra series can be considered as a specialization of the Adomian decomposition. Based on the relationship, the Volterra series can be calculated using an Adomian decomposition method whenever a convergent Volterra series representation exists. A class of nonlinear dynamical systems is considered and a new algorithm is introduced to compute the Volterra series representation for this class of nonlinear systems. The new method significantly simplifies the computation of the Volterra kernels and provides a new choice for the study of Volterra series of nonlinear dynamical systems.     

A new frequency domain representation and analysis for subharmonic oscillation

For a weakly nonlinear oscillator, the frequency domain Volterra kernels, often called the generalized frequency response functions, can provide accurate analysis of the response in terms of amplitudes and frequencies, in a transparent algebraic way. However, a Volterra series representation based analysis will become void for nonlinear oscillators that exhibit subharmonics, and the problem of finding a solution in this situation has mainly been treated by traditional analytical approximation methods. In this paper, a novel method is developed, by introducing a frequency domain subharmonic kernel representation for subharmonic systems subject to a single tone excitation frequency, to allow the advantages and the benefits associated with the traditional frequency domain representations to be applied to severely nonlinear systems that exhibit subharmonic behavior.     

Identification of Nonlinear Aeroelastic Systems Based on the Volterra Theory: Progress and Opportunities

The identification of nonlinear aeroelastic systems based on the Volterra theory of nonlinear systems is presented. Recent applications of the theory to problems in computational and experimental aeroelasticity are reviewed. Computational results include the development of computationally efficient reduced-order models (ROMs) using an Euler/Navier–Stokes flow solver and the analytical derivation of Volterra kernels for a nonlinear aeroelastic system. Experimental results include the identification of aerodynamic impulse responses, the application of higher-order spectra (HOS) to wind-tunnel flutter data, and the identification of nonlinear aeroelastic phenomena from flight flutter test data of the active aeroelastic wing (AAW) aircraft.     

Nonlinear analytical multi-dimensional convolution solution of the second order system

In this paper, a procedure to analytically develop an approximate solution for the prototypical nonlinear mass–spring–damper system based on multi-dimensional convolution expansion theory is offered. The nonlinearity herein is mathematically considered in quadratic and bilinear terms. A variational expansion methodology, one of the most efficient analytical Volterra techniques, is used to develop an analytical two-term Volterra series. The resultant model is given in the form of first and second kernels. This analytical solution is visualized in the time domain followed by a parametric study for understanding the influence of each nonlinear/linear term appearing in the kernel structure. An analytical nonlinear step response is also conducted to characterize the overall system response from the fundamental components. The developed analytical step response provides an illumination for the source of differences between nonlinear and linear responses such as initial departure time, settling time, and steady value. Feasibility of the proposed implementation is assessed by numerical examples. The developed kernel-based model shows the ability to predict, understand, and analyze the system behavior beyond that attainable by the linear-based model.     

基于非线性状态空间辨识的气动弹性模型降阶

高维、非线性气动弹性系统的模型降阶是当前气动弹性力学与控制领域的研究热点之一.然而国内外现有的非线性模型降阶方法仍存在辨识算法复杂、精度有待提高等问题.本研究提出了一种基于非线性状态空间辨识的跨音速气动弹性模型降阶方法.首先,该方法基于非定常空气动力的单位脉冲响应数据,采用特征系统实现算法对非线性状态空间模型的线性动力学部分进行系统辨识.其次,引入状态和控制输入的非线性函数,采用优化算法对非线性函数的系数矩阵进行优化,进而得到考虑非线性效应的空气动力降阶模型.为了验证该降阶模型在预测跨音速气动弹性力学行为的精确性,本文以三维机翼为研究对象,分别从基于非线性降阶模型的气动力辨识、跨声速颤振边界计算和极限环振荡预测三方面进行了算例验证,并与现有的模型降阶方法进行了对比,进一步说明本文所提出方法的有效性.研究结果表明,该降阶模型对上述三类问题的计算精度与直接流-固耦合方法相吻合,可用于高效预测飞行器跨声速气动弹性力学行为.     

