Spectral analysis of non-linear systems involving square-law operations

This paper contains a number of useful theoretical formulas to analyze the frequency domain properties of Gaussian input data passing through non-linear square-law systems. Special bispectral density functions are defined and applied that are functions of a single variable. From measurements of input data and output data only, results are obtained to identify the separate frequency response functions for two models of linear systems in parallel with non-linear square-law systems. Non-linear coherence functions are defined from these models which determine the proportion of the output spectrum due to the non-linear operations. Together with ordinary coherence functions, a measured output spectrum for these models can be decomposed into three components representing the linear operations, the non-linear operations, and the remaining uncorrelated noise effects. This material indicates also how to analyze other types of non-linear models by employing similar techniques.     

Experimental and Numerical Study of Structural Identification Using Non-Linear Resonant Decay Method

One of the practical approaches in identifying structures is the non-linear resonant decay method which identifies a non-linear dynamic system utilizing a model based on linear modal space containing the underlying linear system and a small number of extra terms that exhibit the non-linear effects. In this paper, the method is illustrated in a simulated system and an experimental structure. The main objective of the non-linear resonant decay method is to identify the non-linear dynamic systems based on the use of a multi-shaker excitation using appropriated excitation which is obtained from the force appropriation approach. The experimental application of the method is indicated to provide suitable estimates of modal parameters for the identification of non-linear models of structures.     

SOME GENERAL EFFECTS OF STRONG HIGH-FREQUENCY EXCITATION: STIFFENING, BIASING AND SMOOTHENING

Mechanical high-frequency (HF) excitation provides a working principle behind many industrial and natural applications and phenomena. This paper concerns three particular effects of HF excitation, that may change the apparent characteristics of mechanical systems: (1) stiffening, by which the apparent linear stiffness associated with an equilibrium is changed, along with derived quantities such as stability and natural frequencies; (2) biasing by which the system is biased towards a particular state, static or dynamic, which does not exist or is unstable in the absence of the HF excitation; and (3) smoothening, referring to a tendency for discontinuities to be effectively “smeared out” by HF excitation. Illustrating first these effects for a few specific systems, analytical results are provided that quantify them for a quite general class of mechanical systems. This class covers systems that can be modelled by a finite number of second order ordinary differential equations, generally non-linear, with periodically oscillating excitation terms of high frequency and small amplitude. The results should be useful for understanding the effects in question in a broader perspective than is possible with specific systems, for calculating effects for specific systems using well-defined formulas, and for possibly designing systems that display prescribed characteristics in the presence of HF excitation.     

Variational estimates in electrostatics. Gauss's principle and the method of equivalent charges

A variational modification of the method of equivalent charges is developed using the variational Gauss's principle. It is shown how one can use it to obtain simple analytic formulas for upper estimates of the matrices of the potential coefficients of systems of conductors. Estimates are given for systems of any number of nonintersecting spheres, parallel conductors of circular cross section above a grounded infinite plane, coaxial thin conducting circular discs, a sphere and an oblate ellipsoid of rotation. The effectiveness of the variational estimates obtained is illustrated with specific examples.Krasnoyarsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 30–36, February, 1995.     

Comment on “Monte Carlo optimisation of correlated wave functions for normal two-electron systems” by Kar and Mandal

Kar and Mandal [Phys. Lett. A 240 (1998) 301] reported a remarkably accurate, three non-linear parameter correlated wavefunction for a variety of two-electron systems. Unfortunately, due to linear dependences in the trial function, the secular equations to obtain the mixing coefficients and the variational energy are numerically unstable. Employing Monte Carlo procedures we provide consistent and accurate estimates for these quantities for the case of He.     

