具有运动边界与含时频率的一类环状非球谐振子模型势的精确解(英文)

在球坐标系中研究了一类具有运动边界与含时频率的环状非球谐振子模型势的Schrdinger方程.应用坐标变换将运动边界转化为固定边界,从而获得了系统的精确波函数.研究表明,系统的角向波函数是一个推广的缔合勒让德多项式,径向波函数可以表示为贝赛耳函数.最后我们简单讨论了指数运动边界和指数含时频率这一特殊情况.     

The use of moment equations for calculating the mean square response of a linear system to non-stationary random excitation

The moment equations approach is used to calculate the mean square response of a linear system to non-stationary random excitation which is expressed as a product of a deterministic envelope function and a Gaussian stationary non-white noise. The moment equations are derived by performing single integrations in the time domain and are solved numerically by digital computer. Numerical examples are given for the response of single and two degree-of-freedom systems which are excited by noise with an exponentially decaying harmonic correlation function. It is shown that an overshoot, in the sense that the transient response exceeds its stationary value, may occur even in the case of an exponential envelope function, but that the response does not exhibit overshoot when the natural frequency of the system is almost coincident with the dominant frequency of the input.     

Ground-borne vibration due to static and dynamic axle loads of InterCity and high-speed trains

In predictions of railway-induced vibrations, a distinction is generally made between the quasi-static and dynamic excitation. The quasi-static excitation is related to the static component of the axle loads. The dynamic excitation is due to dynamic train–track interaction, which is generated by a large number of excitation mechanisms, such as the spatial variation of the support stiffness and the wheel and track unevenness. In the present paper, the quasi-static excitation and the dynamic excitation due to random track unevenness are evaluated by means of numerical predictions. A solution strategy is presented that allows for the evaluation of the second-order statistics of the response due to dynamic excitation based on the power spectral density function of the track unevenness. Due to the motion of the train, the second-order statistics of the response at a fixed point in the free field are non-stationary and an appropriate solution procedure is required. The quasi-static and dynamic contribution to the track and free-field response are analysed for the case of InterCity and high-speed trains running at a subcritical train speed. It is shown how the train speed affects the quasi-static and dynamic contribution. Finally, results of numerical predictions for different train speeds are compared with field measurements that have been performed at a site along the high-speed line L2 Brussels–Köln within the frame of homologation tests.     

Vibration control of beams on elastic foundation under a moving vehicle and random lateral excitations

The formulation of three-dimensional dynamic behavior of a Beam On Elastic Foundation (BOEF) under moving loads and a moving mass is considered. The weight of the vehicle is modeled as a moving point load, however the effect of the lateral excitation is considered by modeling: (case 1) a lateral moving load with random intensity for wind excitation and (case 2) a moving mass just in lateral direction of the beam for earthquake excitation. A Dirac-delta function is used to describe the position of the moving load and the moving mass along the beam. The beam foundations are considered as elastic Winkler-type in two perpendicular transverse directions. This model is proposed to investigate the bending response of the rails under the effect of traveling vehicle weight while a random excitation such as earthquake or wind takes place. The results showed the importance of considering the effect of earthquake/wind actions as in bending stress of the beam on elastic foundations. The effect of different regions (different support stiffness) and different velocities of the vehicle on the response of the beam are investigated in mentioned directions. At the end, a linear optimal control algorithm with displacement–velocity feedback is proposed as a solution to suppress the response of BOEFs. By the method of modal analyses and taking into account enough number of vibration modes, state-space equation is obtained, then sufficient number of actuators was chosen for each direction. Stochastic analyses were performed in lateral direction in order to illustrate a comprehensive view for the response of the beam under the random moving load in both controlled and uncontrolled systems. Furthermore, the efficiency of control algorithm on critical velocities is verified by parametric analyses in the vertical direction with the constant moving load for different regions.     

Time-dependent variance and covariance of responses of structures to non-stationary random excitations

In this paper techniques for the analysis of non-stationary random responses of linear structures, discretized by the finite element method so that they can be analyzed as multi-degree of freedom systems, subjected to non-stationary random excitation are developed. The non-stationary random excitation is represented as a product of (a) an exponentially decaying function and a white noise process, and (b) a modulating function in the form of an exponential envelope and a white noise process. Closed form expressions for the time-dependent variance and covariance of response of structures are presented. Application of these expressions is made for the analysis of non-stationary random responses of a physical model of a class of mast antenna structures subjected to base excitation. It is concluded that (a) the coupling terms do have a definite influence on the response; the magnitude of the influence is proportional to the amount of damping in the structure and proximity of the modes excited; (b) the non-stationary random excitations considered are general in that the modulating functions are not necessarily identical, and therefore the influence of various modulating functions of the excitations applied to different locations of the structure on responses can be examined quantivatively; and (c) for a given damping parameter the magnitudes of the modulating function parameters cannot be chosen arbitrarily though the shapes of normalized modulating functions can be selected to best fit the excitation realizations.     

