The spectral density of a nonlinear damping model: multi-DOF case

In this work, the spectral density of the following multi-DOF nonlinear damping model is investigated: Mx¨+D₀x˙+γD(x, x˙)+Kx=σn(t) where γ>0 is a small parameter. A formula for the spectral density is established with O(γ2) accuracy based upon the Fokker-Planck technique and perturbation. One of the features of the multi-DOF oscillation system is that x and x˙ are generally correlated in stationary state. This is true even for linear systems. Necessary and sufficient conditions for uncorrelatedness are given for linear systems. Since the first-order statistics R_xx(0) and R_xy(0), where y=x˙, appear in the spectral density formula, it is desirable to have the explicit stationary probability density for the purpose of evaluating R_xx(0) and R_xy (0). However, in general, as in the single DOF case, an expression for the stationary density is not available. This note gives the explicit stationary density of an “energy”-type nonlinear damping model Mx¨+μ(E_D)Dx˙+Kx=σn(t) in which the “energy” E_D is defined as E_D=1/2(x ^TKDx+y^TMDy) where D>0 is assumed to commute with K and M. In the end, an energy-type nonlinear damping model is worked out completely as an illustration     

The vibration transmissibility of a single degree of freedom oscillator with nonlinear fractional order damping

Motivated by the theoretical analysis of the effects of nonlinear viscous damping on vibration isolation using the output frequency response function approach, the output frequency response function approach is employed to investigate the effects of the nonlinear fractional order damping on vibration isolation based on Volterra series in the frequency domain. First, the recursive algorithm which is proposed by Billings et al. is extended to deal with the system with fractional order terms. Then, the analytical relationships are established among the force transmissibility, nonlinear characteristic coefficients and fractional order parameters for the single degree of freedom oscillator. Consequently, the effects of the nonlinear system parameters on the force transmissibility are discussed in detail. The theoretical analysis reveals that the force transmissibility of the oscillator is suppressed due to the existence of the fractional order damping, but presents different effects on suppressing the force transmissibility of the oscillator over the frequency region by varying the fractional order parameters. Moreover, the fractional order parameters, which affect the force transmissibility, the bandwidth of the frequency region and the resonance frequency, can be used as designing parameters for vibration isolation systems. At last, numerical studies are presented to illustrate the theoretical results.     

A compliant contact model with nonlinear damping for simulation ofrobotic systems

Contact modeling is an important aspect of simulation of many robotic tasks. In the paper, a compliant contact model with nonlinear damping is investigated, and many previously unknown characteristics of the model are developed. Compliance is used to eliminate many of the problems associated with using rigid body models with Coulomb friction, while the use of nonlinear damping eliminates the discontinuous impact forces and most sticky tensile forces which arise in Kelvin-Voigt linear models. Two of the most important characteristics of the model are the dependence of the coefficient of restitution on velocity and damping in a physically meaningful manner, and its computational simplicity. A full mathematical development for an impact response is given, along with the effects of the system and model parameters on energy loss. A quasistatic analysis gives results which are consistent with energy loss characteristics of a more complex distributed foundation model under sustained contact conditions. A foot contact example for a walking machine is given which demonstrates the applicability of the model for impact on foot placement, sustained contact during the support phase, and the breaking of the contact upon liftoff of the foot     

一类允许缺货的易变质物品的随机非线性存贮模型

研究了货物存贮问题．考虑到在仓库出空期间一般人们有耐烦与不耐烦两种反应，研究中假设仓库出空期丢失顾客量服从正态分布，据此建立允许缺货的易变质物品的非线性存贮模型，使模型更接近于实际．并给出数值例子，运用Matlab软件求解最优存贮控制策略，得到近似的最佳进货量和最佳出空期长度．     

The limiting density of a nonlinear system

The RC filter-hard limiter-RC filter nonlinear system is the subject of this paper. Because of computational difficulties implicated in the analysis of the above system, only its response to the zero mean Gaussian system input has been analytically investigated. An approximate output density has also been found for nonzero mean Gaussian, while verified to be “close” to the real one for finite means. In the present paper, a close form of the system output density is obtained when the input mean tends to infinity. For that, ε-upcrossing methods were used.     

On the factorization of discrete-time rational spectral density matrices

For a discrete-time rational spectral density matrixPhi(z), the relationship between the factorizationsPhi(z) = Z(z) + Z^{T} (z^{-1})andPhi(z) = W(z)QW^{T}(z), in terms of minimal state space realizations ofZ(z)andW(z), are derived in a straightforward way, without resorting to the use of the bilinear transformation. The obvious application of this result to performing a spectral factorizafion ofPhi(z)is discussed.     

Modal analysis of nonlinear systems with classical and non-classical damping

In the dynamic analysis and design of structures and equipment, the behavior of various components beyond the linear range is often of interest. A nonlinear vibration analysis is time consuming, particularly if many configurations or loading conditions have to be considered in order to arrive at representative values for design. The selection of an efficient and yet accurate analysis procedure is therefore extremely important.