Low-redshift formula for the luminosity distance in a LTB model with cosmological constant

We calculate the low-redshift Taylor expansion for the luminosity distance for an observer at the center of a spherically symmetric matter inhomogeneity with a non-vanishing cosmological constant. We then test the accuracy of the formulas comparing them to the numerical calculation for different cases for both the luminosity distance and the radial coordinate. The formulas can be used as a starting point to understand the general non-linear effects of a local inhomogeneity in the presence of a cosmological constant, without making any special assumption as regards the inhomogeneity profile.     

High intensity laser plasma interaction

Basic mechanisms, linear and non-linear, are reviewed. Light absorption may take place linearly through inverse bremsstrahlung. Together with usual heat transport by electrons it leads to a linear gas dynamical regime whose main formulas are given. When absorption occurs non-linearly, by resonance in a density gradient or through wave-wave couplings, several non-linear regimes may show up. They are investigated in connection with the ponderomotive force, soliton formation, wavebreaking, … The processes responsible for harmonic generation are given special developments.     

On the sonic fatigue life estimation of skin structures at room and elevated temperatures

Fighter/trainer empennages and STOL (Short Take-Off and Landing) aircraft flap systems are subjected to severe acoustic pressure levels as high as 150–170 dB. As a result, acoustic fatigue has become one of the major factors in design. Empennages and flap systems are also subject to high temperatures and thus the influence of thermal buckling on fatigue life must be taken into consideration. To estimate the sonic fatigue life of skin structures, combined use is made of the Monte Carlo method of non-linear panel response analysis and local stress-strain simulations with rheological models. Calculations were conducted and comparison of the results with experimental data shows that the method estimates the sonic fatigue life accurately when adequate values of fatigue notch factors are chosen. Example calculations with thermal as well as static pressure effects were also carried out and the effect of thermal buckling on sonic fatigue life is clarified.     

不等间距非直线三元阵在水声平面声源被动定位中的应用

根据实际实验环境和布放需求,在对称直线阵被动定位基本理论基础上,推导了不等间距非直线三元阵进行水声被动定位的精确显式公式,将非直线阵形误差角用于修正测距、测向过程,并采用平均法和卡尔曼滤波法对实测数据的中间处理结果和最终结果进行后置处理。海上试验结果验证了算法的正确性,在近距离有效范围内取得较为满意的目标定位结果。     

Gaps and eigenfrequencies in the spectrum of a periodic acoustic waveguide

Sufficient conditions are found for the existence of eigenfrequencies within the spectral gaps which occur as a result of local regular perturbation of the hard walls of a periodic acoustic waveguide. Asymptotic formulas are derived and error estimates shown.     

EXPERIMENTAL AND THEORETICAL INVESTIGATION OF THE LINEAR AND NON-LINEAR DYNAMIC BEHAVIOUR OF A GLARE 3 HYBRID COMPOSITE PANEL

A theoretical model based on Hamilton's principle and spectral analysis is used to study the non-linear free vibration of hybrid composite plates made of Glare 3, a new aircraft structural material. It consists of alternating layers of metal- and fibre-reinforced composites. In previous work, the theoretical model has been used to calculate the first non-linear mode of fully clamped rectangular composite fibre-reinforced plastic (CFRP) laminated plates. This study concerns determination of the linear dynamic properties of the Glare 3 hybrid composite rectangular panel (G3HCRP) such as natural frequencies and mode shapes. The theoretical model is used to calculate the fundamental non-linear mode shape and associated flexural behaviour of the fully clamped G3HCRP. A series of experimental investigations have been conducted using a G3HCRP in order to determine linear dynamic properties. The response due to random excitation was investigated and the experimental measurements are analyzed and discussed. Comparisons are made with finite element predictions and response estimates given by the ESDU method, the latter being a “design guide” approach used by industry. Concerning the non-linear analysis, the results are given for various plate aspect ratios and vibration amplitudes, showing a higher increase of the induced bending stress near the clamps at large deflections. Comparisons between the dynamic behaviour of an isotropic plate and G3HCRP at large vibration amplitudes are presented and good results are obtained.     