Non-stationary random response of a finite cable

Numerous problems of current concern involve the designs of aerodynamic systems which either travel at high speeds or contain structural elements which are excited by moving pressure fluctuations. In a number of recent papers responses of dynamic systems to random excitation have been considered. The appropriate theory for calculating the mean square response of linear systems to both stationary and non-stationary random excitation is well known [1–7]. In this paper, the mean square response of a finite cable to non-stationary random excitation is considered. The non-stationary random excitation is of the form s(t) = e(t)α(t), where e(t) is a well defined envelope function and α (t) is the Guassian, narrow band, stationary part of the excitation which has zero mean. Both the unit step and rectangular step functions are used for the envelope function, and both white noise and noise with an exponentially decaying harmonic correlation function are used to prescribe the statistical property of the excitation. The results obtained are shown to be a complete expression for the mean square response when checked for accuracy by reduction to expressions previously obtained by Lyon [4]. It is felt that these results will aid the design of both linear and two-dimensional aerodynamic systems excited by random pressure fluctuations.     

A unified model for non-stationary and/or non-Gaussian random processes

An analytical model for non-stationary and/or non-Gaussian random processes described in the paper is based on a normal stationary random process. The non-stationarity is introduced as a deterministic dependence of the parameters of the marginal distribution function or those of the correlation function upon the argument t. Consideration that the mentioned parameters are random variables or stationary random processes results in generating non-Gaussian distributions of the unconditioned process. By combining deterministic and random components of the parameters' dependencies, non-stationary and simultaneously non-Gaussian random processes may be easily specified. The model described may be useful for analytical treatment, for identification of experimentally obtained realizations of random processes and for simulation of random processes on computers as well as in the laboratory.     

Theoretical analysis of potential forces in markets

We show that random walks in a moving potential function, with its center at the moving average of market prices, are represented in the form of the self-modulation model. From this point of view we confirm the existence of non-trivial autocorrelation in real market price changes. By generalizing the formulation of potential function we prove that the ARCH model belongs to the special case of random walk in an asymmetric potential with randomly changing coefficient.     

Mean square response of a duffing oscillator to a modulated white noise excitation by the generalized method of equivalent linearization

The non-stationary random response of non-linear systems is considered. The technique of equivalent linearization is generalized for application to non-stationary non-linear random systems and several approximate methods of solution are presented. The example of a Duffing oscillator is studied in detail and its mean square response is evaluated and discussed.     

Identification of the structure parameters applying a moving load

An approach to the flexural stiffness identification of a linear structure is proposed. The idea of the presented approach is to transform the dynamical problem into a static one by integrating the input and output signals. The output signal is the structure displacement due to different kinds of loads such as a pulse acting at a given point, moving a load of deterministic or random type. The obtained solution for the one-point force can be easily generalized to a set of point forces, which can be a model of the pressure of vehicle axes. The presented method can be applied to the identification of structure parameters of bridges. It allows also to take into account some stochastic disturbances following the movement of vehicles through the pavement roughness.     

Resonance response of a single-degree-of-freedom nonlinear vibro-impact system to a narrow-band random parametric excitation

The resonant response of a single-degree-of-freedom nonlinear vibro-impact oscillator with a one-sided barrier to a narrow-band random parametric excitation is investigated. The narrow-band random excitation used here is a bounded random noise. The analysis is based on a special Zhuravlev transformation, which reduces the system to one without impacts, thereby permitting the applications of random averaging over "fast" variables. The averaged equations are solved exactly and an algebraic equation of the amplitude of the response is obtained for the case without random disorder. The methods of linearization and moment are used to obtain the formula of the mean-square amplitude approximately for the case with random disorder. The effects of damping, detuning, restitution factor, nonlinear intensity, frequency and magnitude of random excitations are analysed. The theoretical analyses are verified by numerical results. Theoretical analyses and numerical simulations show that the peak response amplitudes will reduce at large damping or large nonlinear intensity and will increase with large amplitude or frequency of the random excitations. The phenomenon of stochastic jump is observed, that is, the steady-state response of the system will jump from a trivial solution to a large non-trivial one when the amplitude of the random excitation exceeds some threshold value, or will jump from a large non-trivial solution to a trivial one when the intensity of the random disorder of the random excitation exceeds some threshold value.     