Two mode-superposition procedures are presented for the dynamic analysis of nonlinear structures with classical (proportional) and non-classical (non-proportional) damping. The nonlinearity at each time step is treated as a pseudo force. Undamped eigensolution and complex modes are used to uncouple the equations of motion for classical and non-classical damping cases. A recursive procedure based on the exact solution of the differential equation is used to obtain the modal responses. The advantages of the proposed method of computing the response over the existing integration techniques are: (a) the simplicity of the procedure, (b) a substantial reduction in computational time, (c) the possibility of using fewer modes to achieve the desired accuracy, and (d) the adaptability of the procedure to parallel processing machines which will further reduce the computational time.     

Sufficient conditions for the preservation of the boundedness in a numerical method for a physical model with transport memory and nonlinear damping

Departing from a finite-difference scheme to approximate solutions of a nonlinear, hyperbolic partial differential equation which generalizes the Burgers–Huxley equation from fluid dynamics, we investigate conditions on the model coefficients and the computational parameters under which positive and bounded initial data evolve into positive and bounded new approximations. The model under investigation includes nonlinear coefficients of damping and advection, and the reaction term extends the reaction law of the classical Fisher–Kolmogorov–Petrovsky–Piscounov equation. The method can be expressed in vector form in terms of a multiplicative matrix which, under certain parametric conditions, becomes an M-matrix. Using the fact that every M-matrix is non-singular and that the entries of its inverse are positive, real numbers, we establish sufficient conditions under which the method provides new, positive and bounded approximations from previous, positive and bounded data and boundary conditions. The numerical results confirm the fact that the conditions derived here are sufficient for the positivity and the boundedness of the approximations; moreover, computational experiments evidence the fact that the method still preserves these properties for values of the model and the numerical parameters outside of the analytic regions of positivity and boundedness. We point out that our simulations show a good agreement between the numerical approximations computed through our method and the corresponding, analytical solutions.     

阻尼对泡沫夹芯梁非线性振动响应的影响

基于Euler—Bernoulli梁理论、Hamilton原理以及Galerkin方法，建立了大变形悬臂夹芯梁在横向周期载荷作用下的二阶动力学方程；通过考虑外周期激励的不同频率与幅值，详细分析了材料阻尼比对泡沫铝夹芯梁的振动响应的影响．结果表明，泡沫夹芯结构具有较好的阻尼性能，可有效抑制梁的混沌振动．     

Topology optimization of nonlinear structures with damping under arbitrary dynamic loading

This study presents an extended unit load method in which the displacement of a chosen degree of freedom (DOF) in a nonlinear structure under arbitrary dynamic loading is expressed as an integration of mutual strain energy density over a continuum domain. This new integral formulation for the displacement of a chosen DOF is developed by using the virtual work principle and can be used for linear or nonlinear structural behaviours. The integral form of the displacement is then used to develop new formulations for structural topology optimization involving arbitrary dynamic loading using the moving iso-surface threshold (MIST) method. Presented are two specific topology optimization problems with two objective functions: (a) to minimize the peak of a chosen displacement; or (b) to minimize the average power spectral density (PSD) of the chosen displacement over a finite time interval. New MIST formulations and algorithms are developed for solving two damping topology optimization problems of a structure under arbitrary dynamic loading, with or without large displacements, and having cellular damping materials with multi-volume fractions. Several numerical examples are presented to demonstrate the validity and efficiency of the presented unit load method and the MIST formulations and algorithms.     

The spectral accuracy of a fully-discrete scheme for a nonlinear third order equation

A time-discrete pseudospectral algorithm is suggested for the numerical solution of a nonlinear third order equation arising in fluidization. The nonlinear stability and convergence of the new scheme are analyzed. Numerical comparisons with available finite-difference methods are also reported which clearly indicate the superiority of the new scheme.     

An energy-balancing perspective of interconnection and damping assignment control of nonlinear systems

Stabilization of nonlinear feedback passive systems is achieved assigning a storage function with a minimum at the desired equilibrium. For physical systems a natural candidate storage function is the difference between the stored and the supplied energies—leading to the so-called energy-balancing control, whose underlying stabilization mechanism is particularly appealing. Unfortunately, energy-balancing stabilization is stymied by the existence of pervasive dissipation, that appears in many engineering applications. To overcome the dissipation obstacle the method of Interconnection and Damping Assignment, that endows the closed-loop system with a special—port-controlled Hamiltonian—structure, has been proposed. If, as in most practical examples, the open-loop system already has this structure, and the damping is not pervasive, both methods are equivalent. In this brief note we show that the methods are also equivalent, with an alternative definition of the supplied energy, when the damping is pervasive. Instrumental for our developments is the observation that, swapping the damping terms in the classical dissipation inequality, we can establish passivity of port-controlled Hamiltonian systems with respect to some new external variables—but with the same storage function.     