Spectral and temporal changes of optical pulses propagating through time-varying linear media

We present universal formulas for the spectral and temporal output optical fields from a linear traveling-wave medium whose refractive index changes during its propagation within the medium. These formulas agree with known changes in central wavelength and energy that are associated with adiabatic wavelength conversion (AWC). Moreover, they reveal new changes to the optical pulses that have not been noticed, such as pulse compression and spectral broadening. Most significantly, we find that AWC alters the pulse power, pulse chirp, and pulse delay. All of these effects depend on whether the central wavelength is blueshifted or redshifted, the first sign of asymmetry to be reported for AWC. These findings impact the applications of AWC to optical signal processing in microphotonic and nanophotonic structures as well as in lightwave systems.     

Component mode analysis of a simple non-linear,non-conservative system

A component mode analysis is carried out based upon the use of constraint conditions and Lagrange multipliers to treat exemplary physical systems with non-linear damping. Both viscous and dry friction dampers attached to a linear elastic (beam) system are examined. The method is shown to possess conceptual and computational advantages by its ability to reduce the analysis of a multimode system to a small number of degrees of freedom equal to that associated with the non-linear component.     

EFFECTS OF BASE POINTS AND NORMALIZATION SCHEMES ON THE NON-LINEAR NORMAL MODES OF CONSERVATIVE SYSTEMS

In this paper, two factors that affect the behaviors of the non-linear normal modes (NNMs) of conservative vibratory systems are investigated. The first factor is the base points (which are equivalent to Taylor series expanding points) of the non-linear normal modes and the second one is the normalization schemes of the corresponding linear modes. For non-linear systems, in general only the approximated NNM manifolds are obtainable in practice, so different base points may lead to different forms of NNM oscillators and different normalization schemes lead to different forward and backward transformations which in turn affect the numerical computation errors. Three different kinds of base points and two different normalization schemes are adopted for comparison respectively. Two examples of non-linear systems with two and three degrees of freedom, respectively, are given as illustration. Simulations for various cases are made. The analysis and the simulation results indicated that, the best base points are the abstract base points determined via the linear normal mode, which would eliminate the third order terms containing velocity (for cubic systems) or quadratic terms (for quadratic systems) in equations of the NNM oscillators. A better invariance of the NNMs would also be maintained with such base points. The best scheme of normalization is the norm-one scheme that would minimize the numerical errors.     

The tensor product of one-particle representations of an infinite-dimensional Lie algebra

The recently proposed infinite-dimensional Lie algebra as a model of a symmetry scheme is studied from the point of view of its representations. We construct the tensor product of two one-particle representations of this algebra and study the reduction problem. A new series of representations having non-linear mass formulas is found. Some physical consequences for two-particle states are also discussed.The authors wish to thank Prof. V. Votruba for constant interest shown throughout this work and J. Formánek for valuable discussions.     

Use of the Hilbert transform in modal analysis of linear and non-linear structures

An initial study into the application of the Hilbert transform in modal analysis procedures is presented. It is shown that typical structural non-linearities such as non-linear damping and stiffness can be detected and identified directly without the need to generate explicit models. No assumptions regarding the degree of non-linearity are made, which is a restriction in the classical methods for dealing with non-linearities. The properties of the Hilbert transform are discussed with respect to linear and non-linear dynamical systems, and a discrete transform, developed from the continuous functions, is derived in the frequency domain and adapted to modal analysis data in the form of mobility transfer functions. Truncation effects arising from limited frequency ranges of the mobility transfer functions are accounted for by employing correction terms in the frequency domain. Several examples are studied of single and multi-mode systems with non-linearities such as friction, clearance and non-linear stiffness. These examples indicate that the Hilbert transform offers a new method for extending modal analysis to the domains of non-linear systems.     