Effect of a pulse entrapping on weak localization of waves in a resonant random medium

We consider theoretically a new physical effect in coherent backscattering enhancement (CBE) of electromagnetic or acoustic non-stationary waves from a discrete random medium under condition of Mie resonant scattering. The effect manifests itself as an angle-cone broadening of a short pulsed signal CBE from the resonant random medium, compared with the case of a non-resonant random medium. The cone broadening is associated with a pulse-entrapping effect when the pulse, while propagating within the resonant random medium, spends most of the time being 'entrapped' inside scatterers. A theory for the predicted effect is based on, first, the well known relation between the contributions of the ladder and cyclical diagrams to the time spectral density of the wave electric field coherence function and, second, a recently derived radiative transfer equation with three Lorentzian kernels of delay describing a pulse entrapping in an ensemble of resonant point-like scatterers. Using the generalized Chandrasekhar H-function, we obtain an exact analytic expression for the non-stationary albedo of the semi-infinite resonant random medium, taking into account the phenomena of a pulse CBE and entrapping. A simple analytic asymptotics is found for the albedo of the later part of the scattered pulse. This asymptotics shows quantitatively how the entrapping affects the peak amplitude and peak line shape of the CBE of a short pulse.     

Energy harvesting under excitations of time-varying frequency

The design and optimization of energy harvesters capable of scavenging energy efficiently from realistic environments require a deep understanding of their transduction under non-stationary and random excitations. Otherwise, their small energy outputs can be further decreased lowering their efficiency and rendering many critical and possibly life saving technologies inefficient. As a first step towards this critical understanding, this effort investigates the response of energy harvesters to harmonic excitations of time-varying frequency. Such excitations can be used to represent the behavior of realistic vibratory environments whose frequency varies or drifts with time. Specifically, we consider a piezoelectric stack-type harvester subjected to a harmonic excitation of constant amplitude and a sinusoidally varying frequency. We analyze the response of the harvester in the fixed-frequency scenario then use the Jacobi-Anger's expansion to analyze the response in the time-varying case. We obtain analytical expressions for the harvester's response, output voltage, and power. In-depth analysis of the attained results reveals that the solution to the more complex time-varying frequency can be understood through a process which “samples” the fixed-frequency response curve at a discrete and fixed frequency interval then multiplies the response by proper weights. Extensive discussions addressing the effect of the excitation parameters on the output power is presented leading to some initial suggestions pertinent to the harvester's design and optimization in the sinusoidally varying frequency case.     

Statistical modelling of the backscattered field in a one-dimensional non-stationary stochastic problem

Within the framework of an exact wave approach in the spatial-time domain, the one-dimensional stochastic problem of sound pulse scattering by a layered random medium is considered. On the basis of a unification of methods which has been developed by the authors, previously applied to the investigation of non-stationary deterministic wave problems and stochastic stationary wave problems, an analytical-numerical simulation of the behaviour of the backscattered field stochastic characteristics was carried out. Several forms of incident pulses and signals are analysed. We assume that random fluctuations of a medium are described by virtue of the Gaussian Markov process with an exponential correlation function. The most important parameters appearing in the problem are discussed; namely, the time scales of diffusion, pulse durations, the medium layer thickness or the largest observation time scale in comparison with the time scale of one correlation length for the case of a half-space. An exact pattern of the pulse backscattering processes is obtained. It is illustrated by the behaviour of the backscattered field statistical moments for all observation times which are of interest. It is shown that during the time interval when the main part of the pulse energy leaves the medium, the backscattered field is a substantially non-stationary process, having a non-zero mean value and an average intensity that decays according to a power law. There are various power indices for the different duration incident pulses, however, they are not the same as those of previous papers, which were obtained on the basis of an approximate and asymptotic analysis. We have also verified that the Gaussian law is valid for the probability density function of the backscattered field in the case of any incident pulse duration.     