多输入多输出非线性系统的一种函数级数模型

提出一种表示多输入多输出非线性多自由度系统的输入与输出关系的函数级数模型，并对模型作了讨论，证明了单输入单输出Ｖｏｌｔｅｒａ级数模型只是本文的一种特殊情况。用这个模型对多自由度多项式刚度非线性振动系统进行分析，得出这一类系统的高阶核和多维频率响应函数。文中模型为具有多输入多输出的非线性系统分析及响应预测与谱分析提供了一种基础工具。     

A Volterra series approach to the frequency domain analysis of non-linear viscous Burgers’ equation

In this paper, higher order frequency response functions, based on the Volterra series, are employed to characterise the input-output behaviour of the non-linear viscous Burgers?? equation subject to sinusoidal excitation. First, a formal Volterra series representation for each spatial location is derived for the solution of Burgers?? equation with a boundary condition as the input to the system. Then a systematic method is presented to obtain the higher order frequency kernels of the Volterra series at each spatial location by solving a series of ordinary differential equations. It is shown that the convergence region of the individual harmonics with respect to the magnitude of the input excitation can be estimated by using these higher order kernels. The frequency characteristics of Burgers?? equation is investigated and compared with numerical simulation.     

A Volterra Series Approach to the Approximation of Stochastic Nonlinear Dynamics

A response approximation method for stochastically excited, nonlinear, dynamic systems is presented. Herein, the output of the nonlinear system isapproximated by a finite-order Volterra series. The original nonlinear system is replaced by a bilinear system in order to determine the kernels of this series. The parameters of the bilinear system are determined by minimizing, in a statistical sense,the difference between the original system and the bilinear system. Application to a piecewise linear modelof a beam with a nonlinear one-sided supportillustrates the effectiveness of this approach in approximatingtruly nonlinear, stochastic response phenomena in both the statistical momentsand the power spectral density of the response of this system in case ofa white noise excitation.     

Finite memory quadratic Volterra model for the response prediction of a slender marine structure under a Morison load

A quadratic Volterra model with a finite nonlinear memory effect was introduced and applied to the time series prediction of a slender marine structure exposed to the Morison load. First, the unknown nonlinear single-input–single-output dynamic system was identified using the nonlinear autoregressive with exogenous input (NARX) technique based on the prepared datasets of the wave elevation and system response, which was obtained by running nonlinear time domain analysis for a certain short term sea state. The structure of NARX was designed in such a way that the linear part had infinite memory, whereas the nonlinear part had finite memory of a certain length. Second, the frequency domain Volterra kernels, both linear and quadratic, were derived analytically by applying the harmonic probing method to the identified system. To derive the frequency response functions, the sigmoidal function used in NARX to realize the nonlinear relationship between the input and output was expanded to polynomials based on the Taylor series expansion, so that the harmonics of same frequencies were easily matched between the input and output. Finally, the time series of the system response under arbitrarily given short term sea states were predicted using the quadratic Volterra series. The proposed methodology was used to predict the nonlinear dynamic response of a 2-dimentional free standing catenary riser exposed to a random ocean wave load, and the comparison between the prediction and simulation results was made on the probability distribution of the maximum excursion of riser top. The results show that the proposed methodology can successfully capture the nonlinear effects of the dynamic response of a slender marine structure induced by the quadratic term of the Morison formula.     

Identification of a localized non-linearity

The great range of non-linearity sources had led to a high number of identification techniques. In this paper we deal with a particular case: the identification of non-linearity localized in structures. The approach is based on the Volterra kernels which will be expressed analytically in the case of multidimensional systems in non-linear feedback. Using some examples, we will show that, by means of random tests, it is possible to find all the specific transfer functions of the problem, together with the non-linearities brought into play. The main advantage of such a method lies in the fact that the first three Volterra kernels are enough to get a complete identification of the system. This asset makes the Volterra series approach very efficient by removing its main drawback, i.e. heavy calculations.     

Third-order nonlinear singularly perturbed boundary value problem

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A WAVELET THRESHOLDING DENSITY-BASED ADAPTIVE IMPORTANCE SAMPING METHOD

提出了一种基于小波阈值密度估计的结构可靠性分析高效自适应重要抽样方法.该方法利用非线性小波收缩方法对结构失效域样本进行密度估计,并以此作为重要抽样密度进行可靠性分析.与传统基于核密度估计的重要抽样方法比,由于非线性小波阈值密度估计具有较好局部适应性和最优收敛速度,且克服了核密度估计中计算精度严重依赖于参数选择的缺陷,因此以较少的预抽样样本就能获得与传统方法相当的精度,有效提高计算效率.数值算例表明所提方法对工程中常遇到的多设计点及噪音功能函数可靠性问题具有良好适应性.     