Geometric Variational Inference

Efficiently accessing the information contained in non-linear and high dimensional probability distributions remains a core challenge in modern statistics. Traditionally, estimators that go beyond point estimates are either categorized as Variational Inference (VI) or Markov-Chain Monte-Carlo (MCMC) techniques. While MCMC methods that utilize the geometric properties of continuous probability distributions to increase their efficiency have been proposed, VI methods rarely use the geometry. This work aims to fill this gap and proposes geometric Variational Inference (geoVI), a method based on Riemannian geometry and the Fisher information metric. It is used to construct a coordinate transformation that relates the Riemannian manifold associated with the metric to Euclidean space. The distribution, expressed in the coordinate system induced by the transformation, takes a particularly simple form that allows for an accurate variational approximation by a normal distribution. Furthermore, the algorithmic structure allows for an efficient implementation of geoVI which is demonstrated on multiple examples, ranging from low-dimensional illustrative ones to non-linear, hierarchical Bayesian inverse problems in thousands of dimensions.     

Optimum vibration absorber (tuned mass damper) design for linear damped systems subjected to random loads

Optimum design of dynamic vibration absorbers (DVAs) installed on linear damped systems that are subjected to random loads is studied and closed-form design formulas are provided. Three cases are considered in the optimization process: Minimizing the variance of the displacement, velocity and acceleration of the main mass. Exact optimum design parameters for the velocity case, which to the best knowledge of the author do not exist in the literature, are derived for the first time. Exact solutions are found to be directly applicable for practical use with no simplification needed. For displacement and acceleration cases, a solution for the optimum absorber frequency ratio is obtained as a function of optimum absorber damping ratio. Numerical simulations indicate that optimum absorber damping ratio is not significantly related to the structural damping, especially when the displacement variance is minimized. Therefore, optimum damping ratio derived for undamped systems is proposed for damped systems for the displacement case. When acceleration variance is minimized, however, the optimum damping ratio derived for undamped systems is found not as accurate for damped systems. Therefore, a more accurate approximate expression is derived. Numerical comparisons with published approximate expressions at the same level of complexity indicated that proposed design formula yield more accurate estimates. Another important finding of the paper is that for specific applications where all of the response parameters are desired to be minimized simultaneously, DVAs designed per velocity criteria provide the best overall performance with the least complexity in the design equations.     

Diffusion of hydrogen in heterogeneous systems

The effective long-range long-time tracer diffusivity D_eff for interstitial diffusion of hydrogen through heterogeneous systems was studied theoretically for model systems consisting of isolated grains of material G embedded in a matrix of material M. Different solubilities of hydrogen in these two materials as well as different diffusivities are allowed for. Additionally, modified diffusion barriers at the phase boundaries were included in the diffusion model. The effect of different sizes, arrangements, and forms of the grains was also considered. D_eff was determined by Monte Carlo (MC) simulations on simple lattice models of the systems described above. An equilibrium distribution of hydrogen atoms among the two constituent materials was assumed. Our main interest was focused on whether and how D_eff may be related to mesoscopic or macroscopic quantities characterizing the heterogeneous system and its constituent materials, such as the volume fractions of the two materials, the fraction of lattice sites in the immediate vicinity of the phase boundary, the hydrogen concentrations c_G and c_M in the grains and in the matrix and the respective hydrogen diffusivities D_G(c_G) and D_M(c_M). In order to obtain good estimates for these relations in terms of analytic formulas, we attempted to model a heterogeneous system by a network of diffusion elements connected in series and in parallel, in analogy to an electric network. The properties of the basic connections, in parallel and in series, were studied on layered structures, for which analytic expressions for D_eff could be derived. The network formulas for different grain–matrix systems were tested by comparing with results of MC simulations. In general, the network formulas describe the corresponding MC results for D_eff fairly well. It was found that differences in the hydrogen solubilities in the two phases as well as modified energy barriers at the phase boundaries may have dramatic effects on D_eff. Received: 19 September 2000 / Accepted: 6 November 2000 / Published online: 9 February 2001