Signal recognition: Both components of the short time Fourier transform vs. power spectral density

A new feature representation approach was generalised and used for Gaussian recognition. The generalised approach consists of simultaneously using two new recognition features — real and imaginary Fourier components —taking into account the covariance between features. Generalisation of the approach improves recognition effectiveness. An advanced time-frequency technique, the short time Fourier transform, was considered. Covariance and the correlation coefficient between the proposed features were obtained for the first time for arbitrary stationary signals. The recognition effectiveness between the generalised approach and power spectral density was compared. It was shown that power spectral density is not an optimal feature, and represents only a particular case of the generalised approach. The use of power spectral density is optimal if simultaneously the correlation coefficient between Fourier components is equal to zero, and the standard deviations of components are equal. Use of the generalised approach provides an increase in effectiveness in comparison with power spectral density.     

Simulation of noise figure of nonlinear amplifiers using the orthogonalization of the nonlinear model

A simulation procedure of noise figure (NF) of nonlinear amplifiers is developed. NF is defined in terms of the effective signal‐to‐noise ratio (SNR) at the output of a nonlinear amplifier. The effective output SNR when the input consists of a communication signal plus Gaussian noise is evaluated through the identification of the effective output noise and nonlinear distortion power using the orthogonalization of the nonlinear model. The approach is useful for the assessment of noise performance of low‐noise amplifiers in wireless systems. © 2009 Wiley Periodicals, Inc. Int J RF and Microwave CAE 2009.     

A note on the accuracy of spectral method applied to nonlinear conservation laws

Fourier spectral method can achieve exponential accuracy both on the approximation level and for solving partial differential equations if the solutions are analytic. For a linear PDE with discontinuous solutions, Fourier spectral method will produce poor point-wise accuracy without post-processing, but still maintains exponential accuracy for all moments against analytic functions. In this note we assess the accuracy of Fourier spectral method applied to nonlinear conservation laws through a numerical case study. We have found out that the moments against analytic functions are no longer very accurate. However the numerical solution does contain accurate information which can be extracted by a Gegenbauer polynomial based post-processing.Research supported by ARO Grant DAAL03-91-G-0123 and DAAH04-94-G-0205, NSF Grant DMS-9211820, NASA Grant NAG1-1145 and contract NAS1-19480 while the first author was in residence at ICASE, NASA Langley Research Center, Hampton, Virginia 23681-0001, and AFOSR Grant 93-0090.     

The density function reconstruction of surface sources from a single Cauchy measurement

The inverse problem of reconstructing sources is explored when a single boundary Cauchy data is postulated on the potential. We are particularly involved in sources supported by (hyper-)surfaces. Mild assumptions are required on the location of these supports and the calculation of the charge density function is then aimed. We consider a variational formulation, based on a duplication artifice of the potential and we check the symmetry and the positive definiteness of the weak problem. Because of the severe ill-posedness, the use of a regularization is mandatory for a safe approximation of the solution. Lavrentiev’s method is therefore recommended in the context owing to the symmetry and the positivity. We check why that regularization turns out to be a Tikhonov method for some underlying shadow equation that is not needed in computations and is therefore never explicitly constructed. Results stated in a wide literature for the Tikhonov regularization applies as well to our variational problem. An important consequence is that the Morozov Discrepancy Principle, we use for the selection of the regularization parameter yields a convergent strategy. Now, that the Discrepancy Principle requires the residual of that inaccessible ‘shadow equation’, we explain how the Kohn–Vogelius function allows for the computation of that residual.     

A spectral meteorological model on the ICL DAP

The key points to the implementation of a meteorological spectral model on the ICL Distributed Array Processor (DAP) are presented. Spectral models involve transforming the variables between spectral and gridpoint space and these spectral transforms comprise of Legendre and fast Fourier transforms.

The storage format of the data is discussed and the algorithms used for the Legendre transforms presented. Timings of these algorithms are compared with those from a serial machine.     

基于功率谱密度的人体运动状态检测

提出了一种视频图像序列中人体运动状态检测的新方法。针对传统人体状态检测方法利用数据信息的复杂性,对图像中人体外接矩形区域提取位置和尺度稳定的非结构化数据信息——关键点,计算关键点的功率谱密度,在序列视频图像中可获得关键点功率谱密度的分布,利用其分布特点判断人体运动状态。实验证明方法的有效、可行和简易。     

基于社团检测的复杂网络中心性方法

论证了社团检测函数模块密度的优化进程能转化为核矩阵的特征谱分.基于核矩阵最大特征值对应的特征向量,提出了一种新的中心性方法,称为模块密度中心性方法.与以往中心性度量方法不同,这种方法以模块密度检测复杂网络中的社团结构为基础,度量了第一个节点到它分配社团上的贡献,对社团的贡献越大,该节点的中心性值越高,反之亦然.通过合成网络和标准数据集网络,验证了该方法,并同其他中心性方法进行了比较,实验表明提出的模块密度中心性方法对网络中关键节点有更好的解和稳定性.进一步在计算机产生的两个大的随机网络和来自现实世界的两个大的复杂网络中,研究了模块密度中心性方法的统计分布.结果表明了提出的中心性方法能够刻画复杂网络的拓扑结构属性.