基于小波阈值密度的自适应重要抽样方法

提出了一种基于小波阈值密度估计的结构可靠性分析高效自适应重要抽样方法.该方法利用非线性小波收缩方法对结构失效域样本进行密度估计,并以此作为重要抽样密度进行可靠性分析.与传统基于核密度估计的重要抽样方法比,由于非线性小波阈值密度估计具有较好局部适应性和最优收敛速度,且克服了核密度估计中计算精度严重依赖于参数选择的缺陷,因此以较少的预抽样样本就能获得与传统方法相当的精度,有效提高计算效率.数值算例表明所提方法对工程中常遇到的多设计点及噪音功能函数可靠性问题具有良好适应性.      

The integral representation of fractionally exponential functions and their application to dynamic problems of linear visco-elasticity

Fractionally exponential functions are written in the integral form and distribution functions with an Abelian singularity are obtained for the corresponding relaxation and retardation spectra. A principle is stated, defining the dynamic problems for which weakly singular functions can be used as the kernels of the integral operators. A one-dimensional sound wave traveling in a semiinfinite visco-elastic medium is considered. The generalized exponential functions of fractional order, proposed by Yu. N. Rabotnov [1, 2] as the kernels of Boltzmann-Volterra integral relations, have found wide applications in theory of linear visco-elasticity. This is explained partly by the great mathematical flexibility of the F-operators when applying the Volterra principle to the solution of elastically hereditary problems and partly by the fact that almost all weakly singular kernels possessing an Abelian singularity are connected in some way or other with the F-functions. For example, the resolvent of the elementary weakly singular Abelian kernel is an F-function. The product of an exponential function with an Abelian kernel represents a particular case of the product of two F-functions with different fractional parameters, while the resolvent of such a kernel is the product of an exponential function with an F-function [3, 4]. Since the e-functions are defined by slowly convergent series, their various asymptotic forms [2, 5–8] are commonly used in practical calculations. The theory of F-functions can be developed further in the context of their integral representations, which enables a more exact physical interpretation to be given to their parameters and on occasion simplifies computational operations.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 11, No. 1, pp. 103–110, January–February, 1970.In conclusion, the author thanks Yu. N. Rabotnov for discussion.     

An improved nonlinear reduced-order modeling for transonic aeroelastic systems

In this study, an improved nonlinear reduced-order model composed of a linear part and a nonlinear part is explored for transonic aeroelastic systems. The linear part is identified via the eigensystem realization algorithm and the nonlinear part is obtained via the Levenberg–Marquardt algorithm. The impulsive signal is chosen as the training signal for the linear part and the sinusoidal signal is used to determine the order of the linear part. The training signal for the nonlinear part is selected as the filtered white Gaussian noise with the maximal amplitude and frequency range to be designed via the aeroelastic responses. An NACA64A010 airfoil and an NACA0012 airfoil are taken as illustrative examples to demonstrate the performance of the presented reduced-order model in modeling transonic aerodynamic forces. The aeroelastic behaviors of the two airfoils are obtained via computational fluid dynamics to solve the Euler equation and the Navier–Stokes equation, respectively. The numerical results demonstrate that the presented reduced-order model can successfully predict the nonlinear aerodynamic forces with and without viscous flows. Moreover, the presented reduced-order model is capable of capturing the flutter velocity and modeling complex aeroelastic behaviors, including limit-cycle oscillations, beat phenomena and nodal-shaped oscillations at the transonic Mach numbers with high accuracy.     

Effective properties of a linear viscoelastic composite

The relaxation moduli of a composite are determined. The relaxation of its components is described by various few-parameter kernels: Mittag-Leffler functions of different orders and Rzhanitsyn kernel. It is assumed that the composite components are made of model materials with volume relaxation. The Laplace transform and fractional rational approximation are used to develop an algorithm for reducing the relaxation functions of the composite to one class (series of decreasing exponents or exponents of fractional order). The relaxation moduli of a unidirectionally reinforced composite are determined as an example     

Parametric characteristic of the random vibration response of nonlinear systems

Volterra series is a powerful mathematical tool for nonlinear system analysis,and there is a wide range of nonlinear engineering systems and structures that can be represented by a Volterra series model.In the present study,the random vibration of nonlinear systems is investigated using Volterra series.Analytical expressions were derived for the calculation of the output power spectral density(PSD) and input-output cross-PSD for nonlinear systems subjected to Gaussian excitation.Based on these expressions,it was revealed that both the output PSD and the input-output crossPSD can be expressed as polynomial functions of the nonlinear characteristic parameters or the input intensity.Numerical studies were carried out to verify the theoretical analysis result and to demonstrate the effectiveness of the derived relationship.The results reached in this study are of significance to the analysis and design of the nonlinear engineering systems and structures which can be represented by a Volterra series model.