A technique for measuring the sound of a moving tire

A measurement and analysis technique has been developed to determine the narrow band spectra and the radiation patterns of the sound emitted by a moving tire. The sound is measured by a semicircular array of stationary microphones as the tire passes by the array and is recorded on a multi-channel tape recorder. In the analysis procedure corrections are made for effects associated with a moving sound source, such as the non-stationarity of the signal due to the time-dependent transmission path and the Döppler frequency shifts. In this way the power spectra and the radiation pattern of the sound signal are determined as if the receiver were moving with the tire at a fixed distance. A relationship between the Döppler effect and the frequency resolution associated with the finite Fourier transform is presented. This relation is used as a basis for the Döppler correction procedure.     

Dynamic response of a prestressed,orthotropic thick plate strip to a moving line load

This paper is a study of the steady state response of an orthotropic plate strip to a moving line load. The plate is of infinite length and subjected to initial in-plane stresses parallel and perpendicular to the edges. The solution is obtained on the basis of a thick plate theory which takes into account the effects of shear deformation and rotatory inertia. The critical speed of the load which brings about a resonance effect in the system is determined. Further, the bending moment in the plate is calculated for several values of the load speed and the initial stress parameters and shown graphically as a function of the space variable moving with the load.     

Multispeckle diffusing wave spectroscopy of colloidal particles suspended in a random packing of glass spheres

We use a multispeckle diffusing wave spectroscopy (MSDWS) method to study the ensemble-averaged dynamics of the fluctuating speckle pattern when illuminating colloidal particles suspended in a static and opaque porous medium with a coherent light source. Experiments were performed with Brownian latex particles in a random packing of glass spheres. The mixing of the light scattered by the moving colloidal particles and the porous matrix gives rise to a plateau value of the intensity autocorrelation function in the long-waiting-time limit. From the plateau in the correlation function, we can determine the fraction of light scattered from moving particles and estimate the photon mean free path in the colloidal solution. The method opens up promising possibilities to probe the static fraction in semisolid materials.     

Statistical modelling of the backscattered field in a one-dimensional non-stationary stochastic problem

Abstract

Within the framework of an exact wave approach in the spatial-time domain, the one-dimensional stochastic problem of sound pulse scattering by a layered random medium is considered. On the basis of a unification of methods which has been developed by the authors, previously applied to the investigation of non-stationary deterministic wave problems and stochastic stationary wave problems, an analytical-numerical simulation of the behaviour of the backscattered field stochastic characteristics was carried out. Several forms of incident pulses and signals are analysed. We assume that random fluctuations of a medium are described by virtue of the Gaussian Markov process with an exponential correlation function. The most important parameters appearing in the problem are discussed; namely, the time scales of diffusion, pulse durations, the medium layer thickness or the largest observation time scale in comparison with the time scale of one correlation length for the case of a half-space. An exact pattern of the pulse backscattering processes is obtained. It is illustrated by the behaviour of the backscattered field statistical moments for all observation times which are of interest. It is shown that during the time interval when the main part of the pulse energy leaves the medium, the backscattered field is a substantially non-stationary process, having a non-zero mean value and an average intensity that decays according to a power law. There are various power indices for the different duration incident pulses, however, they are not the same as those of previous papers, which were obtained on the basis of an approximate and asymptotic analysis. We have also verified that the Gaussian law is valid for the probability density function of the backscattered field in the case of any incident pulse duration.     

Invariant power law distribution of langevin systems with colored multiplicative noise

The random multiplicative process is studied for the case of a colored multiplicative noise with exponentially decreasing autocorrelation function. We observe the power law exponent of probability distribution in a statistically steady state numerically to clarify the effect of finite correlation time. The renormalization procedure is applied to derive the power law exponent theoretically. The power law exponent is inversely proportional to the autocorrelation time of the multiplicative noise.     

A Vector-Valued Almost Sure Invariance Principle for Sinai Billiards with Random Scatterers

Understanding the statistical properties of the aperiodic planar Lorentz gas stands as a grand challenge in the theory of dynamical systems. Here we study a greatly simplified but related model, proposed by Arvind Ayyer and popularized by Joel Lebowitz, in which a scatterer configuration on the torus is randomly updated between collisions. Taking advantage of recent progress in the theory of time-dependent billiards on the one hand and in probability theory on the other, we prove a vector-valued almost sure invariance principle for the model. Notably, the configuration sequence can be weakly dependent and non-stationary. We provide an expression for the covariance matrix, which in the non-stationary case differs from the traditional one. We also obtain a new invariance principle for Sinai billiards (the case of fixed scatterers) with time-dependent observables, and improve the accuracy and generality of existing results.